36 AG11 Abstracts

IP1

Algebraic Statistics for Social Network Models:Recent Results and Challenges

In collaboration with Petrovic and Rinaldo, I have devel-oped algebraic statistical tools for the study of some dyadicrandom graph models, including Markov bases, that haveimportant implications for the existence of maximum like-lihood estimation and other statistical problems. Thesetools do not extend in a simple fashion to more complexmodels in the class of exponential random graph models. Inthis presentation, I explain why there are difficulties as wemove away from dyadic models and I describe some of thechallenges for algebraic statistics in this area of research.

Stephen FienbergCarnegie Mellon Universityfienberg@stat.cmu.edu

IP2

Complexity of the Separation of Variables and Re-lated Observations

Abstract not available at time of publication.

Leonid GurvitsLos Alamos National Laboratorygurvits@lanl.gov

IP3

Calculus and Constructible Sheaves

This talk describes an ingenious integral calculus based onEuler characteristic, stemming from work on constructiblesheaves due to MacPherson and Kashiwara in the 1970s,and connecting back further to classical integral geometry.The talk will emphasize (1) its novel utility in data man-agement, particularly in aggregation of redundant data andinverse problems over sensor networks; and (2) how issuesof numerical computation, inspired by applications, leadsto fascinating connections with Morse theory and compu-tational topology.

Robert W. GhristUniversity of Pennsylvaniaghrist@seas.upenn.edu

IP4

The Geometry of Cell Structures in Foams andMetals

Networks of crystals in a metal and networks of bubblesin a foam are both examples cell complexes that arise innature. Both systems evolve over time according to mathe-matical equations. We believe that, for generic initial con-ditions, they evolve towards a statistically universal state.The topology and geometry of this universal state is greatinterest in applications. This talk will present both math-ematical results and computer investigations on these sys-tems in two and three space dimensions.

Robert MacPhersonInstitute for Advanced StudySchool of Mathematicsrdm@ias.edu

IP5

Advances in Elliptic-Curve Cryptography

The first part of this talk presents results on attackingelliptic-curve cryptography, in particular an ongoing effortto break the Certicom challenge ECC2K-130 and a detailedstudy on the correct use of the negation map in the Pollard-rho method. The second part presents a signature schemewhich on a 390 USD mass-market quad-core 2.4GHz IntelWestmere (Xeon E5620) CPU can create 108000 signaturesper second and verify 71000 signatures per second on anelliptic curve at a 2128 security level. Public keys are 32bytes, and signatures are 64 bytes. These performance fig-ures include strong defenses against software side-channelattacks: there is no data flow from secret keys to array in-dices, and there is no data flow from secret keys to branchconditions.

Tanja LangeTechnische Universiteit EindhovenEindhoven Institute for the Protection of Systems andInformtanja@hyperelliptic.org

IP6

B-splines: A Fundamental Tool for Analysis andComputation

Starting from their inception in approximation theory 60years ago, the use of B-splines has blossomed in diverseareas in science, engineering and electronic arts. These in-clude animation, computational geometry, computer-aideddesign, computer-aided manufacturing, computer graphics,control theory, geometric design, image analysis, medicalvisualization, optimization, partial differential equations,robotics, and statistics. More recently B-splines have beenassuming a fundamental role in the isogeometric analysis,an ambitious effort to unify shape representation and en-gineering analysis. Honoring its interesting historical de-velopment, this talk gives a mathematical introduction tosplines and B-splines in one and several space dimensionsand also considers various generalizations.

Tom LycheUniversity of OsloDepartment of Informatics and CMAtom@ifi.uio.no

IP7

Symbolic-numeric Algorithms for Computing theSingular Solution of Polynomial Systems

In this talk we will describe some recent progress insymbolic-numeric algorithms for computing the singularsolution of polynomial systems. Using the local dual struc-ture of the isolated singular solution with limited accuracy,we present modifications of Newton’s method to restorequadratic convergence. In particular, when the corank ofthe Jacobian matrix at the singular solution is one, basedon the regularized Newton iteration and the computationof partial differential conditions satisfied approximately atthe singular solution, we develop a new approach to com-pute a proper direction and step size of the Newton itera-tion to ensure the quadratic convergence. Finally, we willshow recent results on computing verified error bounds forsingular solutions of polynomial systems based on Rump’sverification method.

Lihong Zhi

AG11 Abstracts 37

Academia Sinicalzhi@mmrc.iss.ac.cn

IP8

Multiview Geometry

The study of two-dimensional images of three-dimensionalscenes is a foundational subject for computer vision, knownas multiview geometry. We present recent work with ChrisAholt and Rekha Thomas on the polynomials defining im-ages taken by n cameras. Our varieties are threefolds thatvary in a family of dimension 11n-15 when the cameras aremoving. We use toric geometry and multigraded Hilbertschemes to characterize degenerations of camera positions.

Bernd SturmfelsUniversity of California , Berkeleybernd@Math.Berkeley.EDU

CP1

Discrete Bounded-curvature Paths and Path Plan-ning

In 1957 Dubins classified shortest bounded curvature pathsin the plane into two classes. We generalize this result toa new class of discrete polygonal paths. The propertiesand construction of discrete Dubins paths are discussed.Properties of discrete bounded-curvature motion are anal-ysed and applied to prove properties of smooth bounded-curvature paths. In particular, the classification of discreteDubins paths is used to obtain Dubins seminal result as alimiting case. Related to these problems the question ofdefining curvature for a polygonal path is addressed. Po-tential research questions, such as generalizing Cauchy’sarm lemma, are discussed.

Sylvester Eriksson-BiqueUniversity of HelsinkiNew York Universityseb522@nyu.edu

David KirkpatrickUniversity of British Columbia, Canadakirk@cs.ubc.ca

Valentin PolishchuckUniversity of Helsinkipolishch@cs.helsinki.fi

CP1

On Middle Universal Weak and Cross InverseProperty Loops With Equal Length of Inverse Cy-cles

This study presents an isotopism under which the weakinverse property(WIP) is isotopic invariant in loops. It isshown that under this isotopism, whenever n is a positiveeven integer, a finite WIPL has an inverse cycle of lengthn if and only if its isotope is a finite WIPL with an in-verse cycle of length n. Explanations and procedures aregiven on how these results can be used to apply CIPLs tocryptography.

Temitope G. JaiyeolaObafemi Awolowo Universityjaiyeolatemitope@yahoo.com

CP1

Extracting Topological Properties Using ManifoldLearning Techniques

High-dimensional, unordered data, is often difficult to an-alyze or visualize. In such cases, manifold learning tech-niques - like PCA, Isomap, Hessian LLE - are quite helpfuland efficient in extracting the low-dimensional representa-tion of the data. When used individually, these methodsjust give the low-dimensional representation of the data.But owing to the complementary nature of the limitationsof these methods, performing a hierarchy of tests and com-paring the outputs can reveal useful topological featureslike the presence of holes and (degree of) non-linearity ofthe data. We develop and showcase such a hierarchicalstrategy on various material science data to provide insightinto process-property-structure relationships.

Sai Kiranmayee SamudralaDepartment of Mechanical Engineering, Iowa StateUniversityAmes, Iowa -50011saikiran@iastate.edu

Baskar GanapathysubramanianIowa State Universitybaskarg@iastate.edu

CP1

Lower Bounds on Stochastic Games

Shapley’s discounted stochastic games are classical modelsof game theory describing two-player zero-sum games ofpotentially infinite duration. We show, based on a gener-alization of Eisenstein criterion, that there exists a gamewith N positions, m actions for each player in each posi-tion, such that its value is an algebraic number of degreemN−1. This strengthens a result of Etessami and Yan-nakakis who considered the case of m = 2 and proved a2Ω(N) lower bound.

Kristoffer A. HansenAarhus University, Denmarkarnsfelt@cs.au.dk

Michal KouckyCzech Academy of Scienceskoucky@math.cas.cz

Niels Lauritzen, Peter Bro MiltersenAarhus University, Denmarkniels@imf.au.dk, bromille@cs.au.dk

Elias TsigaridasUniversity of Aarhuselias@cs.au.dk

MS1

Symmetric Determinantal Representations of Poly-nomials

In a seminal paper (STOC 1979), Valiant expressed thepolynomial computed by an arithmetic formula as the de-terminant of a matrix whose entries are constants or vari-ables. This result was then extended by Malod and Todato weakly-skew circuits. We are interested here in express-ing formulas and weakly-skew circuits by determinants ofsymmetric matrices. In my talk, I will sketch some of these

38 AG11 Abstracts

constructions for fields of characteristic different from 2. Incharacteristic 2, the picture is quite different: First we canprove an impossibility result, showing that some polynomi-als have no symmetric determinantal representation. Onthe other hand, we answer a question by Burgisser aboutthe VNP-completeness of the partial permanent. This talkis based on joint works with E. L. Kaltofen, P. Koiran andN. Portier, and to a smaller extent with T. Monteil and S.Thomasse.

Bruno GrenetLIP, ENS Lyonbruno.grenet@ens-lyon.fr

MS1

Solving the Generalized MinRank Problem withGroebner Bases

Let K be a field and M be a p × k matrix with entries inK[X1, . . . , Xn] of degree D and 0 < r < min(p, k). Thegeneralized MinRank problem consists in finding the al-gebraic points in Cn at which M has rank r. This is ageneralization of the classical eigenvalue problem in Lin-ear Algebra. It has applications in several areas such ascryptology and real geometry. The algebraic nature of theproblem leads us to study the use Grobner bases techniquesto solve it. In this framework, solving the generalized Min-Rank problem is reduced to solve the polynomial systemdefined by the vanishing of all (r + 1, r + 1)-minors of M.Such a polynomial system is highly structured. In the 0-dimensional case, we provide sharp complexity results onsolving such a structured polynomial system with Grobnerbases when the characteristic of K is 0 or when it is largeenough, and when the entries of M are generic. The usedingredients rely on algebraic properties of determinantalideals and classical tools such as Hilbert series to analyzethe complexity of Grobner bases computations.

Jean-Charles FaugereSalsa project, INRIA Paris-Rocquencourtjean-charles.faugere@inria.fr

Mohab Safey El DinUPMC, Univ Paris 6Mohab.Safey@lip6.fr

Pierre-Jean SpaenlehauerUPMC, Univ. Paris 6pierre-jean.spaenlehauer@lip6.fr

MS1

On the Role of Compactness in Algebraic Complex-ity Theory

In various situations in the study of the computationalcomplexity of algebraic problems, it is convenient to re-strict to a compact setting. For instance, real and complexanalogs of Toda’s Theorem of discrete complexity theoryin the Blum-Shub-Smale model are shown for a compactversion of the polynomial time hierarchy. We study thefirst level of this compact polynomial hierarchy over thereals (compact NP) and look for complete problems in it.The problem of deciding whether a real homogeneous poly-nomial has a non-trivial zero is a natural canditate. How-ever, it is surprisingly difficult to prove this hardness result,since a decision procedure of a closed set does in generalnot yield a closed description of it. We report on partialresults and approaches for this problem. In particular, weshow NP-completeness in a restricted model of computa-

tion which by definition accepts closed sets only. Also, weexplore which problems in NP can be reduced to the com-pact feasibility problem using an effective finiteness Theo-rem involving Lojasiewicz’ inequality.

Saugata BasuPurdue Universitysbasu@math.purdue.edu

Peter ScheiblechnerDepartment of Mathematics, Purdue Universitypscheibl@math.purdue.edu

MS1

On the Complexity of Computing with Zero-dimensional Triangular Sets

We study the complexity of some fundamental operationsfor triangular sets in dimension zero. Using Las-Vegas al-gorithms, we prove that one can perform such operations aschange of order, equiprojectable decomposition, or quasi-inverse computation with a cost that is essentially that ofmodular composition. Over an abstract field, this leads toa subquadratic cost (w.r.t. the degree of the underlying al-gebraic set). Over a finite field, in a boolean RAM model,we obtain a quasi-linear running time using Kedlaya andUmans’ algorithm for modular composition. Conversely,we also show how to reduce the problem of modular com-position to change of order for triangular sets, so that allthese problems are essentially equivalent.

Adrien PoteauxLIP6, Universite Paris 6adrien.poteaux@lip6.fr

Eric SchostComputer Science DepartmentUniv. of Western Ontarioeschost@uwo.ca

MS1

On the Complexity of Solving Quadratic BooleanSystems

A fundamental problem in computer science is to find allthe common zeroes of m quadratic polynomials in n un-knowns over the boolean field GF(2). In particular, thecryptanalysis of several modern ciphers reduces to thisproblem. Up to now, the best complexity bound wasobtained by exhaustive search in O(log(n)2n). We givean algorithm that reduces the problem of solving booleanquadratic systems to a combination of exhaustive searchand sparse linear algebra. Under precise algebraic assump-tions, we show that when m = n, the complexity of thedeterministic variant of this algorithm is upper boundedby O(20.841n), while a probabilistic Las Vegas variant hasexpected complexity O(20.792n). Experiments on randomsystems show that the algebraic assumptions are satisfiedwith probability close to 1.

Magali BardetLITIS, Universite de Rouenmagali.bardet@univ-rouen.fr

Jean-Charles FaugereSalsa project, INRIA Paris-Rocquencourtjean-charles.faugere@inria.fr

Bruno Salvy

AG11 Abstracts 39

INRIA, Paris-Rocquencourtbruno.salvy@inria.fr

Pierre-Jean SpaenlehauerUPMC, Univ. Paris 6pierre-jean.spaenlehauer@lip6.fr

MS2

Secant Varieties of Segre-Veronese Varieties

In general, the secant varieties of Segre-Veronese varietiescan be difficult to describe explicitly. In many cases, thedimensions are not even known. I will explain the followingtheorem: for r at most 5, the rth secant variety of P 2 ×Pn embedded by O(∞,∈) is defined by the minors of theflattening and the Pfaffian of the exterior flattening.

Dustin CartwrightYale Universitydustin@math.berkeley.edu

Daniel ErmanStanford Universityderman@math.stanford.edu

Luke OedingUniversity of California BerkeleyUniversita degli Studi di Firenzelukeoeding@gmail.com

MS2

Tensor Complexes

The most fundamental complexes of free modules over acommutative ring are the Koszul complex, which is con-structed from a vector (i.e., a 1-tensor), and the Eagon–Northcott and the Buchsbaum–Rim complexes, which areconstructed from a matrix (i.e., a 2-tensor). I will discussa multilinear generalization of these complexes, which weconstruct from an arbitrary higher tensor. Our construc-tion provides detailed new examples of minimal free reso-lutions, as well as a unifying view on several well-knownexamples.

Berkesch ChristineDuke Universitycberkesc@math.su.se

Daniel ErmanStanford Universityderman@math.stanford.edu

Manoj KumminiPurdue Universitynkummini@math.purdue.edu

Steven SamMITssam@math.mit.edu

MS2

Determinantal Equations for Secant Varieties

We show that if a smooth variety X is re-embedded bya sufficiently large Veronese embedding of its original em-bedding then, set theoretically, the equations of the rthsecant variety of X, are just the equations defining the rth

secant variety of the Veronese embedding of original pro-jective space and the linear equations. This reduces thequestion of finding the equations of the secant variety ofa (sufficiently amply embedded) variety to the finding ofthe equations of the secant variety of a Veronese embed-ding of a projective space. Time permitting, context andother results will be mentioned. This is joint work withJ. Buczynski and J.M. Landsberg.

Adam GinenskyWH Tradingaginensky@whtrading.com

MS2

Border Bases, Hilbert Scheme and Tensor Decom-position

The commutativity is a natural property that we expect inthe context of algebraic geometry. Surprisingly, this sim-ple property is also enough to characterize the solution ofseveral problems. In this talk, we will show how it is in-trinsically related to the description of the Hilbert schemeof points and to the tensor decomposition problem. Wewill analyze the correspondence between these two prob-lems and detailed different characterisation of ranks of atensor. Revisiting an approach of J.J.Sylvester for the de-composition of a binary forms, we will describe an algo-rithm to decompose general tensors, based on techniquesrelated to border basis computation and truncated momentproblems. Some examples will illustrate the approach.

Bernard MourrainGALAADINRIA Sophia AntipolisBernard.Mourrain@sophia.inria.fr

Jerome BrachatGALAAD, INRIA Sophia Antipolisjerome.brachat@inria.fr

MS2

Decomposition of Tensors of Small Rank

We discuss some algorithms to decompose a tensor in A⊗B ⊗ C as a sum of decomposable tensors. The techniqueis connected to the equations of the secant varieties, whichare known only when the rank is small. The results arebetter in the symmetric case. Some sufficient conditionswhich guarantee the uniqueness of the decomposition willbe presented.

Giorgio OttavianiUniversity of Firenzeottavian@math.unifi.it

MS3

Polynomials and Optimization in Computer Vision

Multiview geometry is the study of planar images of pointsin space, an essential basis for computer vision. I will de-scribe the algebraic objects in this field. These includethe multiview variety associated to a fixed set of cameras– along with an explicit universal Groebner basis for itsideal – and also a Hilbert scheme parametrizing all suchmultiview ideals. This data feeds into the polynomial opti-mization problem of triangulating a point from noisy imageobservations.

Chris Aholt

40 AG11 Abstracts

University of Washingtonaholtc@uw.edu

Bernd SturmfelsUniversity of California , Berkeleybernd@Math.Berkeley.EDU

Rekha ThomasUniversity of Washingtonrrthomas@uw.edu

MS3

Lossless Representation of Graphs using Distribu-tions

Complex structures such as composite objects or networksare often represented by weighted graphs. Since many sig-nal processing and analysis techniques require the input totake the form of a vector, it is desirable to find ways to rep-resent such weighted graphs as vectors. For reasons thatwill be explained in the talk, it is important to make surethat any invariance of the structure under a relabelings ofthe node carry over to the vector representation schemechosen. In this talk, I will discuss how one can use distri-butions of certain functions of the weights of the graph torepresent the ”vast majority” of graphs without losing anyinformation about the structure of the graph.

Mireille BoutinSchool of ECEPurdue Universitymboutin@purdue.edu

MS3

Title Not Available at Time of Publication

Abstract not available at time of publication.

Gunnar E. CarlssonStanford Universitygunnar@math.stanford.edu

MS3

A Categorical Approach to Agent Networks

In this research, we continue the work of Lawvere and Giryin developing a categorical approach to conditional prob-abilities. Specifically, we investigate categorical propertiesof the Kleisli category of the Giry monad with the aim ofconstructing relations. Our goal is to then use this cate-gorical framework to model sensor networks in an abstract,relational and probabilistic way.

Jared CulbertsonAir Force Research Laboratoryjared.culbertson@wpafb.af.mil

MS3

Lower Bounds for the Gromov-Hausdorff distanceUsing Persistent Topology

Viewing shapes as metric spaces (or metric measure spaces)permits dealing with different problems in object matchingunder deformations. The Gromov-Hausdorff distance pro-vides one possible measure of dissimilarity between shapes,but its computation leads to NP-hard problems. We willdiscuss a family of lower-bounds for the GH distance that

involve the comparison of certain invariants arising frompersistent topology constructions. The computation ofthese lower bounds can be done in polynomial time.

Facundo MemoliStanford University, Mathematics Departmentmemoli@math.stanford.edu

MS4

Toric Ideals of Hypergraph Models

This talk is about models parametrized by edges of a hy-pergraph. Such models are toric by definition, and the goalis to obtain a Markov basis for the model. Equivalently,we are interested in the defining ideal of the edge subringof a hypergraph. The ideal of the edge subring of a graphgives a Markov bases for the p1 model for random graphs(networks). The algebraic and combinatorial constructionsused for these toric ideals give insight into the geometryof the model, existence of maximum likelihood estimators,and a better way to generate Markov moves. The ideal the-ory of graphs has a rich history, including some results thatare well-known in algebraic statistics. A more complicatedfamily of algebraic models motivates the generalization ofthis construction to hypergraphs. The talk will outline themotivation and the main results.

Sonja PetrovicUniversity of Illinois, Chicagopetrovic@math.uic.edu

Despina StasiMathematics, Statistics, and Computer ScienceUniversity of Ilinois at Chicagostasi@math.uic.edu

MS4

The Polytope of Degree Sequences and MaximumLikelihood Estimation

In this talk, we show how the polytope of degree sequencescan be used to characterize nonexistence of the maximumlikelihood estimator (MLE) in two statistical models: thebeta model for random undirected graphs with fixed degreesequences and the Rasch model from item response theory.We derive new asymptotic results about the existence ofthe MLE in the random graph case and describe numeri-cally feasible ways for checking for existence of the MLEand for computing the associated facial sets.

Alessandro RinaldoCarnegie Mellon Universityarinaldo@stat.cmu.edu

Sonja PetrovicUniversity of Illinois, Chicagopetrovic@math.uic.edu

Stephen FienbergCarnegie Mellon Universityfienberg@stat.cmu.edu

MS4

An Algebraic Statistics Approach to the EcologicalInference Problems

We present an algebraic computational framework thathandles special cases of missing data problems. We focus

AG11 Abstracts 41

on the ecological inference problem where we use aggregate(ecological) data to infer discrete individual-level relation-ships of interest when individual-level data are unavailable.The proposed method can handle multi-dimensional con-tingency tables, deterministically respects bounds and canincorporate information from many different sources. Weillustrate how the analysis can be done with the recentlydeveloped R-4ti2 package.

Aleksandra Slavkovic, Vishesh KarwaPenn State Universitysesa@stat.psu.edu, vishesh@psu.edu

MS4

Estimating the Number of Binary Multi-way Ta-bles via Sequential Importance Sampling

In 2005, Chen et al introduced a sequential importancesampling (SIS) procedure to analyze binary two-way tableswith given fixed marginal sums (row and column sums) viathe conditional Poisson (CP) distribution. They showedthat compared with Markov chain Monte Carlo (MCMC)-based approaches, their importance sampling method ismore efficient in terms of running time and it also providesan easy and accurate estimate of the total number of con-tingency tables with fixed marginal sums. In 2007, then,Chen showed also that we can generalize it to binary two-way tables with structural zeros. In this talk we will reviewtheir results and then we will show that we can extend theirresult to binary multi-way (d-way, d ≥ 2) contingency ta-bles under the no d-way interaction model, i.e., with fixedd− 1 marginal sums.

Jing XiDepartment of StatisticsUniversity of Kentuckyjxi222@uky.edu

Ruriko YoshidaUniversity of Kentuckyruriko.yoshida@uky.edu

MS5

Computing Vanishing Ideals with the Help of Mul-tivariate Polynomial Interpolation

In this paper, the problem of computing vanishing idealsis investigated with the help of multivariate polynomialinterpolation. We generate certain subsets from the givenpoint set whose associated interpolation bases are obtainedsubsequently in theory, with which the processes of knownalgorithms for vanishing ideals can be accelerated to someextent, according to the sizes of the subsets. Algorithmsare implemented and experimental timings are given.

Tian DongSchool of MathematicsJilinUniversitydongtian.jlu@gmail.com

MS5

The Role of Algebraic Geometry in GeometricModeling

Algebraic Geometry and Geometric Modeling both dealwith curves and surfaces generated by polynomial equa-tions. Algebraic Geometry studies theory, algorithms, andcomputation; Geometry Modeling investigates applications

of polynomial, piecewise polynomial, and rational curvesand surfaces to industrial design and manufacturing. Thepurpose of this talk is to elucidate the role that AlgebraicGeometry plays in Geometric Modeling. We will discussseveral topics of current research, including eliminationtheory, syzygies, and the analysis of singularities.

Ron GoldmanDepartment of Computer ScienceRice Universityrng@cs.rice.edu

MS5

Wachspress Varieties

We will examine an algebraic variety parametrized bybarycentric coordinates of a polygon. We will make useof the combinatorics of the polygon to understand the ge-ometry of this variety. In addition, we will observe thatthis variety can be seen as the blowup of certain points inthe projective plane.

Corey IrvingDepartment of MathematicsTexas A&M Universitycirving@math.tamu.edu

MS5

Splines, Spectral Sequences, and Polyhedral Com-plexes

Piecewise polynomial functions on a polyhedral complex(splines) are fundamental objects in mathematics, with ap-plications ranging from approximation theory and numer-ical analysis to algebraic geometry, where they appear asthe equivariant Chow ring of a toric variety. In joint workwith T. McDonald [Advances in Applied Math, 2009], wefound an (asymptotic) version of the Alfeld-Schumaker di-mension formula for planar splines. I will also describerecent work [Trans. A.M.S, to appear] which uses theCartan-Eilenberg spectral sequence to obtain results onsplines on polyhedral complexes of dimension greater thantwo.

Hal SchenckMathematics DepartmentUniversity of Illinoisschenck@math.uiuc.edu

MS5

Multivariate Interpolation and Algebraic Geome-try

I will describe connections between multivariate interpo-lation and algebraic geometry and present two counterex-amples to problems in the field. First is to a conjecture ofCarl de Boor regarding the existence of certain error for-mula for multivariate interpolation. Second is to a conjec-ture of Tomas Sauer regarding restrictions of interpolationprojector to polynomials of lesser degree.

Boris ShekhtmanDepartment of Mathematics and StatisticsUniversity of South Floridashekhtma@usf.edu

42 AG11 Abstracts

MS6

Persistence Based Signatures for Metric Spaces

We introduce a family of signatures for compact metricspaces, possibly endowed with real valued functions, basedon the persistence diagrams of suitable filtrations built ontop of these spaces. We prove the stability of these signa-tures under Gromov-Hausdorff perturbations of the spaces.We illustrate their use through an application in shape clas-sification.

Frederic ChazalINRIAfrederic.chazal@inria.fr

MS6

Rigid Rips Complexes and Topological Data Anal-ysis

In recent years, persistence diagrams have become an inte-gral part of topological data analysis, providing a tool tomeasure the scale of topological features. From the theo-retical point of view, persistence diagrams can be seen aseffectively computable invariants of metric spaces. A clas-sical way to construct a persistence diagram is by way ofthe so-called Rips complexes. In this paper we define a1-parameter family of simplicial complexes which we callrigid Rips complexes. This family includes the regular Ripscomplex at infinity and yields a 1-parameter family of per-sistence diagrams - the data which we call rigid persistence.In addition to measuring the scale of topological features,this family provides a tool to analyze the evolution of thesefeatures when the rigidity changes. While rigid persistencecontains the regular persistence, we give an example show-ing that the reverse of that statement is not true, and thatrigid persistence contains strictly more information thatthe regular one. (joint with E.-M. Feichtner, J. Lehmann)

Dmitry Feichtner-KozlovUniversity of Bremendfk@math.uni-bremen.de

MS6

Topology, Geometry and Statistics: MergingMethods for Data Analysis

Dimension reduction and shape description for scientificdata sets are difficult problems, ones that continue to growin importance within the statistical, mathematical andcomputer science communities. Powerful new methods ofTopological Data Analysis and Diffusion Geometry haveemerged in the last 10 years, and these have added signif-icantly to the data analysis toolbox. In this talk we willgive an overview of these methods and describe some earlyefforts to make them work together with Statistics. Inparticular we will discuss how random walk methods givedistance functions that enhance topological features, andhow one goes about defining the mean and variance of acollection of persistence diagrams.

John HarerDuke Universityharer@math.duke.edu

MS6

Using Persistent Homology to Compute Probabal-istic Failure of a Sensor Network

We consider the question of sensor network coverage for a

2-dimensional domain. We seek to compute the probabilitythat a set of sensors fails to cover given only non-metric,local (who is talking to whom) information and a probabil-ity distribution of failure of each node. This builds off thework of de Silva and Ghrist who analyzed this problem inthe deterministic situation. We first show that the problemas stated is #P-complete, and thus fast algorithms likelydo not exist unless P=NP. We then give a deterministicalgorithm which is feasible in the case of a small set of sen-sors or at least a set of sensors which is not very dense,and discuss other methods to approximate the probabilityof failure. These algorithms build on the theory of topo-logical persistence.

Elizabeth MunchDuke Universityliz@elizabethmunch.com

MS6

Topology of Spaces of Micro-images and Applica-tions

One of the most celebrated success stories in topologicaldata analysis is perhaps that of the Klein bottle, as a modelspace for relevant 3×3 patches coming from natural images.I will explore in this talk how the Klein bottle model can beextended to include other meaningful patches, and describean application to texture discrimination.

Jose PereaStanford Universityjoperea@stanford.edu

MS7

Riemann Theta Functions: An Applied Introduc-tion

Riemann’s theta function is the fundamental building blockfor function theory on compact Riemann surfaces andtheir Jacobians. In this talk I will present an incompleteoverview of different applications where theta function playan important role.

Bernard DeconinckUniversity of Washingtonbernard@amath.washington.edu

MS7

Tropical Curves, Tropical Theta Functions and In-tegrable Systems

Tropical geometry is the combinatorial algebraic geome-try over the min-plus algebra. For the last dozen years,tropical geometry has been rapidly developed and widelyapplied not only in complex algebraic geometry, but also inmathematical physics. The aim of this talk is to introducean application of tropical curve theory to an integrablepiecewise-linear map which originates from the Toda latticeequation. First we recall basic notions of tropical curves,tropical Jacobian and tropical theta functions by follow-ing the work of Mikhalkin and Zharkov in 2006. Second,a tropical version of Fay’s trisecant identity is formulated.As a special case of this identity, we finally obtain a generalsolution to the piecewise-linear map. (This talk is basedon the collaborative works with T. Takenawa.)

Rei InoueFaculty of ScienceChiba University

AG11 Abstracts 43

reiiy@math.s.chiba-u.ac.jp

MS7

Algebraic Curves and Riemann Surfaces in Matlab

We present a fully numerical approach to Riemann surfacesdefined via plane algebraic curves. The code in Matlabcomputes for a given algebraic equation in two variablesthe branch points and singularities, the holomorphic dif-ferentials and a base of the homology. The monodromygroup for the surface is determined via analytic continu-ation of the roots of the algebraic equation on a set ofcontours forming the generators of the fundamental group.The periods of the holomorphic differentials are computedwith Gauss-Legendre integration along these contours. TheAbel map is obtained in a similar way. The performance ofthe code is illustrated for many examples. As an applica-tion we study quasi-periodic solutions to certain integrablepartial differential equations.

Christian KleinInstitut de Mathematiques de Bourgogne9 avenue Alain Savary, BP 47870 21078 DIJON Cedexchristian.klein@u-bourgogne.fr

MS7

Thetanulls of Algebraic Curves of Genus 3 (A Pre-liminary Report)

In this talk we will discuss thetanulls of genus 3 hyperel-liptic curves. The case of genus 3 hyperelliptic curves hasbeen studied by Shaska/Wijesiri and relations among themhave been determined for cyclic curves of small genus. Inthis talk the more general case for genus 3 will be discussed.

Tanush ShaskaDepartment of MathematicsOakland Universityshaska@oakland.edu

MS7

Determinantal representations, Cauchy kernels,and theta functions

Determinantal representations of plane algebraic curveswere the object of numerous studies during the last cen-tury and a half. They became important for optimiza-tion in the last decade since representing a convex set asa feasibility set for semidefinite programming boils downto constructing a positive symmetric determinantal repre-sentations (for certain multiples of the Zariski closure ofthe boundary). One can obtain explicit formulae for de-terminantal representations using theta functions. Theseformulae are closely related to the famous trisecant iden-tity of Fay. Similarly to Fay’s identity, these formulae canbe easily derived tracing the poles and residues of variousexpressions involving Cauchy kernels for line bundles ona compact Riemann surface. Here the Cauchy kernel is asection in two arguments and is holomorphic except for asimple pole with residue one along the diagonal; it can beeasily written down using theta functions (and the primeform). The purpose of the talk is to describe the variousinterrelations between the different objects: determinan-tal representations, theta functions, and Cauchy kernels.Many of the results are based on joint work with DanielAlpay, Joe Ball, Bill Helton, and Dmitry Kerner.

Victor VinnikovDepartment of Mathematics

Ben Gurion Universityvinnikov@cs.bgu.ac.il

MS8

Closed Form Solutions of Difference Equations

In this talk we present an algorithm that finds closedform solutions for homogeneous linear recurrence equa-tions. The key idea is transforming an input operator Linp

to an operator Lg with known solutions. The main prob-lem of this idea is how to find a solved equation Lg to whichLinp can be reduced. To solve this problem, we use localdata of a difference operator, that is invariant under thetransformation.

Yongjae ChaRISC Linzycha@risc.uni-linz.ac.at

MS8

On the Structure of Compatible Rational Functions

A finite number of rational functions are compatible if theysatisfy the compatibility conditions of a first-order linearfunctional system involving differential, shift and q-shiftoperators. We present a theorem that describes the struc-ture of compatible rational functions. The theorem en-ables us to decompose a solution of such a system as aproduct of a rational function, several symbolic powers, ahyperexponential function, a hypergeometric term, and aq-hypergeometric term. We outline an algorithm for com-puting this product, and present an application.

Shaoshi ChenResearch Institute for Symbolic ComputationJohannes Kepler University Linzschen@amss.ac.cn

Ruyong Feng, Guofeng Fu, Ziming LiKey Laboratory of Mathematics MechanizationAMSS, Chinese Academy of Sciencesryfeng@amss.ac.cn, fuguofeng@mmrc.iss.ac.cn,zmli@mmrc.iss.ac.cn

MS8

The Q=0 Expansion of Q-Holonomic Sequences

I will discuss recent results on the degree, leading term andq=0 expansion of a q-holonomic sequence of polynomials.The results are of interest in Knot Theory and QuantumTopology, as well as the physics of fivebranes.

Stavros GaroufalidisGeorgia Techstavros@math.gatech.edu

MS8

Restricted Lattice Walks and Creative Telescoping

We start with a combinatorial problem: determining thegenerating functions of certain restricted lattice walks.This can be done with support from computer algebra.The most efficient techniques are however not ’rigorous’(joint work with A. Bostan; FPSAC 2009). It is possibleto make them rigorous by evaluating certain definite inte-grals via Zeilberger’s method of creative telescoping. Butthe computational cost for doing so is quite high (ongoingjoint work with A. Bostan, F. Chyzak, M. van Hoeij). This

44 AG11 Abstracts

leads to the general question whether creative telescopingcan be done in a more efficient way. In the end, we sketchan idea for minimizing the cost for creative telescoping byusing a Cylindrical Decomposition computation as a pre-processing step (ongoing joint work with S. Chen).

Manuel KauersRISC Linzmanuel@kauers.de

MS8

Computing Belyi Maps

Let L(y)=0 be a Heun equation with no Liouvillian solu-tions. We consider the question of deciding if there existsa rational function f for which L can be solved in termsof 2F1(a,b;c;f). Here a,b,c are constants, and 2F1 is thehypergeometric function. Computationally, the most non-trivial case occurs when f is a Belyi map. In a joint workwith R. Vidunas, we have computed a complete table withall Belyi maps f that can occur in this context. It con-tains 48 + 366 Belyi maps (48 parametric cases and 366non-parametric cases). The degrees in this table reach upto degree 60, finding these Belyi maps and verifying com-pleteness required new techniques.

Mark van HoeijFlorida State Universityhoeij@mail.math.fsu.edu

MS9

Report on the Recent Progress on the Study ofSecant Defectivity of Segre-Veronese Varieties

In this talk, I report the recent progress on the secant de-fectivity of Segre-Veronese varieties. The primary goal ofthe talk is to present the conjecturally complete list of de-fective secant varieties of Segre-Veronese varieties with twofactors. If time permits, I will discuss defective secants toSegre-Veronese varieties with three or more factors. Thistalk is based on joint work with Chiara Brambilla.

Hirotachi AboUniversity of Idahoabo@uidaho.edu

Chiara BrambillaDip. di Scienze MatematicheUniversita‘ Politecnica delle Marche, Ancona, Italybrambilla@math.unifi.it

MS9

Kruskal’s Uniqueness Inequality is Sharp

Kruskal proved that a tensor in V1⊗V2⊗· · ·⊗Vm of rank rhas a unique decomposition as a sum of r pure tensors if acertain inequality is satisfied. We will show the uniquenessfails if the inequality is weakened.

Harm DerksenUniversity of Michiganhderksen@umich.edu

MS9

A Proof of the Set-theoretic Version of a SalmonConjecture

We show that the irreducible variety of 4× 4× 4 complex

valued tensors of border rank 4 at most is given by poly-nomial equations of degree 5, (the Strassen commutativeconditions), of degree 6, (the Landsberg-Manivel polyno-mials), and of degree 9, (the symmetrization conditions).

Shmuel FriedlandDepartment of MathematicsUniversity of Illinois, Chicagofriedlan@uic.edu

MS9

Tensor Decomposition and Dimensions of SecantVarieties of Segre Varieties

I will discuss joint work with M.V. Catalisano (Genoa) andA. Gimigliano (Bologna) giving the dimensions of ALL thesecant varieties of the Segre Varieties P 1 × · · · × P 1. Thissolves the problem of finding the number of summands inthe decomposition of a general tensor in V ⊗ · · · ⊗ V (anynumber of factors) when dimV = 2. Similar results for thecases when V has higher dimension will also be mentioned.

Anthony GeramitaQueen’s Universitytony@mast.queensu.ca

MS9

Representations with Finitely Many Orbits

In this talk I will report on the joint work with WitoldKraskiewicz on singularities of orbit closures in the repre-sentations of reductive groups with finitely many orbits.We are interested in Cohen-Macaulay, Gorenstein and ra-tional singularities properties as well as in defining idealsof the orbit closures.

Jerzy WeymanNortheastern Universityj.weyman@neu.edu

MS10

Computing Explicit Optimal Value Functions by aSymbolic-numeric Cylindrical Algebraic Decompo-sition

We consider to solve parametric optimization problems.Our purpose is to compute optimal values as a functionof the parameters, which is called an optimal value func-tion. This is very useful for solving reactive/online op-timizations, dynamic programming, hierarchical optimiza-tions and so on. Thus constructing explicit optimal valuefunctions is very important and has many applications inengineering. We present en efficient method, which is basedon a specialized cylindrical algebraic decomposition usingsymbolic-numeric computation, to construct optimal valuefunctions explicitly for polynomial parametric optimizationproblems. Moreover, we also show several practical appli-cation examples in control.

Hirokazy AnaiKyushu Universityanai@jp.fujitsu.com

MS10

Cylindrical Algebraic Decomposition without Aug-

AG11 Abstracts 45

mented Projection

Cylindrical algebraic decomposition is a fundamental toolin computational real algebraic geometry. It takes a set ofmultivariate polynomials and decomposes the real spaceinto cylindrically arranged cells which can be describedby boolean expressions of polynomial equations and in-equalities. It begins by repeated projection (elimination ofvariables) and follows by repeated lifting. During the lift-ing, the descriptions of cells are constructed using Thomslemma. This necessitates the so-called augmented projec-tion (the derivatives are also considered). It often makesthe projection time-consuming and in turn increases thenumber of cells. In this talk, we show how to avoid aug-mented projection by describing cells using Sturm-Habichttheorem (instead of Thoms lemma).

Hoon HongNorth Carolina State Universityhong@math.ncsu.edu

MS10

Composing Hybrid Automata

We address the problem concerning the decidability of cer-tain hybrid automata that may be built hierarchically asis the case in many exemplar systems, be it natural or en-gineered. Parallel composition can be considered a funda-mental tool in such costructions. Somewhat surprisingly,this operation does not always preseve the decidability ofreachability problem i.e., even if we prove the decidabil-ity of reachability over component automata, we cannotguarantee the decidability over their parallel composition.Despite these limitations,it is still possible to provide a re-duction for the reachability problem over parallel compo-sition of first-order constant reset automata (FOCoRe) tothe satisfiability of a particular linear Diophantine system.Moreover, by proving that such satisfiability problem isdecidable for systems with semi-algebraic coefficients, thispaper presents an interesting class of hybrid automata forwhich the reachability problem of parallel composition isdecidable. The resulting hybrid automata appear in sys-tems biological modeling, and hence could be applied whenone is interested in understanding a complex biological sys-tem composed of smaller self-organizing systems. Jointwork with Alberto Casagrande, Pietro Corvaja, and CarlaPiazza, expanding on some earlier preliminary results.

Bud MishraNew York Universitymishra@nyu.edu

MS10

The Critical Point Method into Practice : State ofthe Art and Perspectives

The Critical Point Method has been designed initially toprovide an algorithmic alternative to Cylindrical Alge-braic Decomposition for computing sample points in semi-algebraic sets or for performing quantifier elimination overthe reals. One feature is that it provides a framework toobtain algorithms running in asymptotically optimal de-gree bounds. In this talk, I will present a new algorithmfor computing sample points in semi-algebraic sets definedby equations and inequalities when the set of equations sat-isfies some regularity assumptions. Practical experimentsshow that it outperforms previously known algorithms onsome examples. Some preliminary results on its complexity

will be given also.

Mohab Safey El DinUniversity Pierre and Marie CurieMohab.Safey@lip6.fr

MS10

A Divide-and-conquer Algorithm for RoadmapComputation

We consider the problem of constructing roadmaps of realalgebraic sets. This problem was introduced by Canny toanswer connectivity questions and solve motion planningproblems. Given s polynomials with rational coefficients,of degree D in n variables, Canny’s algorithm has a Monte

Carlo cost of sn log(s)DO(n2) operations in Q. An im-provement by Basu, Pollack and Roy has a deterministic

cost sO(n)DO(n2), for the more general problem of comput-ing roadmaps of a semi-algebraic set. We recently obtained

a probabilistic algorithm of cost (nD)O(n1.5) for the prob-lem of computing a roadmap of a compact and smoothhypersurface of degree D. In this talk, we show how to im-prove this algorithm using divide-and-conquer techniques,and extend it to compact and smooth algebraic sets. Thisrequires us to introduce, and control the degree of, multi-homogeneous systems corresponding to the iterated intro-duction of Lagrange multipliers.

Eric SchostComputer Science DepartmentUniv. of Western Ontarioeschost@uwo.ca

Safey El Din MohabUniversity Pierre and Marie Curiemohab.safey@lip6.fr

MS11

Extensions and Applications of Parametric Align-ment

We review extensions and applications of parametric se-quence alignment, including recently published and sub-mitted works. Parametric alignment (or RNA folding,etc.) gives rise to sum-product algorithms over semiringsof polytopes. Generalizing parametric alignment to thek-best alignment setting leads to k-set polytopes and re-lated objects. General results about lattice polytopes givebest-known output complexity and running time boundsfor several extensions of parametric alignment and relatedproblems.

Peter HugginsUniversity of Kentuckypeter.huggins@gmail.com

MS11

Algebraic Models in Systems Biology

Progress in systems biology relies on the use of mathemat-ical and statistical models for system level studies of bi-ological processes. Several different modeling frameworkshave been used successfully, including traditional differen-tial equations based models, a variety of stochastic models,agent-based models, and Boolean networks, to name somecommon ones. This talk will focus on several types of dis-crete models, and will describe a common mathematicalapproach to their comparison and analysis, which relies

46 AG11 Abstracts

on computational algebraic geometry. A software package,Analysis of Dynamic Algebraic Models (ADAM) will bepresented.

Reinhard LaubenbacherVirginia BioinformaticsInstitutereinhard@vbi.vt.edu

MS11

A Semi-algebraic Description of the of the GeneralMarkov Model on Phylogentic Trees

The phylogenetic variety for the general Markov model(GM) on a phylogenetic tree is the closure of the image ofa parameterization map over the complex numbers of thisk-state substitution model. Since the variety arises froma closure operation over the complex numbers, there arepoints in the variety that satisfy the defining polynomials,but do not correspond to real stochastic parameters for themodel. Therefore a semi-algebraic description of the imageis needed. We begin with the case of 3-dimensional tensorsand 2 states and describe several approaches for giving thesemi-algebraic conditions that a three dimensional tensor isin the image of the real parameterization map for the GMmodel. Then we discuss generalizations of this descriptionto more leaves and more states.

Amelia TaylorColorado Collegeamelia.taylor@coloradocollege.edu

Elizabeth S. AllmanUniversity of Alaska Fairbankse.allman@alaska.edu

John RhodesUniversity of Alaska, Fairbanksjarhodes2@alaska.edu

MS11

Reverse Engineering of Regulatory Networks UsingAlgebraic Geometry

Discrete models have been used successfully in modeling bi-ological processes such as gene regulatory networks. Whencertain regulation mechanisms are unknown it is importantto be able to identify the best model with the availabledata. In this context, reverse engineering of finite dynami-cal systems from partial information is an important prob-lem. In this talk we will present a framework and algorithmto reverse engineer the possible wiring diagrams of a finitedynamical system from data. The algorithm consists onusing algebraic sets to encode all possible wiring diagrams,and choose those that are minimal using the irreduciblecomponents.

Alan Veliz-CubaUniversity of Nebraska Lincolnaveliz-cuba2@unl.edu

MS11

Why Do We Care about Inequalities in (Algebraic)Statistics?

Graphical models with hidden nodes usually have a compli-cated description involving inequality constraints. A stan-dard approach in algebraic statistics is to ignore inequal-

ities which allows to use classical algebraic tools withoutreferring to the real algebraic geometry. This led for ex-ample to the method of phylogenetic invariants. Only re-cently some effort has been put to understand the full semi-algebraic structure of graphical models with hidden data.In my talk I will make a small step back trying to explainwhy do we care about inequality constraints. I will showhow they may improve our understanding of statistical in-ference. My analysis is very simplistic and focuses on thesimplest naive Bayes model. However, the main ideas willgeneralize to more complicated models.

Piotr ZwiernikMittag-Leffler InstituteStockholm, Swedenpiotr.zwiernik@gmail.com

MS12

Algebraic Geometry for Analysis of MicroscopeImages

This talk will describe potential applications for algebraicgeometry in the analysis of microscope images.

Mary ComerSchool of ECEPurdue Universitycomerm@ecn.purdue.edu

MS12

Signal Registration via Polynomial System Solu-tion Method and Image Recognition via MomentMethod

We will introduce a new method for solving over-constrained polynomial systems of equations that can beapplied to register a query object in 1D or 2D onto a cor-responding template. Also we will introduce the ”shapedictionary”, which is setted up by extracting a set of mo-ment features from the new moments we define and can beapplied in image recognition.

Shanshan HuangSchool of ECEPurdue Universityhuang94@math.purdue.edu

MS12

Finding the Non-reconstructible Locus

Given the unordered collection of the pairwise distancesof a finite point configuration in affine space, Boutin andKemper proved that one can recover the configuration upto Euclidean motion from these distances if the configu-ration is generic. In special cases, non-equivalent config-urations may give the same distance set, but these non-reconstructible cases are contained in an algebraic hyper-surface as Boutin and Kemper show also.

David Knott, Uli WaltherDepartment of MathematicsPurdue Universitydave.knott@gmail.com, walther@math.purdue.edu

Michael BurkhartPurdue Universitymcburkha@purdue.edu

AG11 Abstracts 47

MS12

Object-image Correspondence Under Projections

We provide criteria for deciding whether a given planarcurve is an image of a given spatial curve, obtained by aparallel or central projection with unknown parameters.These criteria reduce the projection problem to a certainmodification of the equivalence problem of planar curvesunder affine and projective transformations. The latterproblem can be addressed using Cartan’s moving framemethod. The computational advantage of the methodpresented here, in comparison to algorithms based on astraightforward solution, lies in a significant reduction of anumber of real parameters that has to be eliminated in or-der to establish existence or non-existence of a projectionthat maps a given spatial curve to a given planar curve.The same technique can be applied to solve object imagecorrespondence for finite lists of points.

Joseph Burdis, Irina KoganNorth Carolina State Universityjoe.burdis@gmail.com, iakogan@ncsu.edu

MS12

Invariant Histograms and Signatures for ObjectRecognition and Symmetry Detection in Images

I will survey recent developments in the use of group-invariant histograms, using distances, areas, etc., and sig-natures, using differential invariants, joint invariants, in-variant numerical approximations, etc., for object recogni-tion and symmetry detection in images.

Peter OlverMathematics DepartmentUniversity of Minnesotaolver@math.umn.edu

MS13

Title Not Available at Time of Publication

Abstract not available at time of publication.

Martin AvendanoTexas A&M Universitymavendar@yahoo.com.ar

MS13

Bounding the Sum of Square Roots via Lattice Re-duction

Let k and n be positive integers. Define R(n, k) to be theminimum positive value of

|ei√s1 + e2√s2 + · · ·+ ek

√sk − t|

where s1, s2, · · · , sk are positive integers no larger than n,t is an integer and ei ∈ {1, 0,−1} for all 1 ≤ i ≤ k. It isimportant in computational geometry to determine a goodlower and upper bound of R(n, k). In this talk we willpresent an algorithm to find lower bounds based on lat-tice reduction algorithms. It produces lower bounds muchbetter than the root separation technique does.

Qi ChengUniversity of Oklahamaqcheng@ou.edu

MS13

Chamber Cones and Fast Solving over Local Fields

Abstract not available at time of publication.

J. Maurice RojasTexas A&M Universityrojas@math.tamu.edu

MS13

Symbolic-numeric Methods for Near-singular Sys-tems

In this talk we will discuss recent developments on handlingmultivariate polynomial systems which are near singularones, using hybrid symbolic-numeric methods. First wewill consider structure preserving iterative methods whichfind the distance to- and the nearest element on certaindiscriminant varieties. Second, we will consider methodsthat turn ill-conditioned systems (with clusters of roots)into well-conditioned ones such that the roots of the newsystem are at the center of gravity of the clusters.

Agnes SzantoNorth Carolina State Universityaszanto@ncsu.edu

MS14

Morse Theory in Topological Data Analysis

We introduce a method for analyzing high-dimensionaldata. Our approach is inspired by Morse theory and usesthe nudged elastic band method from computational chem-istry. As output, we produce an increasing sequence of cellcomplexes modeling the dense regions of the data. We testthe method on data sets arising in social networks, in imageprocessing, and in microarray analysis, and we obtain smallcell complexes revealing informative topological structure.

Henry AdamsStanford Universityhenrya@math.stanford.edu

Atanas AtanasovHarvard Universitynasko@math.harvard.edu

Gunnar E. CarlssonStanford Universitygunnar@math.stanford.edu

MS14

Expansion of Random Simplicial Complexes

Expander graphs are widely studied in theoretical com-puter science and discrete mathematics and have nowfound a host of applications. There has been recent interestin defining and exhibiting higher-dimensional analogues ofexpander graphs, by Gromov and others. I will discuss re-cent joint work with Dotterrer, where we show that manykinds of random cell complexes satisfy a certain ”cobound-ary expansion” property which is a generalization of edgeexpansion of graphs to higher dimensions. Time permit-ting, I will also discuss recent work with Babson, Hoffman,and Paquette on the threshold for Kazhdan’s Property Tin the fundamental groups of random complexes, and how

48 AG11 Abstracts

this ties in with the expansion picture.

Matthew KahleInstitute for Advanced Studymkahle@gmail.com

Dominic DotterrerUniversity of Toronto”dominic dotterrer” ¡d.dotterrer@gmail.com¿

MS14

Alexander Duality for Functions

This work contributes to the point calculus of persistenthomology by extending Alexander duality to real-valuedfunctions. Given a perfect Morse function f : Sn+1 → [0, 1]and a decomposition Sn+1 = U∪V such that M = U∩V isan n-manifold, we prove elementary relationships betweenthe persistence diagrams of f restricted to U , to V , and toM .

Herbert EdelsbrunnerDepartment of Computer ScienceDuke Universityedels@cs.duke.edu

Michael KerberInstitute of Science and Technology (Austria)mkerber@ist.ac.at

MS14

The Optimality of the Interleaving Distance onMultidimensional Persistence Modules

Building on an idea of Chazal et al. [1], we define andstudy the interleaving distance, a pseudometric on multidi-mensional persistence modules generalizing the bottleneckdistance. We present several results about the interleav-ing distance. Our main result is that when the underlyingfield is Q or Z/pZ for a prime p, the interleaving distanceis (in a certain sense) optimal. This result is new even for1-D persistence. As a byproduct of our results, we obtaina converse to the algebraic stability theorem of [1]. Thisanswers a question posed in that paper.

Reference

[1] Chazal, Cohen-Steiner, Glisse, Guibas, and Oudot.Proximity of persistence modules and their diagrams.

Michael LesnickStanford Universitymlesnick@stanford.edu

MS14

Algebraic Well Groups

Given a mapping f : X → M from a topological space toan orientable manifold and U a subset of M , we identifya subgroup of the homology of f−1(U) that is stable tohomotopic perturbations of the mapping.

Amit PatelINRIAamit@akpatel.org

MS15

Theta Functions, Curves and Monopoles

The modern approach to integrability proceeds via a Rie-mann surface, the spectral curve, and solutions of the in-tegrable system may be built from theta (and allied) func-tions on the curve. In many applications this curve isspecified by transcendental constraints in terms of peri-ods and implementing these requires a good understandingof the curve, for example its homology and period matrix.Computational algebraic geometry facilitates this. In somecases, including the construction of magnetic monopoles,physical symmetries are inherited by the spectral curveand there may be consequent simplifications of both thefunction theory and transcendental constraints. We shalllook at this interplay of ideas.

Harry BradenSchool of MathematicsUniversity of Edinburghhwb@staffmail.ed.ac.uk

MS15

Singularities of Theta Divisors

Abstract: We will survey the current state of knowledgein complex algebraic geometry of the structure of the sin-gularities of the theta divisor of a principally polarizedabelian variety. We will discuss the dimension of the sin-gular locus of the theta divisor, the multiplicity of pointson it, the rank of the tangent cone, and other recent resultsand the remaining classical open questions.

Samuel GrushevskyDepartment of MathematicsSuny Stony Brooksam@math.sunysb.edu

MS15

Some Computational Problems Using RiemannTheta Functions in Sage

Computational tools in algebraic geometry are useful forgenerating new conjectures and providing a means for solv-ing problems in applications such as optimization and in-tegrable systems. Recent features in Sage for performingcomputations with Riemann theta functions and algebraiccurves provide steps towards solving a large class of theseproblems. In this talk we will discuss current and futuredevelopments in Sage for computational algebraic geom-etry and examine two applications in particular: gener-ating genus two and three solutions to the Kadomtsev–Petviashvili equation and computing determinantal repre-sentations of homogenous plane curves.

Christopher SwierczewskiApplied MathematicsUniversity of Washingtoncswiercz@gmail.com

MS16

Asymptotics of D-finite Sequences and the CheetahAlgorithm

This talk tackles the following problem:input : recurrence of a D-finite sequenceoutput: asymptotics an ∼ CAnnα with closed forms forC,A, α. Several experimental approaches are existing (e.g.packages by Doron Zeilberger, Manuel Kauers, Marc Mez-

AG11 Abstracts 49

zarobba & Bruno Salvy, see also recent work on G-functionsby Tanguy Rivoal & Stphane Fischler), but several prob-lems remain difficult (choosing the right branch, singular-ity, etc) and to decide if C = 0 is conjectured by somepeople to be undecidable. We use here a mixture of sym-bolic approach (local behaviour of solution of linear differ-ential equations, as well described by works of Frobenius,Fuchs...) and numerical analysis (the so-called cheetah al-gorithm) to get C for a lot of problems coming from numbertheory, physics, probability theory, combinatorics.

Cyril BanderierUniversite Paris-Nordcyril.banderier@lipn.univ-paris13.fr

MS16

Lattice Green’s Functions of theHigher-Dimensional Face-Centered Cubic Lattices

We study the lattice Green’s functions of the face-centeredcubic lattice (fcc) in up to six dimensions. We givecomputer algebra proofs of results that were conjecturedby Guttmann and Broadhurst for the four- and five-dimensional fcc lattices. Additionally we derive a differ-ential equation for the six-dimensional fcc lattice, a resultthat was not believed to be achievable with current com-puter hardware. We also present some conjectures concern-ing the nature of the lattice Green’s function in arbitrarydimensions.

Christoph KoutschanRISC Linzkoutschan@risc.jku.at

MS16

The Method of Brackets: A Heuristic Method forIntegration

The method of brackets consists of a small number of rulesthat provide an effective procedure for the evaluation ofmany integrals appearing in Feynman diagrams. Examplesand heuristic rules will be presented. This is joint workwith Ivan Gonzalez, Karen Kohl and Armin Straub.

VIctor MollTulane Universityvhm@tulane.edu

MS16

Recent Progress in Automation of Asymptotics

It is well known that coefficient asymptotics depend mainlyon the geometry of the algebraic surface Q=0. One of thechallenges in moving from theorems that handle most casesin practice to automated asymptotics is to combinatorializethe geometric data. This talk concerns some of the infras-tructure necessary to carry out the combinatorialization.A completely automated and rigorous algorithm for thesmooth bivariate case is described in a forthcoming paper.

Robin PemantleUniversity of Pennsylvanniapemantle@math.upenn.edu

MS16

Hypergeometric Evaluations of the Densities of

Short Random Walks

We consider random walks in the plane which consist of nsteps of fixed length each taken into a uniformly randomdirection. Our interest lies in the probability density func-tion of the distance travelled by such a walk. While LordRayleigh’s limiting density is an excellent approximationfor moderately large n, we seek closed forms for the den-sities in the case of small n. One of the goals of the talkwill be to show that in the cases n=3 and n=4 hypergeo-metric evaluations can be given. The basic ingredients arecombinatorial properties of the associated even moments,computer algebra as well as a surprising modularity of thedensities. This is joint work with Jonathan M. Borwein,James Wan, and Wadim Zudilin.

Armin StraubTulane Universitymail@arminstraub.com

MS17

Spectrahedra and Determinantal Representations

Spectrahedra are the feasible sets of semidefinite program-ming. It is an important task to classify spectrahedra. Thealgebraic problem behind this question is to write polyno-mials as determinants of linear matrix polynomials. I willdiscuss recent results concerning this problem. In particu-lar I will show how such determinantal representations areclosely linked to sums of squares decompositions of an as-sociated Hermite matrix. The results are joint work withDaniel Plaumann and Andreas Thom.

Tim NetzerUniversity of Leipzigtim.netzer@math.uni-leipzig.de

MS17

Jacobian SDP Relaxation for Polynomial Opti-mization

Consider the optimization problem of minimizing a poly-nomial function subject to polynomial equalities and/or in-equalities. Jacobian SDP Relaxation is new type semidef-inite programming method that would solve the problemexactly. Its basic idea is to add some new polynomial equal-ities from the Jacobian of the polynomials, and then applythe hierarchy of Lasserre type sum of squares relaxations.The main result is that if we apply a sufficiently high re-laxation order, then the relaxation will be exact and globaloptimal solutions for the original polynomial optimizationcould be obtained.

Jiawang NieUniversity of California, San Diegonjw@math.ucsd.edu

MS17

Positivity of Piecewise Polynomials

Real algebraic geometry provides certificates for the pos-itivity of real polynomials under polynomial constraintsby expressing them as a suitable combination of sums ofsquares and the defining inequalitites. We show how Puti-nar’s theorem for strictly positive polynomials on compactsets can be applied in the case of strictly positive piecewisepolynomials on a simplicial complex. In the 1-dimensionalcase, we improve this result to cover all non-negative piece-

50 AG11 Abstracts

wise polynomials and give explicit degree bounds.

Daniel PlaumannUniversitat Konstanzdaniel.plaumann@uni-konstanz.de

MS17

Semidefinite Programs with Rank Constraints

In 1953 Grothendieck worked on the theory of Banachspaces where he proved the fundamental theorem inthe metric theory of tensor product, nowadays calledGrothendieck inequality. This inequality is a fundamen-tal and unifying tool in many areas of mathematics andcomputer science (functional analysis, combinatorics, ma-chine learning, system theory, quantum information the-ory, numerical linear algebra). With hindsight one canview Grothendiecks inequality and its proof (which is algo-rithmic) as the first randomized approximation algorithmbased on semidefinite programming. In the talk I want toextend Grothendiecks inequality so that it can be used togive approximation algorithms for finding ground states ofthe n-vector model in statistical mechanics. Grothendiecksinequality itself together with the best known constant(due to Krivine) gives a 0.56 approximation algorithm forthe Ising model on the integer lattice. For the three-dimensional Heisenberg model the algorithm achieves a ra-tio of 0.78. (based on joint work with Jop Briet, Fernandode Oliveira Filho)

Frank VallentinDelft University of Technologyf.vallentin@tudelft.nl

MS17

Reformulating Polynomial Programs as CopositivePrograms

We present a canonical convexification procedure whichyields an equivalent formulation of polynomial program-ming problems as linear conic programs over the dual ofthe cone of copositive forms. This formulation is inspiredby Burer’s dual copositive formulation of binary quadraticprogramming, which can be recovered as a special case ofour procedure. The convexification procedure is based onnew certificates of non-negativity for polynomials over theintersection of an unbounded closed domain and the zeroset of a given polynomial.

Javier PenaCarnegie Mellon Universityjfp@andrew.cmu.edu

Juan C. VeraTilburg School of Economics and ManagementTilburg Universityj.c.veralizcano@uvt.nl

Luis F. ZuluagaFaculty of Business AdministrationUniversity of New Brunswicklzuluaga@unb.ca

MS18

Refined Bounds on Connected Components of SignConditions on a Variety

In this talk I will talk about a recent result bounding the

number of connected components of all realizable sign con-ditions of a family of real polynomials in Rk, restricted toa real variety of dimension smaller than k. Unlike previousresults in this direction, our bound distinguishes the rolesof the degrees of the polynomials defining the variety fromthat of the other polynomials. This distinction appears tobe crucial in certain applications – particularly, in certainproblems of discrete geometry on bounding the number ofincidences. I will also point out connections to previousalgorithmic work that led to the new proof.

Saugata Basu, Barone SalPurdue Universitysbasu@math.purdue.edu, sbarone@math.purdue.edu

MS18

Fast Estimates of Hankel Matrix Condition Num-bers and Numeric Sparse Interpolation

We investigate our early termination criterion for sparsepolynomial interpolation when substantial noise is presentin the values of the polynomial. Our criterion in the exactcase uses Monte Carlo randomization which introduces asecond source of error. We harness the Gohberg-Semenculformula for the inverse of a Hankel matrix to compute inquadratic arithmetic time estimates for the structured con-dition numbers of all the arising Hankel matrices, and ex-plain how false ill-conditionedness can arise from our ran-domizations. Finally, we demonstrate by experiments thatour condition number estimates lead to a viable termina-tion criterion for polynomials with about 20 non-zero termsand of degree about 100, even in the presence of noise ofrelative magnitude 0.00001. Joint work with Wen-shin Lee(Univ. Antwerp) and Zhengfeng Yang (ECNU, Shanghai)

Erich KaltofenNorth Carolina State UniversityMathematics Departmentkaltofen@math.ncsu.edu

MS18

On the Computing Time of the Continued Frac-tions Method

The maximum computing time of the continued fractionsmethod for polynomial real root isolation is shown to beat least quintic in the degree of the input polynomial.This computing time is realized for an infinite sequenceof polynomials of increasing degrees, each having the samecoefficients. This is the first non-trivial lower bound forthe maximum computing time of the continued fractionsmethod. Until recently such large computing times werenot even observed. For each input polynomial under con-sideration, the continued fractions method recursively callsitself on polynomials that become ever harder to process.In the analysis, the coefficients of those polynomials aretraced indirectly, by tracing images of the real and nonrealroots of the input polynomial under certain linear frac-tional transformations. The recursion tree is completelydescribed; its height is more than half the degree of the in-put polynomial. The quintic lower bound is proven usinga series of about eighty theorems and lemmas. The proofalso uses well-known theorems that have not been used incomputing time analyses before. Key words: Polynomialreal root isolation, computing time lower bounds, symmet-ric functions, subadditivity, Fibonacci numbers, Mignottepolynomials, loxodromic transformations. This is a report

AG11 Abstracts 51

on recent joint work with G. E. Collins, Madison, WI.

Werner KrandickDrexel Universitykrandick@cs.drexel.edu

MS18

Connectivity in Semialgebraic Sets

A semialgebraic set is a subset of real space defined by poly-nomial equations and inequalities. A semialgebraic set is aunion of finitely many maximally connected components.In this talk, we consider the problem of deciding whethertwo given points in a semialgebraic set are connected, thatis, whether the two points lie in a same connected com-ponent. In particular, we consider the semialgebraic setdefined by f not equal 0 where f is a given bivariate poly-nomial. The motivation comes from the observation thatmany important/non-trivial problems in science and engi-neering can be often reduced to that of connectivity. Dueto it importance, there has been intense research effort onthe problem. We will describe a method based on gradientfields and provide a sketch of the proof of correctness basedMorse complex.

James Rohal, Hoon HongNorth Carolina State Universityjjrohal@ncsu.edu, hong@ncsu.edu

Mohab Safey El DinUPMC, Univ Paris 6Mohab.Safey@lip6.fr

MS18

A Divide-and-conquer Method for ComputingCylindrical Algebraic Decomposition

Cylindrical algebraic formulas (CAF) provide an explicitrepresentation of semialgebraic sets as finite unions ofcylindrically arranged disjoint cells bounded by graphs ofalgebraic functions. For a quantified system of polynomialequations and inequalities, cylindrical algebraic decompo-sition computes a CAF representation of its solution set. Ifthe system is given as a Boolean combination of formulas,its cylindrical algebraic decomposition can be computed ei-ther directly, using the cylindrical algebraic decomposition(CAD) algorithm, or in two steps, by first finding a CAFrepresentation of the solution set of each formula and thencomputing the required Boolean combination of the CAFrepresentations using the CAFCombine algorithm of [1].In my talk I will discuss the problem of deciding automati-cally which of the two methods to use and how to subdividethe input system into a Boolean combination of subformu-las for the two-step method. I will describe how graphtheory methods can be used for this purpose. [1] Compu-tation with Semialgebraic Sets Represented by CylindricalAlgebraic Formulas, Proceedings of the International Sym-posium on Symbolic and Algebraic Computation, ISSAC2010, 61-68, Munich, Germany, July 25-28, 2010. ACM,Stephen M. Watt, ed.

Adam StrzebonskiWolfram Research Inc.adams@wolfram.com

MS19

Reachability Approach to the Persistence of Reac-

tion Networks

For a reaction network, persistence is the property that nospecies tend to extinction if all species are initially present.We call vacuous persistence a stronger property: the sameasymptotic feature when all species are implicitly present.We will present a necessary and sufficient condition forvacuous persistence in terms of reachability, describe twoclasses of vacuously persistent networks relevant to bio-chemistry, and relate our condition to known sufficient con-ditions for persistence.

Gilles GnacadjaAmgengilles.gnacadja@gmail.com

MS19

Identifiability of Species Phylogenies Under theCoalescent Model

A phylogenetic tree is a graph that displays evolution-ary relationships among a collection of organisms. Thesequence data available for phylogenetic inference often in-clude samples taken from multiple genes within each organ-ism. This necessitates modeling of the evolutionary processat two distinct scales. First, given an overall phylogeny rep-resenting the actual evolutionary history of the species, in-dividual genes evolve their own histories, called gene trees.Then, along each gene tree, sequence data evolve, leadingto the observed data that is used for inference. The coales-cent model provides the link between the evolution of thegene trees given the species tree, and the evolution of thesequence data given the gene trees. Phylogenetic invariantshave been proposed as a tool for inferring phylogenies usingdata from a single gene, and their mathematical propertieshave been widely studied. In this talk, we consider thedevelopment of methods based on phylogenetic invariantsdeveloped specifically for species trees, as opposed to genetrees. In particular, we use methods from algebraic statis-tics to establish identifiability of the species phylogeny.

Laura KubatkoThe Ohio State Universitylkubatko@stat.osu.edu

Julia ChifmanMathematical Biosciences InstituteThe Ohio State Universityjulia.chifman@gmail.com

MS19

Algebraic Results on the Multispecies CoalescentModel

Phylogenetic models relate a gene tree to sequence dataarising on it. To study evolutionary relationships betweenspecies, one also needs the coalescent to relate a speciestree to the gene trees arising on it. Without strong as-sumptions, the ’time’ units in these models are unrelated,so the distribution of topological gene trees is of interest.In this setting, we describe several recent results on iden-tifiability of species trees which have used a viewpoint ofalgebraic geometry.

Elizabeth S. AllmanUniversity of Alaska Fairbankse.allman@alaska.edu

James Degnan

52 AG11 Abstracts

University of Canterburyjamdeg@gmail.com

John A. RhodesUniversity of Alaska Fairbanksj.rhodes@alaska.edu

MS19

New Sufficient Conditions for Ruling out MultipleSteady States in Chemical Reaction Systems

In a chemical reaction system, the concentrations of chem-ical species evolve in time, governed by the polynomialdifferential equations of mass-action kinetics. This talkconcerns the problem of determining whether a chemicalreaction network admits multiple steady states. In gen-eral, establishing the existence of (multiple) steady statesis challenging, as it requires the solution of a large systemof polynomials with unknown coefficients. However, fornetworks that have special structure, various easy criteriacan be applied. This talk will highlight a new criterionfor ruling out multiple steady states based on the JacobianCriterion of Craciun and Feinberg, and will present workon a classification of small multistationary chemical reac-tion networks. Relevant examples from biochemistry willbe given.

Anne Shiu, Badal JoshiDuke Universityannejls@math.duke.edu, joshi@math.duke.edu

MS20

Certification and Complexity in Solving NonlinearSystems of Equations.

In this talk I will present some of the recent progress in theproblem of solving systems of equations with two require-ments: the solutions must be certified, and the complexityof the algorithms must be understood.

Carlos BeltranDepartamento de MatematicasUniversidad de Cantabriabeltranc@unican.es

MS20

Numerical Algorithms for Dual Bases of Positive-Dimensional Ideals

An ideal of a local polynomial ring can be described bycalculating a standard basis with respect to a local mono-mial ordering. However if we are only allowed approxi-mate numerical computations, this process is not numeri-cally stable. On the other hand we can describe the idealnumerically by finding the space of dual functionals thatannihilate it. There are several known algorithms for find-ing the truncated dual up to any specified degree, whichis useful for zero-dimensional ideals. I present a stoppingcriterion for positive-dimensional cases based on homoge-nization that guarantees all generators of the initial mono-mial ideal are found. This has applications for calculatingHilbert functions.

Robert KroneSchool of MathematicsGeorgia Institute of Technologyrkrone3@math.gatech.edu

MS20

Applications of Numerical Algebraic Geometry toPure Mathematics

This talk will survey known applications of the homotopycontinuation methods and numerical algebraic geometry tothe problems arising in several areas of pure mathematics. Iwill showcase our joint work with Frank Sottile on comput-ing Galois groups of Schubert problems via homotopy con-tinuation. Rigorous certification of results obtained withapproximate numerical algorithms will be discussed as well.

Anton LeykinSchool of MathematicsGeorgia Institute of Technologyleykin@math.gatech.edu

MS20

Symbolic-Numeric Algorithms for the Computa-tion of Discrete Invariants

This talk will focus on the use of symbolic-numeric methodsto compute discrete invariants of varieties, schemes, andsheaves.

Chris PetersonColorado State Universitypeterson@math.colostate.edu

MS20

Applying Littlewood-Richardson Homotopies toSchubert Problems

Based on Ravi Vakil’s geometric proof of the Littlewood-Richardson rule, we developed a numerical homotopycontinuation algorithm for finding all solutions to Schu-bert problems on Grassmannians. For generic Schu-bert problems the number of paths tracked is optimal.The Littlewood-Richardson homotopy algorithm is imple-mented using the path trackers of the software packagePHCpack.

Jan VerscheldeDepartment of Mathematics, Statistics and ComputerScienceUniversity of Illinois at Chicagojan@math.uic.edu

Frank SottileTexas A&M Universitysottile@math.tamu.edu

Ravi VakilDepartment of MathematicsStanford Universityvakil@math.stanford.edu

MS21

Search Bounds with Respect to Height in Linearand Quadratic Spaces

In this talk, I will discuss a variety of results on existenceof points and subspaces of bounded height, possibly sat-isfying some additional algebraic conditions, in linear andquadratic spaces over global fields and rings. These resultsrepresent some of the recent developments on extensionsand generalizations of such classical Diophantine theorems

AG11 Abstracts 53

as Siegel’s lemma and Cassels’ theorem on small zeros ofquadratic forms.

Lenny FukshanskyClaremont McKenna Collegelenny@cmc.edu

MS21

A Deterministic Polynomial Time Algorithm forFinding Roots of Polynomials Over Complex Num-bers

We present an iterative algorithm that has global conver-gence, that is, for any univariate polynomial of degree nover complex numbers and any initial point, the algorithmwill find a root of the polynomial, and the number of com-plex operations used is bounded by a polynomial in n andthe number of digits for desired accuracy.

Yong FengChengdu Institute of Computer ApplicationsChinese Academy of Sciencesyongfeng@casit.ac.cn

Shuhong GaoClemson Universitysgao@math.clemson.edu

MS21

Higher Mahler Measure and Lehmer’s Question

The k-higher Mahler measure of a nonzero polynomial Pis the integral of logk |P | on the unit circle. I will discussLehmer’s question for k > 1 and show some interestingformulae for 2- and 3-higher Mahler measure of cyclotomicpolynomials.

Matilde LalinUniversity of Montrealmlalin@dms.umontreal.ca

Kaneenika SinhaIndian Institute of Science Education and Research,Kolkatakaneenika@iiserkol.ac.in

MS21

Title Not Available at Time of Publication

Abstract not available at time of publication.

Korben RusekTexas A&M Universitykorben@rusek.org

MS21

The Subset Sum Problem

This will be an introduction to some new results and prob-lems on the subset sum problem, particularly those withadditional algebraic structure.

Daqing WanUniversity of California, Irvinedwan@math.uci.edu

MS22

Hyperdeterminantal Varieties from Tensor Com-plexes

I will discuss a family of hyperdeterminantal varietieswhich include varieties cut out by maximal minors of a ma-trix as well as hypersurfaces given by hyperdeterminants ofthe boundary format. These varieties arise from an explicitconstruction of certain free resolutions, called tensor com-plexes.

Christine BerkeschPurdue Universitycberkesc@gmail.com

Daniel ErmanUniversity of Michigandanerman@gmail.com

Manoj KumminiChennai Mathematical Institutemkummini@cmi.ac.in

Steven V SamMITssam@math.mit.edu

MS22

Markov Bases and Beyond

Describing Markov bases of toric statistical models hasbeen recognized as a rich source of interesting combina-torial problems. We will give an overview over past andongoing research in this area.

Thomas KahleEPDI / Institut Mittag-Lefflerkahle@mis.mpg.de

MS22

Testing Chromosome Proximity Hypothesis UsingLog-linear Models

Studying chromosome proximity and its effect on chromo-some exchange plays important role in understanding ofdevelopment of certain types of cancer. The proximity-effect hypothesis states that number of exchanges is largerbetween chromosomes that are located close to each other.To evaluate previously published data from numerous ex-periments involving 22 human autosomes of respectivelythree data sets of lymphocyte cells that have been subjectto ionizing radiation we used tools of algebraic statisticstogether with asymptotic properties of log linear models.Specifically we used Markov basis of a log-linear modelwhich does not have proximity-effect parameters, to samplefrom a large space of tables that have the same minimal suf-ficient statistics as the experiment data table. The MarkovChain Monte Carlo approach did not provide sufficient ev-idence to the reject null hypothesis of no proximity-effect.We considered a modified model, were proximity effect ofindividual pairs was included as an additional parameter.After using asymptotic tests we could not reject certainmodified models.

Tatsiana MaskalevichSan Francisco State Universitytmaskalevich@gmail.com

54 AG11 Abstracts

MS22

Primary Decomposition of a Class of ConditionalIndependence Ideals

We describe (in a combinatorial way) the primary decom-position of a class of ideals arising in the context of condi-tional independence models. The ideals we consider gener-alize the ideals considered by Fink (2010) in a way distinctfrom that of Herzog, Hibi, Hreinsdottir, Kahle, and Rauh(2010). We give a combinatorial description of the theminimal components, along with the corresponding primeideals (they turn out to be the same, although there areembedded primes) of these conditional independence ide-als. Along the way we introduce an equivalence relationand recover some other interesting algebra and geometryresults as a consequence of the development of the proof ofour main result.

Amelia TaylorColorado Collegeamelia.taylor@coloradocollege.edu

Irena SwansonReed Collegeiswanson@reed.edu

MS22

Pairwise Ranking: Choice of Method Can ProduceArbitrarily Different Order

We showed that for any two of the three popular methodsfor ranking by pairwise comparison (HodgeRank, Tropicaland Principal Eigenvector), and for any pair of rankings ofat least four items, there exists a comparison matrix for theitems such that the rankings found by the two methods arethe prescribed ones. We discuss the implications, study thegeometry and combinatorics, and state some open prob-lems.

Ngoc M. TranUC Berkeleytran@stat.berkeley.edu

MS23

The Berlekamp/Massey Algorithm and CountingSingular Hankel Matrices over a Finite Field

We derive a formula for the number of singular n×n Han-kel matrices over a finite field of q elements by observingthe Berlekamp/Massey algorithm run on the entries, al-lowing some entries above or on the anti-diagonal to befixed. We also derive a formula for the number of n × nHankel matrices whose first r leading principal submatri-ces are non-singular and the rest are singular. This resultgeneralizes to block-Hankel matrices as well.

Matthew Comer, Erich KaltofenNorth Carolina State UniversityMathematics Departmentmcomer@ncsu.edu, kaltofen@math.ncsu.edu

MS23

Exact Linear Algebra and Algebraic Topology

We present exact linear algebra methods for the compu-tation of homology groups of simplical complexes and inalgebraic K-theory for computation of the cohomology ofthe linear group GL7(Z). These compputations are re-

alised through the computation of the ranks and of the in-teger Smith normal forms of the boundary matrices of thegroups. We present novel methods for the efficient compu-tation of these routines, specialized for the large and sparsematrices arising in these applications.

Jean-Guillaume L. DumasMNC-IMAG, UJF Grenoblejean-guillaume.dumas@imag.fr

MS23

Integral Cohomology of Some Arithmetic Groups

The cohomology of a discrete group G is the cohomology ofthe quotient space X/G where X is any contractible spaceadmitting a free action of G. This talk will describe an im-plemented algorithm for computing the integral cohomol-ogy of groups such as PSL4(Z), Sp4(Z) and PSL2(O) forvarious rings O of quadratic integers. The algorithm canalso compute the cohomology of finite index subgroups ofthese three groups. The talk will discuss bottlenecks thatarise when the homology degree or the subgroup index islarge.

Graham EllisNational University of Ireland, Galwaygraham.ellis@nuigalway.ie

MS23

Symbolic-numeric Linear Algebra and Lattice Ba-sis Reduction

LLL basis reduction is an important algorithm in computerscience and mathematics making worthwile efficiency im-provements. Currently fastest reduction algorithms are us-ing both symbolic and numeric linear algebra techniques.Revisiting and improving classical tools from the field ofnumerical analysis, we propose new reduction algorithmsand certificate ([X.-W. Chang, D. Stehle, G. Villard][A.Novocin, D. Stehle, G. Villard]). We describe these devel-opments, we show in particular how approximate compu-tations may be introduced at various levels in the overallreduction process.

Gilles Villard

CNRS, Ecole Normale Superieure de Lyon, U. LyonGilles.Villard@ens-lyon.fr

MS23

Berlekamp/Massey: Implementations in LinBox

The Berlekamp/Massey algorithm and its variants com-prise a key component to many exact linear algebra meth-ods. Often the computation of the minimal generator ofa linearly generated sequence is a vital sub-procedure inmethods like linear solving, smith forms and rank com-putations. A discussion of recent and current work on theimplementation of this algorithm within the LinBox librarywill be given. The algorithm presents interesting choicesfor the implementation and data structures required duringits computations. The different design choices, applicationsand computational results will be presented.

George YuhaszMorehouse CollegeDivision of Science & Mathematicsgyuhasz@morehouse.edu

AG11 Abstracts 55

MS24

Analysis Aware Representations, Parameteriza-tions, and Models

Isogeometric Analysis (IA) has been proposed as a method-ology for bridging the gap between Computer Aided De-sign (CAD) and Finite Element Analysis (FEA). In orderto support design and full 3D IA, new ab initio designmethods must create suitable representations and approxi-mation techniques must include parameterization method-ologies for volumes. This presentation discusses some ofthe challenges in moving from current representations anddatafitting techniques towards this goal and demonstratesinitial results and analyses.

Elaine CohenUniversity of Utahcohen@cs.utah.edu

MS24

Splines and Isogeometric Analysis (IGA)

Finite Element Analysis (FEA) was already established in1970 when Computer Aided Geometric Design (CAGD)community was established. Until the introduction of IGAby T. J. R. Hughes in 2005 little cooperation took place be-tween the FEA and CAGD communities. IGA has the po-tential to drastically improve the interoperability of CAGDand FEA and the quality FEA. To achieve this we need im-proved technology and increased research into spline tech-nology, FEA and visualization.

Tor DokkenSINTEF ICT, Department of Applied MathematicsOslo, Norwaytor.dokken@sintef.no

MS24

Characterization of Hierarchical Spline Spaces

The only requirement which characterizes the hierarchicalmodel is a refinable nature of the underlying basis func-tions defined on nested approximation spaces. The talkwill present recents results on the construction of normal-ized hierarchical spline bases and their properties.

Carlotta GiannelliJohannes Kepler University LinzAustriacarlotta.giannelli@jku.at

Bert JuttlerJohannes Kepler University Linzbert.juettler@jku.at

MS24

Locally Refined B-splines

We will address local refinement of tensor product B-splines specified as a sequence of inserted line segmentsparallel to the knot lines. We obtain a quadrilateral gridwith T-junctions in the parameter domain, and a collec-tion of tensor product B-splines on this mesh here namedan LR-mesh. The approach applies equally well in dimen-sions higher than two. Moreover, in the two dimensionalcase this collection of B-splines spans the full spline spaceon the LR-mesh.

Tom Lyche

University of OsloDepartment of Informatics and CMAtom@ifi.uio.no

MS24

Splines on Rectangular Subdivisions: Dimensionand Basis

Standard parametrisations of surfaces in CAGD are basedon tensor product bspline functions, defined from a gridof nodes over a rectangular domain. This representationis not well-adapted to local refinement, which is particu-larly important in applications such as surface reconstruc-tion or isogeometric analysis. To extend the representa-tion of tensor-product splines while providing local refine-ment capabilities, some recent works have considered T-subdivisions. These are partitions of an axis-aligned boxinto smaller axis-aligned boxes. In this talk, we will an-alyze the space of bivariate functions that are piecewisepolynomial of bidegree (m,m′) and class C(r, r′) over sucha planar T-subdivision. We shall show how algebraic tech-niques yield a good control of its resolution, give a newformula for its dimension and explicit bases for small de-gree and regularity.

Bernard MourrainINRIA Sophia Antipolismourrain@sophia.inria.fr

MS25

Reeb Graph Reconstruction and Zigzag PersistentHomology

Reeb graph reconstruction is an important problem intopological data analysis. We study the error in recon-structing the Reeb graph of a point cloud sampled from amanifold, using the MAPPER algorithm. Using the lan-guage of zigzag persistence, we are able to succinctly cap-ture the difference between the reconstructed and actualReeb graphs. A worst case bound on the error is derivedin terms of the sampling density. We demonstrate practicalapplications such as dataset classification.

Aravindakshan BabuICME, Stanford UniversityPhD Studentakshan@stanford.edu

Daniel MuellnerStanford Math Departmentmuellner@math.stanford.edu

Gunnar E. CarlssonStanford Universitygunnar@math.stanford.edu

MS25

PHIsoMap: Estimating Intrinsic Distance UsingPersistent Homology

Many dimension reduction techniques (including IsoMap)start by constructing a graph on the data in order to es-timate the geodesic distance from the underlying mani-fold. We present a algorithm that uses persistent homology(in dimensions one and zero) to construct an ε-proximitygraph; our initial experiments show that this graph pro-duces strikingly good estimates of the underlying geodesic

56 AG11 Abstracts

distance.

Paul BendichDuke Universitypbendich@gmail.com

Jacob HarerNorth Carolina State Universityjacob.harer@duke.edu

MS25

Covering and Packing by Diagonal Distortion

We address the problem of covering Rn with congruentballs, while minimizing the number of balls that containan average point. Considering the 1-parameter family oflattices defined by stretching or compressing the integergrid in diagonal direction, we give a closed formula for thecovering density that depends on the distortion parameter.We observe that our family contains the thinnest latticecoverings in dimensions 2 to 5. We also consider the prob-lem of packing congruent balls in Rn, for which we give aclosed formula for the packing density as well. Again weobserve that our family contains optimal configurations,this time densest packings in dimensions 2 and 3.

Herbert Edelsbrunner, Michael KerberInstitute of Science and Technology (Austria)edels@ist.ac.at, mkerber@ist.ac.at

MS26

Implicitization of Surfaces via Geometric Tropical-ization

In this talk we discuss recent developments in tropicalmethods for implicitization of surfaces. This study was pi-oneered in the generic case by work of Sturmfels, Tevelevand Yu, and is based on the theory of geometric tropi-calization, developed by Hacking, Keel and Tevelev, Thelatter hinges on computing the tropicalization of subvari-eties of tori by analyzing the combinatorics of their bound-ary in a suitable (tropical) compactification. We enhancethis theory by providing a formula for computing weightson tropical varieties, a key tool for tropical implicitization.Finally, we address the question of tropical implicitizationfor non-generic surfaces and illustrate our techniques withseveral numerical examples in 3-space.

Maria Angelica CuetoUniversity of California, Berkeleymacueto@math.berkeley.edu

MS26

Singular Tropical Hypersurfaces

The concept of a singular point of a tropical variety is notwell established yet. A natural definition is to declare apoint q in a tropical variety V singular if q is the imageunder the valuation of a singular point of a classical alge-braic variety, defined over the field of Puiseux series andwith tropicalization V. We present a purely tropical char-acterization of a singularity of a tropical hypersurface ofarbitrary dimension (with fixed support) in terms of tropi-cal Euler derivatives. We also describe in this setting non-transversal intersections of planar tropical curves. We showan algorithm to compute all singularities and we relate ourapproach to the known tropicalization of the discriminant.

Alicia Dickenstein

Universidad de Buenos Airesalidick@dm.uba.ar

Luis TaberaUniversidad de Cantabriataberalf@unican.es

MS26

Tropical Reparameterisations

It is well known that the tropicalisation of a polynomialmap φ : Am → An maps (R ∪ {∞})m into the tropical-isation of the image X of φ. Typically this tropical mapis not surjective. To remedy this, one may precompose φwith other polynomial maps and then tropicalise the re-sult. Ideally, one would like to prove that finitely manysuch reparameterisations suffice to cover the entire trop-icalisation of X. I will give an overview of when this isknown to be the case.

Jan DraismaT.U.Eindhoven, The NetherlandsDepartment of Mathematicsj.draisma@tue.nl

MS26

Computing Tropical Resultants

Tropical resultant varieties arise when fixing a set of New-ton polytopes and asking for which set of coefficients theirtropical hypersurfaces have a common intersection. In thistalk we compare various methods for computing tropicalresultants. Some of the algorithms involve traversing sub-fans of secondary fans while other works by reconstructiontropical hypersurfaces from projections. Using these tech-niques we also get a new method for computing mixed fiberpolytopes.

Anders JensenUniversitat des Saarlandesanders@soopadoopa.dk

Josephine YuGeorgia Institute of Technologyjosephine.yu@math.gatech.edu

MS26

An Effective Computation of Tropical LinearSpaces, with an Application to A-discriminants

We develop and implement a very fast algorithm for com-puting tropical linear spaces, making some computationalapplications of tropical geometry now viable. For this pur-pose, we study a fan structure on the Bergman fan of amatroid which slightly refines its nested set structure, andwhich is amenable to our computational purposes. Somesoftware implementations will be shown, including an ap-plication to the computation of A-discriminants.

Felipe RinconUC Berkeleyfelipe@math.berkeley.edu

MS27

The Moment Problem for Continuous Linear Func-

AG11 Abstracts 57

tionals

Let V be the countable dimensional real polynomial alge-bra. Let τø be a locally convex topology on V . Let K be aclosed subset of Rn, and let M := M{g1,···,gs} be a finitelygenerated quadratic module in V . We investigate the fol-lowing question: When is the cone Psd(K) (of polynomialsnonnegative on K) included in the τ closure of M?. Wegive an interpretation of this inclusion with respect to rep-resenting continuous linear functionals by measures. Wediscuss several examples; we compute the closure of thecone of sums of squares with respect to weighted norm-p topologies. We show that this closure coincides withthe cone Psd(K) for certain convex compact K. We usethese results to generalize Berg et al. work on exponentiallybounded moment sequences. Joint work with M. Ghasemiand E. Samei

Salma KuhlmannUniversity of Konstanz , Germanysalma.kuhlmann@uni-konstanz.de

MS27

A New Look at Nonnegativity on Closed Sets andPolynomial Optimization

We provide a new characterization of the convex cone CK

of degree-d polynomials that are nonnegative on a closedset K ⊆ Rn, provided that one knows all moments of afinite Borel measure whose support is exactly K. Fromthis we also provide a hierarchy of outer approximations(Pj) of CK where each each convex cone Pj is a spectra-hedron described only in terms of the coefficients of thepolynomials, i.e., with no lifting. As a by-product, withK = Rn

+, we obtain converging outer approximations ofthe cone of copositive matrices. Finally, and also as a by-product, when K is a simple set like e.g., a box, an ellip-soid, the hypercube {−1, 1}n, the positive orthant, or thewhole space Rn, etc., we provide a monotone nonincreasingsequence of upper bounds on the global minimum of anypolynomial f on K. These upper bounds complement thesequence of lower bounds obtained from the moment-sosrelaxations.

Jean B. LasserreLAAS CNRS and Institute of Mathematicslasserre@laas.fr

MS27

On the Inverse Moment Problem for Polytopes

Reconstructing a measure from its moments is known as aninverse moment problem, and has important applicationsfor data processing, e.g. in geophysics, image recognition,etc. The problem is well-studied in dimension 2, wheretools of complex analysis are applicable. The basic in theapproach there are procedures for expressing coordinatesof the vertices V (P ) of a polygon P , given moments ofa measure supported on P . Higher-dimensional problemsare traditionally treated in an approximate way, by workingwith 2-D slices, and only lead to approximate solutions. Wepresent efficient procedures for reconstructing polynomialmeasures dμ, supported on convex d-dimensional polytopesP , given finitely many moments

∫PXadμ, a ∈ Zd

+. Ourprocedures recover the vertices of P and μ, and can be usedin exact (rational inputs, exact rational outputs) as wellas in non-exact setting (inputs with errors, approximateoutputs). We use tools from toric geometry, and symbolic-numeric computations with representations of V (P ) as uni-

variate rational functions.

Nikolay GravinNTU Singaporengravin@pmail.ntu.edu.sg

Jean B. LasserreLAAS CNRS and Institute of Mathematicslasserre@laas.fr

Dmitrii Pasechnik, Sinai RobinsNTU Singaporedima@ntu.edu.sg, rsinai@ntu.edu.sg

MS27

Using Symmetries in Polynomial Optimization

Solving polynomial optimization problems is known to bea hard task in general. The semidefinite methods whichenvolved within recent years give possibilities to calculatethese hard problems. However, as the sizes of the resultingSDPs grow fast, it is helpful to exploit some structure of theproblem. We present approaches for exploiting symmetrieswithin this framework. Our special focus is on problemswhich are invariant by the symmetric group.

Cordian B. RienerUniversity of Konstanz, Germanyriener@math.uni-frankfurt.de

Thorsten TheobaldJ.W. Goethe-UniversitatFrankfurt am Main, Germanytheobald@math.uni-frankfurt.de

Jean B. LasserreLAAS CNRS and Institute of Mathematicslasserre@laas.fr

MS27

Successive Convex Relaxation Methods for Opti-mization over Semi-Algebraic Sets

I will discuss various techniques (new and old) used in con-vergence proofs for successive convex relaxation methodsapplied to optimization problems expressed as maximiza-tion of a linear function over a finite set of polynomial in-equality constraints. I will focus on convergence rates andspecially structured non-convex optimization problems.

Levent TuncelUniversity of WaterlooDept. of Combinatorics and Optimizationltuncel@math.uwaterloo.ca

MS28

Toward a Salmon Conjecture

Methods from numerical algebraic geometry are applied incombination with techniques from classical representationtheory to show that the variety of 3-by-3-by-4 tensors ofborder rank 4 is cut out by polynomials of degree 6 and9. Combined with results of Landsberg and Manivel, thisfurnishes a computational solution of an open problem inalgebraic statistics, namely, the set-theoretic version of All-man’s salmon conjecture for 4-by-4-by-4 tensors of borderrank 4. A proof without numerical computation was given

58 AG11 Abstracts

recently by Friedland and Gross.

Luke OedingUniversity of California BerkeleyUniversita degli Studi di Firenzelukeoeding@gmail.com

MS28

Geometry of Tensors and Numerical Decomposi-tion

This talk will consider tensor parameter spaces from a ge-ometric point of view. The study of tensors from the van-tage point of algebraic geometry leads to symbolic andsymbolic-numeric algorithms related to tensor decompo-sitions, tensor approximations, and generic rank. Both ageometric description and basic implementations of severalof these algorithms will be presented.

Hirotachi AboUniversity of Idahoabo@uidaho.edu

Giorgio OttavianiUniversity of Firenzeottavian@math.unifi.it

Chris PetersonColorado State Universitypeterson@math.colostate.edu

MS28

Some non-minimal Canonical Representations ofForms as a Sum of Powers of Linear Forms

We will give some illustrations of the title phenomenon,especially Reichstein’s beautiful “completion of the cube”from the late 1980s.

Bruce ReznickUniversity of Illinois, Urbana-Champaignreznick@math.uiuc.edu

MS28

Stochastic Models, Tensor Rank, and Inequalities

The hyperdeterminant Δ on 2 × 2 × 2 tensors appears ina semialgebraic characterization of the probability distri-butions arising from binary models of evolution on phylo-genetic trees, through its connection to tensor rank. Tounderstand inequality constraints for non-binary models,we construct an explicit representation space of functionswhich, for k×k×k tensors of border rank k, is analogous toΔ. In particular, their non-vanishing on a tensor of borderrank k indicates it has rank k.

Elizabeth S. AllmanUniversity of Alaska Fairbankse.allman@alaska.edu

Peter D. JarvisUniversity of TasmaniaPeter.Jarvis@utas.edu.au

John A. RhodesUniversity of Alaska Fairbanksj.rhodes@alaska.edu

Jeremy SumnerUniversity of Tasmaniajsumner@utas.edu.au

MS28

Ranks and Generalized Ranks

The Waring rank of a polynomial of degree d is the leastnumber of terms in an expression for the polynomial as asum of dth powers. The problem of finding the rank of agiven polynomial and studying rank in general is relatedto secant varieties, and there are applications throughoutengineering and the sciences, such as in signal processingand computational complexity; and of course, it has beena central problem of classical algebraic geometry. For ex-ample, J.J. Sylvester gave a lower bound for rank in termsof catalecticant matrices in the mid-19th century. Whilecatalecticant matrices and varieties have become objectsof study in their own right, there has been relatively littleprogress in the last 160 years on the problem of bound-ing or determining the rank of a given (not general) poly-nomial. I will describe joint work with J.M. Landsbergwhich gives a new, elementary improvement to the catalec-ticant lower bound for the rank of a polynomial, in termsof the geometry of the polynomial. I will also describe newwork which generalizes this improved bound to multiho-mogeneous polynomials, corresponding to secant varietiesof Segre-Veronese varieties. A further generalization to therecently described Young flattenings (treating secant vari-eties of arbitrary varieties) is work in progress.

J.M. LandsbergTexas A&Mjml@math.tamu.edu

Zach TeitlerBoise State Universityzteitler@boisestate.edu

MS29

A Practical, Principled Approach to Fast Simplifi-cation of Tarski Formulas

This talk focuses on the problem of false simplification ofTarski formulas, which means finding meaningful simplifi-cations that occur in practice within a period of time thatis small relative to most symbolic computations involvingpolynomial systems over the reals. An effective algorithmfor fast simplification can be used to improve other algo-rithms, by pre-processing input, post-processing output,and, perhaps most importantly, by reducing the size of in-termediate results. The talk will describe extensions ofwork I have previously published on the subject that, ul-timately, provide a fast simplification procedure that is ef-fective at improving the performance and quality of resultsfor several other algorithms. The approach makes some in-teresting connections between a sub-class of simplificationson Tarski formulas and coding theory; a connection whichis exploited in complexity analyses.

Christopher BrownUnited States Naval Academywcbrown@usna.edu

MS29

Certified Global Optimization with Exact Sum-Of-

AG11 Abstracts 59

Squares

Given a multivariate polynomial f with real or rational co-efficients, we focus on computing and certifying its lowerbounds or its global infimum. The global optimizationproblem is NP-hard. A well known technique is to use thesum-of-squares(SOS) and semidefinite programming(SDP)to relax it to get lower bounds. We implement the SDPinterior-point method in Maple and get high precision SDPresults which can not be obtained by the fixed precisionSDP solvers in matlab. Then we certify the lower boundsby making projections to get the exact SOS representa-tions. To compute the global infimum, we use the re-sults from the computation of sample points in the smoothreal algebraic set, then relax the problem by computingits infima over some finite semialgebraic sets which canbe reduced to SDPs. From Artin’s affirmative solution ofHilbert’s 17th problem, we know that a nonnegative poly-nomial can be written as a sums of squares of rational func-tions. Given a fixed degree of the denominators, we givethe certification if the polynomial can not be represented assums of squares of rational functions with denominators ofthe fixed degree. This is a joint work with Erich Kaltofen,Mohab Safey El Din, Zhengfeng Yang and Lihong Zhi.

Feng GuoNorth Carolina State Universityfguo@mmrc.iss.ac.cn

MS29

Multivariate Root Bound

Given a system F of n polynomials in n variables, a boundB is obtained such that if F(x) = 0, then x1,...,xn ¡ B, for allreal x. A root bound gives a useful starting point for var-ious domain decomposition methods used to approximatethe solution of systems of polynomials. Such a bound canalso be used to estimate the complexity of root finding al-gorithms. We present a bound which is significantly tighterthan those previously studied. The bound is obtained byintegrating results from Elimination theory and Eigenvalueinclosure theory.

Aaron Herman, Hoon HongNorth Carolina State Universityaherman@ncsu.edu, hong@ncsu.edu

MS29

On Hilbert’s 16th Problem

At the start of the 20th century David Hilbert proposed23 problems to be solved over the coming century. Thefirst portion of his 16th problem asks about the arrange-ment of the connected components of nonsingular algebraiccurves in the real projective plane. Despite progress overthe last century, the problem has remained unsolved in ageneral sense. Efforts to resolve this problem have previ-ously focused on constructions of maximal curves by handthrough slight permutations of singular curves. Approach-ing the problem by hand unfortunately tends to producepolynomials with large cofficients and curves which are vi-sually poor when plotted. Instead this talk will approachthe problem from a computerized perspective after usinggrid processing to compute topologies for many generatedpolynomials. The statistical distribution of curve topolo-gies by degree and coefficient range will be discussed as wellas ’nice’ witness polynomials for known topological cases.

Jason YellickNorth Carolina State University

jason-ncsu-phd@unaddressable.org

MS29

Computing Rational Points in Convex Semi-Algebraic Sets

Let P = {h1, . . . , hs} ⊂ Z[Y1, . . . , Yk], D ≥ deg(hi) for1 ≤ i ≤ s, σ bounding the bit length of the coefficientsof the hi’s, and Φ be a quantifier-free P-formula defininga convex semi-algebraic set. We design an algorithm re-turning a rational point in S if and only if S ∩ Qk �= ∅.It requires σO(1)DO(k3) bit operations. If a rational pointis outputted its coordinates have bit length dominated by

σDO(k3). Using this result, we obtain a procedure decid-ing if a polynomial f ∈ Z[X1, . . . , Xn] is a sum of squaresof polynomials in Q[X1, . . . , Xn]. Denote by d the degreeof f , τ the maximum bit length of the coefficients in f ,D =

(n+dn

)and k ≤ D(D + 1) −

(n+2d

n

). This procedure

requires τO(1)DO(k3) bit operations and the coefficients ofthe outputted polynomials have bit length dominated by

τDO(k3). This is joint work with Mohab Safey El Din

Lihong ZhiAcademia Sinicalzhi@mmrc.iss.ac.cn

MS30

Using Fiber Products and Numerical Algebraic Ge-ometry to Find Exceptional Mechanisms

An exceptional mechanism is a mechanism having specialmovement as compared to other mechanisms in the sameclass. These correspond to points in the parameter space ofa given class, which have fiber dimension greater than thegeneric fiber dimension. Such points are usually obscuredby their containment in an irreducible component. We usenumerical algebraic geometry, starting with the fiber prod-uct method of Sommese and Wampler, to identify thesepoints, thus finding exceptional mechanisms.

Daniel J. BatesColorado State UniversityDepartment of Mathematicsbates@math.colostate.edu

Eric HansonDepartment of MathematicsColorado State Universityhanson@math.colostate.edu

Jonathan HauensteinTexas A&M Universityjhauenst@math.tamu.edu

Charles WamplerGeneral Motors Research Laboratories, EnterpriseSystems Lab30500 Mound Road, Warren, MI 48090-9055, USAcharles.w.wampler@gm.com

MS30

Beyond Parametrization of Algebraic Varieties byMeans of Global Optimization

This research work presents a new global optimization al-gorithm, Evo-Runge-Kutta, to tackle the open problem of

60 AG11 Abstracts

minimizing a positive function on an algebraic variety. Weillustrate the application of Evo-Runge-Kutta, a two-phaseevolutionary optimization algorithm using a multicriteriaapproach, for computing explicit and implicit Runge-Kuttamethods. In particular, we present theoretical motivationsfor the application of the class of evolutionary algorithms:the hardness to find 1) the dimension of the variety and2) feasible solutions for the parametrization. The mappingbetween algebraic geometry and global optimization is di-rect. In order to design methods having specific properties,we expect that many open problems in algebraic geometrywill be modeled as constrained multi-objective optimiza-tion problems.

Ivan MartinoStockholms Universitetmartino@math.su.se

Giuseppe NicosiaUniversity of CataniaDepartment of Mathematics and Computer Sciencenicosia@dmi.unict.it

MS30

Solving Principal Agent Problems by PolynomialProgramming

We present a new way to solve principal agent problems bypolynomial programming techniques. We study the casewhere the agent’s actions are unobservable by the prin-cipal but the outcomes are. We assume that the agent’sactions lie in an interval and the space of outcomes is afinite set. Furthermore the agent’s expected utility is arational function in his actions. The resulting problem isa bilevel optimization problem with the principal’s prob-lem as the upper and the agent’s problem as the lower level.The key idea is to find an exact reformulation of the agent’sproblem as a semidefinite optimization problem. Since thisis a convex optimization problem, we then have necessaryand sufficient global optimality conditions for the agent’sproblem. The reformulation can be done by using classicalresults from real algebraic geometry linking positive poly-nomials and semidefinite matrices. We obtain a nonlinearprogram. If all functions are rational functions, we thencan solve it to global optimality.

Philipp RennerDepartment of Business AdministrationUniversitaet Zuerichphilipp.renner@business.uzh.ch

Karl SchmeddersUniversity of Zuerichkarl.schmedders@business.uzh.ch

MS30

Tumor Growth Models and Numerical AlgebraicGeometry

Numerical Algebraic Geometry is being used to explorefree boundary problems arising from tumor growth models.These methods are used to compute solutions, check theirstability, and track the solutions through bifurcations ascritical parmaters change. The polynomial systems, whichinvolve thousands of variables, pose interesting challenges.

Andrew SommeseDepartment of Mathematics, University of Notre Dame

Notre Dame, IN 46556-4618, USAsommese@nd.edu

Wenrui HaoDept. of Applied and Comp. Mathematics and StatisticsUniversity of Notre Damewhao@nd.edu

Jonathan HauensteinTexas A&M Universityjhauenst@math.tamu.edu

Bei HuDept. of Applied & Computational Mathematics &StatisticsUniversity of Notre Dameb1hu@nd.edu

MS31

Ideals of Graph Homomorphisms and Graph Col-orings

We show how ideals of graph homomorphisms in algbraicstatistics can be employed to study graph colorings.

Alex EngstromUC Berkeleyalex@math.berkeley.edu

MS31

The Binomial Ideal of the Intersection Axiom forConditional Probabilities

The binomial ideal associated with the intersection axiomof conditional probability is shown to be radical and isexpressed as intersection of toric prime ideals. This resolvesa conjecture in algebraic statistics due to Cartwright andEngstrom.

Alex FinkUniversity of California at Berkeleyarfink@ncsu.edu

MS31

Commutative Algebra of Statistical Ranking Mod-els

A model for statistical ranking is a family of probabilitydistributions whose states are orderings of a fixed finite setof items. We represent the orderings as maximal chainsin a graded poset. The most widely used ranking mod-els are parameterized by rational function in the modelparameters, so they define algebraic varieties. We studythese varieties from the perspective of combinatorial com-mutative algebra. One of our models, the Plackett-Lucemodel, is non-toric. Five others are toric.For these modelswe examine the toric algebra, its lattice polytope, and itsMarkov basis.

Volkmar WelkerPhilipps-Universitat MarburgFachbereich Mathematik und Informatikwelker@mathematik.uni-marburg.de

Bernd SturmfelsUC Berkeleybernd@math.bekerley.edu

AG11 Abstracts 61

Lukas KatthanUniversity of Marburgkatthan@mathematik.uni-marburg.de

MS31

Polytopes and Imsets

We study polytopes and imsets of generalized tree graphs.

Patrik NorenKTH Stockholmpnore@math.kth.se

MS31

Conditional Independence Ideals and Robustness

The talk discusses a class of conditional independence (CI)ideals motivated by conceptual studies of robustness [Ay,N., and Krakauer, D. C., “Geometric robustness theoryand biological networks,’ Theory in Biosciences, vol. 125,no. 2, pp. 93 – 121, 2007.]. Consider a collection of n+ 1finite random variables X0, X1, . . . , Xn with the interpre-tation that X0 is the output of the system, computed fromthe inputs X1, . . . , Xn. Such a system is robust if the com-plete knowledge of all input variables is not necessary inorder to compute the output. This notion of robustness canbe modelled as a collection of CI statements, generalizingthe set of CI statements studied in [Fink, A., “The bino-mial ideal of the intersection axiom for conditional proba-bilities,’ Journal of Algebraic Combinatorics, vol. 33, no. 3,pp. 455–463, 2011.]. CI statements give polynomial restric-tions on the joint probability distribution of the system. Ifthe output variable is binary, then the resulting CI idealis a binomial edge ideal, as defined in [Herzog, J., Hibi,

T., Hreinsdottir, F., Kahle, T., and Rauh, J., “Bino-mial edge ideals and conditional independence statements,’Advances in Applied Mathematics, vol. 45, no. 3, pp. 317– 333, 2010.]. The results generalize to the case that theoutput is non-binary. In particular, the CI ideal is radical,and the primary decomposition has a nice interpretation.

Johannes RauhMax Planck Institute for Mathematics in the Sciencesjrauh@mis.mpg.de

Nihat AyMax Planck Institute for Mathematics in the Sciencesnay@mis.mpg.de

MS32

Linear Algebra over Dense Matrices over GF(2)and Small Extensions - Albrecht

Linear algebra over F2 and F2e where e is a small integerhas many applications such as cryptography and codingtheory. Since matrices over these base fields can be effi-ciently bit-packed, dense techniques are often feasible forconsiderable matrix sizes beyond 100,000 x 100,000. Inthis talk we will present our work on matrices over thesefields and discuss algorithms, implementation techniquesand issues.

Martin AlbrechtUniversite Pierre et Marie CurieParis, Francemalb@lip6.fr

MS32

Pivoting Strategies for Sparse Matrices over FiniteFields

Solving a polynomial system of equations over a finite fieldis at the heart of algebraic cryptanalysis and numerousother applications, including the Quadratic Sieve. Severalalgorithms for solving such systems of equations over thosefields involve reducing a matrix over a finite field. In thispaper, we analyze a plethora of pivoting strategies, rangingfrom the naive to the complex, for performing structuredGaussian Elimination to minimize fill-in and field operationcounts. We consider random matrices, as well as matricesof the forms produced by the Quadratic Sieve and also theXL/F4 algorithms. The ability of the various techniques tomaintain sparsity is compared for various sizes and inputmatrices, as well as the running times.

Gregory BardMathematics DepartmentFordham University, The Bronx, NY.gregorybard@mac.com

MS32

Groebner Bases and Linear Algebra

There is a strong interplay between computing efficientlyGrobner bases and linear algebra. In this talk, we focus onseveral aspects of this convergence:

• Algorithmic point of view: algorithms for comput-ing efficiently Grobner bases (F4, F5, . . .) not only relyheavily on linear algebra but were based on linear al-gebra techniques. Recent algorithms to change theordering of a Grobner basis is also related to multivari-ate generalization of the Wiedemann algorithm. Mix-ing Grobner bases methods and linear algebra tech-nique for solving sparse linear systems leads to an ef-ficient algorithm to solve boolean quadratic equations;this algorithm is faster than exhaustive search by anexponential factor

• Application point of view: for instance, a generaliza-tion of the eigenvalue problem to several matrices –the MinRank problem – is at the heart of the securityof many multivariate public key cryptosystems.

• Design of library: when implementing Grobner basesalgorithms the crucial task is the linear algebra part;moreover the specific structure of the matrices hasto be taken into account to speedup even more thecomputations.

Jean-Charles FaugereUniversite Pierre and Marie CurieJean-Charles.Faugere@inria.fr

MS32

Verified Numerical Linear Algebra: Linear SystemSolving

Solving numerically a linear system can be performed veryefficiently, using optimized routines, but it yields an ap-proximate solution without any indication about its accu-racy. Getting an enclosure of the error between the ap-proximate and the exact solutions is called ”verification”.We present a verified algorithm which gets an accurate en-closure, with a moderate overhead both in complexity andin practical performance. Its key ingredients are intervalarithmetic, iterative refinement and well-chosen computing

62 AG11 Abstracts

precision.

Nathalie RevolArenaire project, LIP, Ecole Normale Superieure de Lyonand Lab. ANO, Univ. Lille, FranceNathalie.Revol@ens-lyon.fr

MS32

Exact Solutions to Mixed-integer Linear Program-ming Problems

We present a hybrid symbolic-numeric approach for solvingmixed-integer-programming problems exactly over the ra-tional numbers. By performing many operations using fastfloating-point arithmetic and then verifying and correctingresults using symbolic computation, exact solutions canbe found without relying entirely on rational arithmetic.Computational results will be presented based on an ex-act branch-and-bound algorithm implemented within theconstraint integer programming framework SCIP.

Dan SteffyZuse Institute Berlinsteffy@zib.de

William CookGeorgia Institute of Technologybico@isye.gatech.edu

Thorsten Koch, Kati WolterZuse Institute Berlinkoch@zib.de, wolter@zib.de

MS33

Algebraic Problems in Graphical Modeling

A graphical model is a statistical model associated with agraph whose nodes correspond to random variables. I willgive an introduction to these models and provide statisticalmotivation for algebraic problems that will be addressed bythe speakers in the session.

Mathias DrtonDepartment of StatisticsUniversity of Chicagodrton@uchicago.edu

MS33

Parameter Identifiability via the Half-trek Crite-rion in Mixed Graphs

In the setting of Gaussian graphical models, mixed graphsallow for both linear effects between the nodes (directededges) and covariance among the Gaussian errors on eachnode (bidirected edges). We study parameter identifiabil-ity in these models, that is, we ask for conditions thatensure that the edge coefficients and correlations appear-ing in a linear structural equation model can be uniquelyrecovered from the covariance matrix of the associated nor-mal distribution. In particular, we are interested in genericparameter identifiability, where recovery is possible for al-most every choice of parameters. We give a new graphicalcriterion that is sufficient for generic identifiability. Unlikeprior work on this problem, our criterion does not requirethe directed part of the graph to be acyclic, and addition-ally it improves on existing criteria for the acyclic case. Wealso develop a related necessary condition, and examine the“gap” between sufficient and necessary conditions through

simulations as well as exhaustive algebraic computationsfor graphs with up to five nodes.

Rina FoygelDepartment of StatisticsUniversity of Chicagorina@uchicago.edu

Jan DraismaEindhoven University of Technologyjdraisma@win.tue.nl

Mathias DrtonDepartment of StatisticsUniversity of Chicagodrton@uchicago.edu

MS33

Parameter Identification of Structural EquationModels

Structural equation models (SEMs) are used to formalizea variety of causal queries as certain types of probabilitydistributions. A central problem in SEMs is the analysisof identification. A model is identified if it only admitsa unique parametrization to be compatible with a givendata set. Here, I will introduce a computer algebra soft-ware to test identifiability, and discuss recent advances onglobal identification via the analysis of the Jacobian of theparametrization.

Luis D. Garcia-PuenteDepartment of Mathematics and StatisticsSam Houston State Universitylgarcia@shsu.edu

MS33

Discrete Graphical Models with One Hidden Vari-able

Kruskal’s theorem applies to a simple latent class model, inwhich three observed variables are independent when con-ditioned on a single hidden one. Our aim is to give generalconditions for identifiability using Krukal’s theorem and,when the conditions fail to hold, to characterize the spacewhere identifiability breaks down. We present a techniquethat, given an arbitrary directed graphical model with asingle hidden variables, modifies the model in such a waythat we can apply Kruskal’s theorem. The technique isbased on the following three operations:

• Clump several variables (all hidden or all observed)into a single one, with larger state space.

• Condition on the state of an observed variable,

• Marginalize over an observed variable (making it hid-den).

Each of these operations can be done multiple times, andin combination with one another.

Elizabeth S. Allman, John A. RhodesUniversity of Alaska Fairbankse.allman@alaska.edu, j.rhodes@alaska.edu

Elena StanghelliniDipartimento ti Economia Finanza e StatisticaUniversita’ di Perugiaelena.stanghellini@stat.unipg.it

Marco Valtorta

AG11 Abstracts 63

Department of Computer Science and EngineeringUniversity of South Carolinamgv@cec.sc.edu

MS33

Identifiability of Phylogenetic Tree Models for Bi-nary Data

Our focus in this talk is on undirected discrete graphicaltree models when all the variables in the system are binary,where leaves represent the observable variables and whereall the inner nodes are unobserved. A novel approach,based on the theory of partially ordered sets, allows us toobtain a convenient parametrization of this model class. Asimple product-like form of the resulting parameterizationgives insight into identifiability issues associated with thismodel class. In particular, we provide necessary and suffi-cient conditions for such a model to be identified up to theswitching of labels of the inner nodes. When these con-ditions hold we give explicit formulas for the parametersof the model. Whenever the model fails to be identifiedwe use the new parameterization to describe the geometryof the unidentified parameter space. We illustrate theseresults using a simple example.

Piotr ZwiernikMittag-Leffler InstituteStockholm, Swedenpiotr.zwiernik@gmail.com

MS34

Evasion Paths in Mobile Sensor Networks

In “Coordinate-free Coverage in Sensor Networks withControlled Boundaries via Homology,” Vin de Silva andRobert Ghrist use the local connectivity data of a mobilesensor network to determine, in some cases, whether anevasion path exists in the network. We consider exam-ples that show the existence of an evasion path dependsnot only on the network’s connectivity data but also on itsembedding, and we search for invariants of the embeddingthat provide sharper criteria for the existence of an evasionpath.

Henry AdamsStanford Universityhenrya@math.stanford.edu

MS34

Mapping Spaces in Computational AlgebraicTopology

A basic aspect of the modern perspective on homotopytheory is that the set of maps between spaces can itselfbe regarded as a topological space, where paths in thismapping space represent homotopies. All of the homotopytheory of spaces can be encoded in the data of the mappingspace. This talk will present recent work on combinatorialsimplicial mapping spaces in computational topology, em-phasizing connections between simplicial subdivision andsampling.

Andrew BlumbergUniversity of Texasblumberg@math.utexas.edu

Michael MandellUniversity of Indiana

mmandell@indiana.edu

MS34

Homotopy Theory of Dynamical Systems

A dynamical system is a space X with a pairing from X x Sto X for some parameter space S, and a map of such dynam-ical systems is an S-equivariant map. There is an injective

and a projective Quillen model structure for the result-ing category of spaces with S-action, and both are easilyderived. Simultaneously varying the parameter space Salong with the space X up to weak equivalence is more in-teresting, and requires a new model structure having weakequivalences defined by homotopy colimits, as well as a gen-eralization of Thomason’s model structure for small cate-gories. These techniques give methods of detecting when

one dynamical system is close to another, either spatiallyor combinatorially.

Rick JardineUniversity of Western Ontariojardine@uwo.ca

MS34

Detecting Persistent Knotting

Given a filtration of spaces X0 ⊂ X1 ⊂ · · · ⊂ Xn ⊂ R3,we discuss what it means for any knotting of the spaceXi be persist in the larger space Xi+p. This generalizestechniques from classical knot theory. In particular, thisknotting can be characterized by the persistent homotopygroups of its complement. We will discuss techniques todetermine if knotting of a complex persists in a larger spaceor if the knotting is transient.

David LetscherSaint Louis Universityletscher@slu.edu

MS35

Rank 4 Premodular Categories

By relaxing the non-degeneracy condition of modular cat-egories one obtains the more general class of premodularcategories. These are of interest for several reasons: firstly,as spherical categories they lead to modular categories viathe Drinfeld center construction. Secondly, they give riseto link invariants. Finally, recent constructions suggestthat to get non-trivial 4-D TQFTs the modularity condi-tion should be removed. We will discuss recent work onthe classification of rank 4 premodular categories.

Paul BruillardMath. Dept.Texas A&M Universitypjb2357@gmail.com

MS35

Principle of Maximum Entropy and Ground Spacesof Local Hamiltonians

Correlations are of key importance in understanding manyfundamental phenomena of many-body quantum physics.Motivated by the concept of irreducible correlations stud-ied by [Linden et al., PRL 89, 277906] and [Zhou, PRL101, 180505], which is in turn based on the principle ofmaximum entropy, we present that, some quantum states

64 AG11 Abstracts

are with certain k-particle correlation if and only if theyspan a ground space of some k-local Hamiltonian. It es-tablishes a better understanding of k-body correlations andbuilds an intimate link of correlations of quantum states toground spaces of local Hamiltonians.

Bei ZengDept. of Math. and Stats.University of Guelphzengb@uoguelph.ca

Zhengfeng JiPerimiter Institutejizhengfeng@gmail.com

MS35

Localizing Topological Quantum Computers

The quantum circuit and topological models for quantumcomputation are computationally equivalent in the sensethat suitably chosen models in each setting can efficientlysimulate a universal model in the other setting. I will dis-cuss several recent attempts to refine this theoretical resultto a more practical, hybrid model. This is based upon jointwork with Zhenghan Wang.

Eric RowellTexas A&M Universityrowell@math.tamu.edu

Zhenghan WangMicrosoft ResearchStation Qzhenghwa@microsoft.com

MS35

Hidden Symmetry Subgroup Problems in Quan-tum Computing

We advocate a new way of addressing hidden structureproblems and of finding efficient quantum algorithms. Wedefine and investigate the Hidden Symmetry SubgroupProblem (HSSP), a generalization of the well studied Hid-den Subgroup Problem (HSP). Given a group acting ona set and an oracle whose level sets define a partition ofthe set, the task is to recover the subgroup of symmetriesof this partition inside the group. The HSSP provides aunifying framework that, beside the HSP, encompasses awide range of oracle problems, including hidden polyno-mial problems. While we show that the HSSP can haveinstances with exponential quantum query complexity, forvarious instances we obtain efficient quantum algorithms.To achieve that we design a general framework for reducingthe HSSP to the HSP which works efficiently in many in-teresting cases related to symmetries of polynomials. TheHSSP therefore connects in a rather surprising way cer-tain hidden polynomial problems with the HSP. Using thisconnection, we obtain the first ever efficient quantum al-gorithm for the hidden polynomial problem, for quadraticpolynomials over fields of constant characteristic.

Pawel WocjanComputer Science Dept.U. Central Floridawocjan@eecs.ucf.edu

MS35

SIC-POVMs in Dimension Three via Singular El-

liptic Curves

It is not known whether each complex dimension d containsa set of equiangular lines with the maximum possible car-dinality of d2 (a SIC-POVM). Zauner conjectured that onealways exists as an orbit of a finite Heisenberg group. In di-mension 3, Hughston observed that the inflection points ofany Hesse cubic give a particular Heisenberg SIC-POVM. Igive a geometric interpretation for the entire infinite familyas the 3-torsion points of singular elliptic curves.

Jon YardLos Alamos National Laboratoryjtyard@lanl.gov

MS36

Convexity and SOS-Convexity

A multivariate polynomial p(x) = p(x1, ..., xn) is sos-convex if its Hessian H(x) can be factored as H(x) =MT (x)M(x) with a possibly nonsquare polynomial matrixM(x). The notion of sos-convexity is a sufficient conditionfor convexity of polynomials that can be checked efficientlywith semidefinite programming. We will present three re-sults on the subject: (i) We will show that two other nat-ural sos relaxations for convexity based on the definitionof convexity and its first order characterization turn out tobe equivalent to sos-convexity. (ii) We will present an ex-plicit example of a convex but not sos-convex polynomial insix variables and of degree four, whose construction comesdirectly from our recent proof of NP-hardness of decidingconvexity of quartic polynomials. (iii) We will show thatfor polynomials in n variables and of degree d, the notionsof convexity and sos-convexity are equivalent if and onlyif n = 1 or d = 2 or (n, d) = (2, 4). Remarkably, theseare exactly the cases where nonnegative polynomials areguaranteed to be sums of squares, as proven by Hilbert.

Amir Ali AhmadiMITa a a@mit.edu

MS36

Positive Semidefinite Gorenstein Ideals

I will explain how positive semidefinite Gorenstein idealsnaturally arise as objects that separate nonnegative poly-nomials from sums of squares. It is possible to deduce allof the cases where nonnegative polynomials are equal tosums of squares as the cases where psd Gorenstein idealsdo not exist. I will present applications to decompositionsof ternary forms as sums of powers of rational functionsand symmetric tensor decompositions

Greg BlekhermanUniversity of California, San Diegogrrigg@gmail.com

MS36

Strong Nonnegativity and Sums of Squares on RealVarieties

Motivated by scheme theory, we introduce strong nonneg-ativity on real varieties, which has the property that a sumof squares is strongly nonnegative. We show that this al-gebraic property is equivalent to nonnegativity for nonsin-gular real varieties. Moreover, for singular varieties, wereprove and generalize obstructions of Gouveia and Netzerto the convergence of the theta body hierarchy of convex

AG11 Abstracts 65

bodies approximating the convex hull of a real variety.

Mohamed OmarUC Davis and Caltechmomar@math.ucdavis.edu

Brian OssermanUC Davisosserman@math.ucdavis.edu

MS36

The Algebraic Boundary of SO(2)-orbitopes

An SO(2)-orbitope is the convex hull of an orbit undersome linear action of SO(2) on a finite dimensional realvector space. Such a set is always a compact, convex, semi-algebraic set. We will characterize the Zariski closure of itsboundary in the 4-dimensional case. One component is thesecant variety to the Zariski closure of the orbit, which isa real algebraic curve, and for the other components, wegive explicit equations. Our answer will have implicationson the semi-algebraic geometry of these orbitopes and onthe question, whether or not they are spectrahedra. Thisis work in progress.

Rainer SinnUniversity of KonstanzRainer.Sinn@uni-konstanz.de

MS36

The Central Curve of a Semidefinite Program

The central curve of a semidefinite program is an algebraiccurve specified by an affine linear space of symmetric ma-trices and a cost vector. Here we examine the algebraicproperties and beautiful geometry of this curve. We willfocus on examples and special cases coming from appli-cations. In the special case of diagonal matrices (linearprogramming) we will see connections to hyperplane ar-rangements and matroid theory.

Cynthia VinzantUniversity of California, Berkeleycvinzant@math.berkeley.edu

MS37

Ideal Forms of Coppersmith’s Theorem andGuruswami-Sudan List Decoding, Part 2

Abstract not available at time of publication.

Henry CohnMicrosoft Research New Englandhenry.cohn@microsoft.com

MS37

Ideal Forms of Coppersmith’s Theorem andGuruswami-Sudan List Decoding

Coppersmith’s algorithm is a celebrated technique for find-ing small solutions to polynomial equations modulo inte-gers. It has many important applications in cryptogra-phy, particularly in the cryptanalysis of RSA. In this talk,we show how the ideas of Coppersmith’s theorem can beextended to a more general framework encompassing theoriginal number-theoretic problem, list decoding of Reed-Solomon and algebraic-geometric codes, and the problemof finding solutions to polynomial equations modulo ideals

in rings of algebraic integers.

Nadia HeningerPrinceton Universitynadiah@cs.princeton.edu

MS37

List Decoding for AG Codes Using Grobner Bases

Finding efficient algorithms for the interpolation step inGuruswami-Sudan list decoding remains a problem of in-terest. We will discuss how the recent algorithms of [Lee-O’Sullivan, List Decoding of Hermitian Codes, JSC 44(2009), 1662-1675] and [Trifonov, Efficient Interpolationin the Guruswami-Sudan Algorithm, preprint 2010], whichfind the desired interpolating polynomial in a certain mod-ule Grobner basis, can be extended to codes from othercurves. We will also compare their methods with an FGLMapproach.

John B. LittleCollege of the Holy Crossjlittle@holycross.edu

MS37

General Bounds for Generalized Toric Codes

Toric codes are algebraic evaluation codes correspondingto linear systems on toric varieties. Their parameters arestrongly related to the geometry of the lattice polytopedefining a liner system. Recently Diego Ruano introduced,so called, generalized toric codes (GTC) when one takesarbitrary subspaces of linear systems on a toric variety. Itis harder to capture their properties as they are defined byarbitrary finite lattice point configurations S which lackthe geometry of lattice polytopes. We introduce a gen-eral lower bound for the minimum distance of a GTC. Thebound involves the minimum distance of GTCs defined byprojections of the configuration S and the fibers of the pro-jection.

Ivan SoprunovCleveland State Universityi.soprunov@csuohio.edu

MS37

Toric codes and Minkowski Length of 3D Polytopes

The Minkowski sum of two polytopes is the set of all pair-wise sums of their points. The central object of my talk isthe Minkowski length L(P ) of a lattice polytope P whichis defined to be the largest number of primitive latticesegments whose Minkowski sum is in P . The Minkowski

length represents the largest possible number of factors ina factorization of polynomials with exponent vectors in Pand comes up in lower bounds for the minimum distanceof toric codes. I will explain some combinatorial resultsabout L(P ) where P is a 3D lattice polytope in connectionwith 3D toric codes.

Jenya SoprunovaKent State Universitysoprunova@math.kent.edu

Olivia BeckwithHarvey Mudd Collegeobeckwith@gmail.com

66 AG11 Abstracts

Matthew GrimmKent State Universitymgrimm2@kent.edu

Bradley WeaverGrove City Collegeweaverbr1@gcc.edu

MS38

A Domain Decomposition Algorithm for Comput-ing Multiple Steady States of Differential Equa-tions

Problems in engineering often lead to systems of partialdeferential equations, for which the only hope of solutionis to compute numerical solutions. This talk will describedomain decomposition method for solving polynomial sys-tems derived from the discretization of differential equa-tions based on homotopy continuation. This method di-vides the domain into subdomains and solves the polyno-mial system arising from each subdomain. Homotopy con-tinuation then uses these solutions to build solutions forthe original domain.

Wenrui HaoDept. of Applied and Comp. Mathematics and StatisticsUniversity of Notre Damewhao@nd.edu

Jonathan HauensteinTexas A&M Universityjhauenst@math.tamu.edu

Bei HuDept. of Applied & Computational Mathematics &StatisticsUniversity of Notre Dameb1hu@nd.edu

Andrew SommeseDepartment of Mathematics, University of Notre DameNotre Dame, IN 46556-4618, USAsommese@nd.edu

MS38

Numerical Algebraic Geometry Applied to VariousTheoretical Physics Problems

Nonlinear equations arise in theoretical physics naturallyand frequently. In general, it is always difficult to solve(i.e., get ALL solutions, either exactly or numerically)them. However, if the non-linearity of the equations ispolynomial-like, then one can use the computational andnumerical algebraic geometry methods to solve the equa-tions exactly or numerically, respectively. In the talk, thiswill be explained in more detail by taking a few examplesfrom theoretical physics, namely, in statistical mechanics,string theory and lattice field theories, polynomial equa-tions arise naturally and it will be shown how the alge-braic geometry methods can be used to get very interestingphysics out.

Dhagash MehtaDepartment of PhysicsSyracuse Universitydbmehta@syr.edu

MS38

Using Homotopy Continuation to Model Rate-dependent Magnetic Phenomena

Rate-dependent magnetic phenomena occur because ther-mal fluctuations can drive a system of magnetic momentsover energy barriers between stable states. A new methodfor modeling these transitions uses the Bertini homotopycontinuation package. Systems of polynomial equations aresolved for the stable states and energy barriers. The so-lutions are incorporated in a master equation for the timeevolution of the probability of each stable state.

Andrew NewellNorth Carolina State Universityandrew newell@ncsu.edu

Matt NiemergColorado State UniversityFort Collins, COmatthew.niemerg@gmail.com

Daniel J. BatesColorado State UniversityDepartment of Mathematicsbates@math.colostate.edu

MS38

Point Clouds of Varities and Persistent Homology

Computing topological invariants of algebraic objects is si-multaneously challenging, from an algebraic standpoint,and of large interest to application fields. We approachthis challenge with a two-pronged method: first, usingtechniques from homotopy continuation and numerical al-gebraic geometry, we generate point clouds – finite samplesets – on the variety, and subsequently, we approach thepoint cloud with techniques from applied algebraic topol-ogy. Numerical algebraic geometry provides techniques tosolve systems of polynomial equations and retrieve solutionpoints that are guaranteed to contain the isolated pointsin the variety defined by the equational system. By inter-secting the variety of interest with generic hyperplanes, wecan produce equation systems with isolated points locatedon all components of the original variety. Furthermore,the techniques of persistent homology were originally de-veloped in order to acquire topological information fromfinite point samples, from point clouds. We apply thesetechniques to gain computational indications of the topo-logical structure of the point cloud generated from homo-topy continuation, and describe some ways to decomposeand re-assemble topological information for better compu-tational performance.

Mikael Vejdemo-JohanssonStanford Universitymik@math.stanford.edu

Jonathan HauensteinTexas A&M Universityjhauenst@math.tamu.edu

David EklundInstitut Mittag-Lefflerdaek@math.kth.se

Martina ScolamieroPolitecnico di Torinomartina.scolamiero@polito.it

AG11 Abstracts 67

Chris PetersonColorado State Universitypeterson@math.colostate.edu

MS39

Approximate Implicitization using ChebyshevPolynomials

Whereas traditional approaches to implicitization of ratio-nal parametric curves have focussed on exact methods, thepast two decades have seen increased interest in the appli-cation of approximate methods for implicitization. In thistalk we will discuss how the properties of the Chebyshevpolynomial basis can be used to improve the speed, sta-bility and approximation quality of existing algorithms forapproximate implicitization. We will also look at how thealgorithm is well suited to parallelization.

Oliver BarrowcloughSINTEFNorwayoliver.barrowclough@sintef.no

MS39

Blending Natural Quadrics and Associated Vol-umes

Natural quadrics are simplest primitive shapes used inCAD: spheres, right circular cones or right circular cylin-ders. We construct exact rational fixed and variable ra-dius blends of minimal possible parametrization degreesbetween two natural quadrics using canal surfaces in allcases where it is possible. Bounds on parameters that en-sure non-singular blends and associated volumes will bepresented.

Heidi E. I. DahlVilnius UniversityHeidi.Dahl@mif.vu.lt

MS39

Why Approximate Algebraic Methods?

The talk will address the potential of the use of approx-imate algebraic methods in CAGD systems, with a focuson intersection and self-intersection algorithms. An intro-duction to the original approach to approximate impliciti-zation will be given.

Tor DokkenSINTEF ICT, Department of Applied MathematicsOslo, Norwaytor.dokken@sintef.no

MS39

Application of Numerical Methods in Implicitiza-tion

Relying on methods for implicit support prediction, we re-duce implicitization of rational curves and (hyper)surfacesto interpolating the implicit coefficients. We apply nu-merical methods, including SVD, in order to compute amatrix kernel. Computational experiments in Maple andMatlab are performed to compare different approaches forconstructing a real or complex matrix, either square orrectangular, and for computing its kernel, with respect torobustness and speed. Our methods exploit sparsity and

may compute approximate implicit equations.

Tatjana Kalinka, Ioannis Z. EmirisNational and Kapodistrian University of Athenstatjana.kalinka@gmail.com, emiris@di.uoa.gr

Christos KonaxisNational Kapodistrian University of Athensckonaxis@di.uoa.gr

MS39

Approximate Implicitization of Envelope Surfaces

Given a rational family of parametric surfaces in a cer-tain region of interest, we are interested in computing animplicit representation of the envelope. Although severalmethods for exact implicitization are known, they tend tobe computationally expensive and may even introduce un-wanted branches and singularities. We adapt the conceptof approximate implicitization to envelopes and formulatean algorithm for computing a piecewise algebraic approxi-mation of low degree.

Tino SchulzJohannes Kepler UniversityLinz, Austriatino.schulz@jku.at

Bert JuttlerJohannes Kepler Univ. LinzDepartment of Applied Geometrybert.juettler@jku.at

MS40

Dynamics of Weakly Reversible Reaction Networksand the Global Attractor Conjecture

I will discuss some recent work pertaining to weakly re-versible chemical reaction networks and, in particular, theGlobal Attractor Conjecture. I will present some details ofthe proof of the conjecture in the setting where the under-lying reaction diagram consists of a single linkage class, orconnected component.

David AndersonDepartment of MathematicsUniversity of Wisconsin Madisonanderson@math.wisc.edu

MS40

QSSA and Algebraic Elimination in Chemical Ki-netics

Chemical mechanisms for reaction network dynamics in-volve many highly reactive and short-lived species (inter-mediates), present in small concentration, in addition tothe main reactants and products, present in larger concen-tration. A classic model reduction method known as thequasi-steady-state approximation (QSSA) is often used toeliminate the highly reactive intermediate species and re-move the large rate constants that cannot be determinedfrom concentration measurements of the reactants andproducts. We show that the program usually taught tostudents for applying the 100 year-old approach of clas-sic QSSA model reduction cannot be carried out in many,perhaps most, relevant kinetics problems. In particular, byusing Galois theory, we prove that the required algebraicequations cannot be solved for as few as five bimolecular

68 AG11 Abstracts

reactions between five species. We also describe algorithmsthat can test any mechanism for solvability, and proposea new strategy for dealing with unsolvable systems, basedon rescaling the reactive intermediate species.

Gheorghe CraciunDepartment of Mathematics, University ofWisconsin-Madisoncraciun@math.wisc.edu

Casian Pantea, James RawlingsUniversity of Wisconsin, Madisonpantea@math.wisc.edu, rawlings@engr.wisc.edu

MS40

Catalysis in Reaction Networks

We define catalytic networks as chemical reaction networkswith an essentially catalytic reaction pathway: one whichis “on’ in the presence of certain catalysts and “off’ in theirabsence. We show that examples of catalytic networksinclude synthetic DNA molecular circuits that have beenshown to perform signal amplification and molecular logic.Recall that a critical siphon is a subset of the species in achemical reaction network whose absence is forward invari-ant and stoichiometrically compatible with a positive point.Our main theorem is that all weakly-reversible networkswith critical siphons are catalytic. Consequently, we obtainnew proofs for the persistence of atomic event-systems ofAdleman et al., and normal networks of Gnacadja. We de-fine autocatalytic networks, and conjecture that a weakly-reversible reaction network has critical siphons if and onlyif it is autocatalytic.

Manoj GopalkrishnanTata Institute of Fundamental ResearchMumbai, Indiamanoj@tcs.tifr.res.in

MS40

Linear Conjugacy of Chemical Reaction Networks

Under suitable assumptions, the dynamic behaviour of achemical reaction network is governed by an autonomousset of polynomial ordinary differential equations over con-tinuous variables representing the concentrations of the re-actant species. It is known that two networks may possessthe same governing mass-action dynamics despite disparatenetwork structure. To date, however, there has only beenlimited work exploiting this phenomenon even for the caseswhere one network possesses known dynamics while theother does not. In this presentation, I will present resultswhich bring these known results together into a broaderunified theory which we have called conjugate chemical re-action network theory. In particular, I will present a theo-rem which gives conditions under which two networks withdifferent governing mass-action dynamics may exhibit thesame qualitative dynamics and use it to extend the scopeof the well-known theory of weakly reversible systems.

Matthew D. Johnston, David SiegelUniversity of Waterloomdjohnst@math.uwaterloo.ca, dsiegel@math.uwaterloo.ca

MS40

Persistence of Chemical Kinetics Systems

It is known that a solution to a chemical kinetics system

can only approach the boundary of the non-negative or-thant at points belonging to a semi-locking set. We general-ize results of Angeli, De Leenheer and Sontag (2007) whichassume that every semi-locking set satisfies a conservationcondition or is dynamically non-emptiable by introducing anotion of a weakly dynamically non-emtiable semi-lockingset. Systems whose semi-locking sets are weakly dynami-cally non-emptiable are persistent. The facet result of An-derson and Shiu (2010) also fits into this framework.

David Siegel, Matthew JohnstonUniversity of Waterloodsiegel@math.uwaterloo.ca,mdjohnston@math.uwaterloo.ca

MS41

Pentads and Minors Define the Gaussian Two-factor Model

In 1928, educational psychologist Truman Lee Kelley pub-lished Crossroads in the Mind of Man, which contained anew pentad test for Gaussian factor analysis with two fac-tors. Pentads are degree-five polynomials that vanishes onall symmetric n × n-matrices that can be expressed as arank-two matrix plus a diagonal matrix. In their recent al-gebraic approach to factor analysis, Drton, Sturmfels, andSullivant raise the problem of determining all such polyno-mials. I will report on a finite computation that determinesa generating set of polynomials for all n. This is joint workwith Andries Brouwer.

Jan DraismaT.U.Eindhoven, The NetherlandsDepartment of Mathematicsj.draisma@tue.nl

Andries BrouwerT.U. Eindhoven, The NetherlandsDepartment of Mathematics and Computer Scienceaeb@cwi.nl

MS41

Geometry of Deep Belief Networks

We report on recent progress on the algebraic statisticsof deep belief networks. These are graphical models builtfrom layered restricted Boltzmann machines, which in turncorrespond to Hadamard products of secant varieties ofSegre varieties.

Jason MortonPennsylvania State Universitymorton@math.psu.edu

MS41

Parametrization of Causal Models Associated withAcyclic Directed Mixed Graphs

Acyclic directed mixed graphs may be used to repre-sent causal directed acyclic graph (DAG) models in whichsome variables are unmeasured (latent). In this talk wepresent a parametrization of the observed distribution inthe multivariate binary case. The resulting model satisfiesMarkov constraints, and so-called Verma constraints. Theparametrization leads to efficient algorithms for computingintervention distributions, as well as allowing scoring-basedmodel search. [Joint work with Robin Evans (Universityof Washington), James Robins and Ilya Shpitser (Harvard

AG11 Abstracts 69

School of Public Health).]

Thomas RichardsonStatisticsUniversity of Washingtonthomasr@u.washington.edu

MS41

Treks and Determinants of Path Matrices in Gaus-sian Graphical Models

This talk will explore a connection between combinatoricsand algebraic statistics. In particular, we will look at Gaus-sian graphical models, whose covariance matrices can begiven in terms of certain path families called treks. Inspiredby classical results in algebraic combinatorics, we developa graph-theoretic criterion for determining the rank of asubmatrix of the covariance matrix.

Kelli TalaskaDepartment of MathematicsUniversity of California, Berkeleytalaska@math.berkeley.edu

Seth SullivantNorth Carolina State Universitysmsulli2@ncsu.edu

Jan DraismaT.U.Eindhoven, The NetherlandsDepartment of Mathematicsj.draisma@tue.nl

MS41

Geometry of Maximum Likelihood Estimation inGaussian Graphical Models

We study maximum likelihood estimation in Gaussiangraphical models from a geometric point of view. In currentapplications of statistics, we are often faced with problemsinvolving a large number of random variables, but only asmall number of observations. An algebraic eliminationcriterion allows us to find exact lower bounds on the num-ber of observations needed to ensure that the maximumlikelihood estimator exists with probability one. This isapplied to bipartite graphs and grids, leading to the firstinstance of a graph for which the maximum likelihood es-timator exists with probability one even when the numberof observations equals the treewidth of the graph.

Caroline UhlerDepartment of StatisticsUC Berkeleycuhler@stat.berkeley.edu

MS42

Basic Notions and Current Trends in NumericalAlgebraic Geometry

This talk will overview some of the recent advances withinthe field of numerical algebraic geometry without dis-cussing applications (covered in an earlier minisympo-sium). The goal is to describe some of the latest trendsand current open problems in advancing this field. Basicnotions of numerical algebraic geometry (homotopy contin-uation, numerical irreducible decomposition, etc.) will beintroduced very briefly as this is an introductory talk forthis minisymposium. Coming after the 12-15 talks in the

Applications of Numerical Algebraic Geometry minisym-posium, though, I will keep the introductory parts short!

Daniel J. BatesColorado State UniversityDepartment of Mathematicsbates@math.colostate.edu

MS42

Numerical Algebraic Geometry via NumericalPolynomial Algebra

Techniques of numerical polynomial algebra are applied tonumerical algebraic geometry. These techniques includeSylvester and Macaulay matrices, and local and global du-ality. Macaulay’s H-bases are connect the homogeneousand affine cases. The computations are done by use ofSVD based numerical linear algebra using Mathematica.The main goal of these methods is to provide explicit poly-nomials defining the ideals defining algebraic sets which arecomponents of, or unions of, other algebraic sets.

Barry H. DaytonDepartment of Mathematicsb-dayton@neiu.edu

MS42

Certification of Approximate Multiple Roots

Approximation and certification of multiple roots is a realchallenge in numerical computation. In this talk, we willdescribe a symbolic-numeric algorithm for the certificationof singular isolated points. The approach is based on thecomputation of the inverse systems of the isolated singularpoint. We derive a one-step deflation technique, from thedescription of the multiplicity structure in terms of differ-entials. The deflated system can be used in Newton-basediterative schemes with quadratic convergence. Startingfrom a polynomial system and a sufficiently small neighbor-hood, we obtain a criterion for the existence and unique-ness of a singular root of a given multiplicity structure,applying a well-chosen symbolic perturbation. Standardverification methods, based e.g. on interval arithmetic anda fixed point theorem, can be employed to certify that thereexists a unique perturbed system with a singular root inthe domain.

Bernard MourrainINRIA Sophia Antipolismourrain@sophia.inria.fr

Angelos MantzaflarisGALAAD, INRIA Sophia Antipolisangelos.mantzaflaris@inria.fr

MS42

Isosingular Sets and Deflation

This talk will discuss the new concept of isosingular sets,which are pure-dimensional irreducible algebraic subsetsof the set of solutions to a system of polynomial equationssuch that generic points of an isosingular set share a com-mon singularity structure. The definition of these sets de-pends on deflation, a procedure that uses differentiation toregularize solutions. A weak form of deflation has provenuseful in regularizing algebraic sets, making them amenableto treatment by the algorithms of numerical algebraic ge-

70 AG11 Abstracts

ometry. We introduce a strong form of deflation and useit to define deflation sequences, which are similar to thesequences arising in Thom-Boardman singularity theory.Isosingular sets consist of points that have the same defla-tion sequence. From this, one may define the isosingularset of a given point and also the isosingular local dimen-sion. While isosingular sets are of theoretical interest asconstructs for describing singularity structures of algebraicsets, they also expand the kinds of algebraic sets that canbe investigated with methods from numerical algebraic ge-ometry.

Charles WamplerGeneral Motors Research Laboratories, EnterpriseSystems Lab30500 Mound Road, Warren, MI 48090-9055, USAcharles.w.wampler@gm.com

Jonathan HauensteinTexas A&M Universityjhauenst@math.tamu.edu

MS43

|ZKup| = |ZHenn|2 for Lens Spaces

M. A. Hennings and G. Kuperberg defined quantum in-variants ZHenn and ZKup of 3-manifolds based on Hopfalgebras, respectively. We prove that ZKup(L(p, q),H) =

ZHenn(L(p, q)#L(p, q),H) for lens spaces when both in-variants are based on a factorizable finite dimensional rib-bon Hopf algebra H.

Liang ChangMath. Dept.U.C. Santa Barbaraliangchang@math.ucsb.edu

Zhenghan WangMicrosoft ResearchStation Qzhenghwa@microsoft.com

MS43

Quantum Stabilizer Codes from Toric Varieties

The technology to produce classical codes from higher di-mensional algebraic varieties has been known for approxi-mately 10 years. The combinatorial structure of toric va-rieties made them particularly suitable for applying thesetechniques. Nevertheless, the only algebraic varieties fromwhich one could construct quantum codes seemed to becurves. In this talk, we outline how to construct quantumstabilizer codes from higher dimensional toric varieties. Ex-amples of such codes constructed from toric surfaces willbe discussed briefly.

Roy JoshuaMath. Dept.Ohio State U.rrjoshua@gmail.com

MS43

Quantum Error-Correcting Codes over Rings

The discretization of errors in quantum error-correctingcodes typically relies on a generalization of the Pauli matri-ces. As a consequence, most quantum codes have been con-structed with the help of classical codes over finite fields.

We discuss methods of discretization that use matrices thatare parametrized by finite Frobenius rings, and sketch howquantum codes can be constructed in this case.

Andreas KlappeneckerComputer Science Dept.Texas A&M Universityklappi@cse.tamu.edu

MS43

Classifying Small Index Subfactors

I’ll describe recent progress on the classification of subfac-tors with index at most 5, and corollaries for fusion cate-gories.

Scott MorrisonMath. Dept.U.C. Berkeleyscott@tqft.net

MS43

Recent Progress in Exactly Soluble Discrete Mod-els for Topological Phases in Two Dimensions

The study of two-dimensional topological phases in con-densed matter systems is a frontier in the field of condensedmatte theory as well as topological quantum computation.Discrete or lattice models, which are exactly soluble havebeen proposed by Kitaev and by Levin and Wen, respec-tively. Here we present a summary of recent progress, madeby us and other groups, in studying these models and theirgeneralizations. The topics to be reviewed include 1) Dual-ity between the Kitaev and Levin-Wen modeles in certainspecial cases; 2) General procedure for computing groundstate degeneracy when the models are put on a topolog-ically non-trivial surface; 3) More detailed study of thesectors of topolgical excitations; 4) General framework fortopology-preseving renormalization group flow that char-acterizes these models as fixed points; 5) Generalization ofthese models to more general graphs or to incorporatingmore general degrees of freedom. Our approach, thoughclosely related to topological field theory and tensor cate-gory theory, could be understood by physicists.

Yong-Shi WuPhysics Dept.University of Utahwu@physics.utah.edu

Spencer StirlingDept. of Math.University of Utahstirling@physics.utah.edu

Yuting HuDept. of Physics and AstronomyUniversity of Utahyuting@physics.utah.edu

MS44

Minimum Fuel Multi-impulse Orbit Transfer

Most of the equations in orbital mechanics are expressedin terms of trigonometric functions. However, except forthe case of Kepler’s equation, all the remaining instancescorrespond to simple planar or spatial rotations. By rep-resenting these rotations by points in the unit circle (for

AG11 Abstracts 71

2D rotations) or unit quaternions (for 3D rotations), allthe equations become polynomials, hence tractable withalgebraic geometry techniques. In this talk, we will focusmainly in the problem of orbit transfer using a finite se-quence of thrusts, and minimum fuel.

Martin AvendanoTexas A&M Universityavendano@math.tamu.edu

Maurice RojasDepartment of Mathematics, Texas A&M Universityrojas@math.tamu.edu

Daniele MortariTexas A&M Universitymortari@tamu.edu

MS44

Reinterpreting Regularization of Collisionsthrough Real Algebraic Geometry

Regularization of binary and/or simultaneous binary col-lisions in the collinear N-body problem is a common toolused to analyze the geometry of orbits near these kindsof singularities. The Levi-Civita type coordinate and timetransformations typically used in regularization are rein-terpreted in terms of real algebraic geometry. This reinter-pretation applies to Hamiltonians whose potential part isa finite sum of reciprocals of homogeneous polynomials inthe position coordinates.

Lennard F. BakkerBrigham Young Universitybakker@math.byu.edu

MS44

Resonances in Celestial Mechanics

A dynamical system is in resonance p : q if there are twofrequencies ω1 and ω2 such that ω1/ω2 = p/q, with p andq two coprime numbers. Thus, resonances are related withperiodic motions, and periodic motions are inherent to Ce-lestial Mechanics problems. Nature presents many exam-ples of resonances, for instance the orbital-rotational mo-tion of the Moon, but there are many other cases of reso-nances, which appear in the equations of motion and thatcreates many difficulties in the integration, mainly by an-alytic methods. For these cases, we present some waysin tackling this problem by means of appropriate sets ofvariables. Some examples are presented as well as someapplications to the stability of the solutions.

Antonio ElipeUniversity of Zaragozaelipe@unizar.es

Victor LancharesUniversidad de la RiojaSpainvlancha@unirioja.es

MS44

The Algebra and Geometry of Periodic SatellitesGround Tracks

We survey the algebra and geometry of periodic satellitesground tracks for remote sensing observations from space of

the Earth and other planets. The repetition of the groundtracks allows for multiple scientific observations to be takenof the same column of space all the way to a little belowthe planet surface at various times of day. The symmetryand geometry of such orbits are governed by finite groupswhich classify them.

Martin LoJPL-CalTechmartin.w.lo@jpl.nasa.gov

MS44

Evolution of Flower Constellations Theory

The theory of Flower Constellations consists of a generalmethodology to design constellations of Ns satellites. Theoriginal theory evolved over time and a rich literature wasproduced with subsequent insights and mathematical re-formulations. This talk describes these mathematical pro-gresses making now the theory connected with the 2D and3D Lattice theory and the Necklaces theory.

Daniele MortariTexas A&M Universitymortari@tamu.edu

MS45

On the Decodability of Primitive Reed-SolomonCodes

Reed-solomon codes are (list) decodable up to the Johnson-Guruswami-Sudan bound. No polynomial time decodingalgorithm is known when number of errors is larger thanthe JGS bound. It appears hard to establish complexityresults for the primitive Reed-Solomon codes. In this talk,I will present several results on this problem. I will alsodiscuss whether the results can be generalized to algebraicgeometric codes.

Qi ChengUniversity of Oklahamaqcheng@ou.edu

MS45

Interpolation Decoding of a Class of Quasi-cyclicCodes

For a quasi-cyclic code of index �, we transform the de-coding problem into an interpolation problem for a Reed-Solomon code. For each location, � values are assigned(counting multiplicities) and the usual interpolation decod-ing algorithms may be applied. We compute upper boundson the decoding capability of the algorithm, showing thatit can correct well beyond the minimum distance boundwith high probability. Experimental results corroborateour computations.

Fernando Hernando, Michael E. O’SullivanSan Diego State Universityfhernando@mail.sdsu.edu, mosulliv@math.sdsu.edu

Diego RuanoAalborg Universitydiego@math.aau.dk

MS45

Small Bias Sets from General Algebraic Geometry

72 AG11 Abstracts

Codes

Small bias sets are important in a number of applica-tions, from derandomization of algorithms to hash func-tions. There is a connection between small bias sets andlinear codes. In this talk we consider the construction ofsmall bias sets from general algebraic geometry codes.

Gretchen L. MatthewsClemson Universitygmatthe@clemson.edu

MS45

Goppa-like Codes Beating the Best Known Codes

The paper presents a construction of subfield subcodesfrom generalized Reed-Solomon codes that is similar toclassical Goppa codes, but simpler. Following an idea ofRoseiro et al, we use Delsarte’s theorem to create subfieldsubcodes of larger than expected dimension by ensuringtheir duals, which are trace codes, have small dimension.This method produced several codes over F5, F3 and F2

that beat the best known codes, and numerous others thatmatch the best know parameters.

Michael E. O’Sullivan, Fernando Hernando, KyleMarshallSan Diego State Universitymosulliv@math.sdsu.edu, fhernando@mail.sdsu.edu, mar-shall.kyle.d@gmail.com

MS45

Extended Norm-Trace Codes

Consider the function field Fqr (x, y)/Fqr defined by

xu = L(y),

where u| qr−1q−1

and L(y) =d∑

i=0

aiyqi is a linearized poly-

nomial with a0, ad �= 0 and qd distinct roots in qr . Notethat the Hermitian and norm-trace function fields are spe-cial cases of this function field. Recent work has yieldedexplicit bases for certain Riemann-Roch spaces of this func-tion field. In this talk, we apply this progress to construc-tion of codes arising from these spaces.

Justin PeacheyClemson Universityjpeache@clemson.edu

MS46

On the Algebraic Representation of Bisectors forLow Degree Surfaces

Bisectors are geometric constructions appearing very oftenin CAD. For two given low degree parametric surfaces, itwill be shown a new approach to determine an algebraicrepresentation of their bisector by using the so-called gener-alized Cramer rules. The new introduced approach allowsto obtain easily a parametrization of the quadric-plane orquadric-cylinder bisectors, which is rational in most cases(in the remaining cases the parametrization involves onesquare root which is well-suited for approximation pur-poses).

Ibrahim AdamouUniversity of CantabriaSpain

ibrahim.adamou@unican.es

Laureano Gonzalez-VegaUniversidad de CantabriaDpto. Matematicaslaureano.gonzalez@unican.es

Mario Alfredo Fioravanti VillaUniversity of Cantabriamario.fioravanti@unican.es

MS46

Toric Degenerations of Toric Varieties and Appli-cations to Modeling

Toric varieties have interesting combinatorial propertiesthat make them attractive for applications to geometricmodeling. They were employed by R. Krasauskas to definetoric patches that are a generalization of Bezier patches.Toric degenerations were used by L. D. Garcıa-Puente, F.Sottile and C. Zhu to explain some ways that control pointsgovern the shape of toric patches when the weights vary. Inthis talk we will present our efforts to extend these proper-ties to irrational toric patches. This is a joint project withF. Sottile and N. Villamizar.

Elisa PostinghelUniversity of OsloNorwayelisa.postinghel@gmail.com

Frank SottileTexas A&M Universitysottile@math.tamu.edu

Nelly VillamizarCMA University of OsloNorwayn.y.v.v@cma.uio.no

MS46

On the Dimension of Triangular Splines

We consider the space of spline functions defined on a trian-gular subdivision of a polygonal domain. Using homologi-cal techniques we will present a lower and an upper boundto the dimension of this spline space which are, in many orperhaps most of the cases, more general and give better ap-proximations to the exact value of the dimension than thealready existing ones. These results can also be extendedto spline spaces on 3-dimensional (simplicial) complexes.

Nelly VillamizarCMA University of OsloNorwayn.y.v.v@cma.uio.no

Bernard MourrainINRIA Sophia AntipolisBernard.Mourrain@inria.fr

MS47

Determining Multiple Steady States in Mass Ac-tion Networks by Linear Inequality Systems

Ordinary Differential Equations (ODEs) are an importanttool in many areas of Quantitative Biology. For many ODE

AG11 Abstracts 73

systems multistationarity (the existence of ≥ 2 positivesteady states) is a desired feature. For mass action net-works establishing multistationarity is equivalent to estab-lishing the existence of at least two positive solutions ofa polynomial system with unknown coefficients. For net-works with certain structural properties, necessary and suf-ficient conditions for multistationarity that take the formlinear inequality systems are presented.

Carsten ConradiMax-Planck-Institut Dynamik komplexer technischerSystemeconradi@mpi-magdeburg.mpg.de

MS47

Tools From Computational Algebraic Geometry forthe Study of Biochemical Reaction Networks

We will illustrate the use of some tools from computationalalgebraic geometry for the study of (bio)chemical reactionnetworks, extracted from our joint work with Carsten Con-radi, Mercedes Perez Millan, and Anne Shiu.

Alicia DickensteinUniversidad de Buenos Airesalidick@dm.uba.ar

MS47

Complexity of Steady States in Molecular Net-works and Numerical Algebraic Geometry

One of the most pressing questions in modern biology ishow molecular interactions at the cellular level lead tophysiology, particularly disease. The most accessible math-ematical approach uses the law of mass-action to describethe underlying molecular network in terms of a polynomialdynamical system. Using tools from projective algebraicgeometry coupled with high performance numerical analy-sis, we investigate the complexity of steady states in net-works of established biological relevance, such as the Wntpathway, and translate the parameter problem of biologyinto a question about complex structure moduli of the un-derlying algebraic variety.

Robert KarpHarvard Universityrobert karp@hms.harvard.edu

MS47

Persistence and the Global Attractor Conjecture:Recent Approaches

We describe recent approaches to proving the PersistenceConjecture (which describes a class of mass-action systemsfor which variables do not approach zero) and the GlobalAttractor Conjecture (which describes a class of mass-action systems for which trajectories converge to a singlepositive equilibrium). We show that an extended versionof the Persistence Conjecture holds for systems with two-dimensional stoichiometric subspace and bounded trajec-tories and we prove the Global Attractor Conjecture forsystems with three-dimensional stoichiometric subspace.

Casian PanteaUniversity of Wisconsin, Madisonpantea@math.wisc.edu

Gheorghe CraciunDepartment of Mathematics, University of

Wisconsin-Madisoncraciun@math.wisc.edu

Fedor NazarovUniversity of Wisconsin-Madisonnazarov@math.wisc.edu

MS47

Sequential and Distributive Multisite Phosphory-lations Have Toric Steady States

We show that the steady states of the sequential anddistributive multisite phosphorylations system, are de-scribed by binomial equations, and can thus be explic-itly parametrized by monomials. This result is implicitin [Wang and Sontag, 2008] and it is a particular case of[Thomson and Gunawardena, 2009]. We moreover give suf-ficient conditions for any chemical reaction system to havetoric steady states. This is joint work with Alicia Dicken-stein, Anne Shiu and Carsten Conradi.

Mercedes Perez MillanDto. de Matematica - FCEN - Universidad de BuenosAiresmpmillan@dm.uba.ar

Alicia DickensteinUniversidad de Buenos Airesalidick@dm.uba.ar

Anne ShiuDuke Universityannejls@math.duke.edu

Carsten ConradiMax-Planck-Institut Dynamik komplexer technischerSystemeconradi@mpi-magdeburg.mpg.de

MS48

Finite Groebner Bases in Infinite DimensionalPolynomial Rings and Applications

We discuss the theory of monoidal Groebner bases, a con-cept which generalizes the familiar notion in a polynomialring and allows for a description of Groebner bases of ide-als that are stable under the action of a monoid. The mainmotivation for developing this theory is to prove finitenesstheorems in commutative algebra and its applications. Amajor result of this type is that ideals in infinitely manyindeterminates stable under the action of the symmetricgroup are finitely generated up to symmetry. We dis-cuss how this machinery gives new proofs of some classicalfiniteness theorems in algebraic statistics as well as a proofof the independent set conjecture of Hosten and the secondauthor.

Christopher HillarMathematical Sciences Research Institutechillar@msri.org

MS48

Finiteness Theorems for Chains of Lattice Ideals

We study chains of lattice ideals that are invariant under asymmetric group action. In our setting, the ambient ringsfor these ideals are polynomial rings which are increasing in(Krull) dimension. Thus, these chains will fail to stabilize

74 AG11 Abstracts

in the traditional commutative algebra sense. However,we prove a general theorem which says that ”up to theaction of the group”, these chains stabilize up to monomiallocalization. This gives a partial resolution to a conjectureof Aschenbrenner and Hillar.

Christopher HillarMathematical Sciences Research Institutechillar@msri.org

Abraham Martin del CampoTexas A&M Universityasanchez@math.tamu.edu

MS48

Defining Equations of Secant Varieties to Segre-Veronese Varieties

We describe the defining ideal of the rth secant variety ofP2 ×Pn embedded by O(∞,∈), for arbitrary n and r ≤ 5.We also present the Schur module decomposition of thespace of generators of each such ideal. Our main resultsare based on a more general construction for producingexplicit matrix equations that vanish on secant varieties ofproducts of projective spaces. This extends previous workof Strassen and Ottaviani.

Dustin CartwrightYale Universitydustin@math.berkeley.edu

Daniel ErmanStanford Universityderman@math.stanford.edu

Luke OedingUniversity of California BerkeleyUniversita degli Studi di Firenzelukeoeding@gmail.com

MS48

Algebraic Identification of Binary-Valued HiddenMarkov Processes

The complete identification problem is to decide whether astochastic process (Xt) is a hidden Markov process and ifyes to infer a corresponding parametrization. So far onlypartial answers to either the decision or the inference parthave been given all of which depend on further assumptionson the processes. Here we present a full, general solutionfor binary-valued hidden Markov processes. Our approachis rooted in algebraic statistics hence geometric in nature.We demonstrate that the algebraic varieties which describethe probability distributions associated with binary-valuedhidden Markov processes are zero sets of determinantalequations which draws a connection to well-studied objectsfrom algebra. As a consequence, our solution provides im-mediate algorithmic access where tests come in form ofelementary (linear) algebraic routines.

Alexander SchoenhuthIPAM andCWI Amsterdama.schoenhuth@cwi.nl

MS48

Finiteness on Homogeneous Markov Chain Models

In 2010, Takemura and Hara showed a complete descriptionof a Markov basis for conditional tests of the toric homoge-neous Markov chain model, the envelope exponential fam-ily for the homogeneous discrete time Markov chain model,with the following cases; (1) two-state, arbitrary length;(2) arbitrary finite state space and length of three; and(3) two-state, arbitrary length without initial parameters.Motivated by these results, we study finiteness of Markovbases for the toric homogeneous Markov chain model. Inthis talk we consider the toric homogeneous Markov chainmodel without initial parameters and without any loopsfor the state space S with |S| = 3, 4. A key tool is a stategraph, that is, the directed multigraph G(x) such that ver-tices are given by the states in S and the directed edges aregiven by the transitions from state i to j in x, a summaryof moves in Markov chains.

Ruriko YoshidaUniversity of Kentuckyruriko.yoshida@uky.edu

Abraham Martin del CampoTexas A&M Universityasanchez@math.tamu.edu

David HawsUniversity of Kentuckydchaws@gmail.com

MS49

Mahonian Partition Identities Via Polyhedral Ge-ometry

In a series of papers, George Andrews and various coau-thors successfully revitalized seemingly forgotten, powerfulmachinery based on MacMahon’s Omega operator to sys-tematically compute generating functions of integer parti-tions. Our goal is to geometrically prove and extend manyof the Andrews et al theorems, by realizing a given familyof partitions as the set of integer lattice points in a certainpolyhedron. This is joint work with Matthias Beck andNguyen Le.

Benjamin BraunUniversity of Kentuckybenjamin.braun@uky.edu

MS49

Lattice Points in Polyhedra and the SummationMethod for Mixed-Integer Optimization

We present a new fully polynomial-time approximationscheme for the problem of optimizing non-convex polyno-mial functions over the mixed-integer points of a polytopeof fixed dimension [J.A. De Loera, R. Hemmecke, M. Kppe,R. Weismantel, FPTAS for optimizing polynomials overthe mixed-integer points of polytopes in fixed dimension,Math. Prog. Ser. A 118 (2008), 273–290]. The algo-rithm also extends to a class of problems in varying di-mension. This is the culmination of an effort to extend theefficient theory of discrete generating functions of latticepoints in polyhedra to the mixed-integer case in a practi-cal way, without using discretization, and to bring it to usein optimization.

Matthias Koeppe

AG11 Abstracts 75

Dept. of MathematicsUniv. of California, Davismkoeppe@math.ucdavis.edu

Nicole BerlineCentre de Mathematiques Laurent Schwartz

Ecole Polytechniquenicole.berline@math.polytechnique.fr

Michele VergneInstitut Mathematique de JussieuUniversite Paris 7 Diderotvergne@math.jussieu.fr

MS49

Vector Partition Functions and Applications to LieRepresentation Theory

Given a finite set I of non-zero integral vectors with non-negative coordinates, and a vector γ with coordinates(γ1, . . . , γn), the vector partition function PI(γ) is the num-ber of ways we can split γ as an integral sum with non-negative coefficients of the vectors in I . We formulate analgorithm for computing the vector partition function asquasipolynomials in (γ1, . . . , γn) over a finite set of combi-natorial chambers and demonstrate an on-line implemen-tation. We explain some problems from Lie representationtheory motivating this study.

Todor MilevEduard Czech Centertodor.milev@googlemail.com

MS49

Ehrhart Theory and Lecture Hall Polytopes

For a sequence s = (s1, . . . , sn) of positive integers, definethe s-lecture hall polytope to be the set {x ∈ Rn|0 ≤ x1

s1≤

x2s2

≤ . . . ≤ xnsn

}. We prove a theorem relating the Ehrhart

series of the s-lecture hall polytope to ascent statistics ons-inversion sequences. We show how the theorem can berefined and specialized to yield known results and derivenew ones. This includes joint work with Michael Schuster,Gopal Viswanathan, and Thomas Pensyl.

Carla D. SavageNorth Carolina State UnivDepartment of Computer Sciencesavage@csc.ncsu.edu

MS49

An Efficient Calculation of the Top Ehrhart Coef-ficients for the Knapsack.

Let A be a sequence of N + 1 positive integers. For tan integer, let E(A)(t) be the number of solutions in non

negative integers xi of the equation∑N+1

i=1Aixi = t. Then

E(A)(t) =∑N

i=0Ei(t)t

i where Ei(t) is a periodic functionof t. Fixing k, we show that the residue theorem in onevariable, and the signed unimodular decomposition of acone, leads to an algorithm of polynomial complexity tocompute the top k-coefficients of E(A)(t).

Michele VergneInst. Math. Jussieuvergne@math.jussieu.fr

Velleda BaldoniU. Roma Tor Vergatabaldoni@mat.uniroma2.it

Nicole BerlineEcole Polytechniqueberline@math.polytechnique.fr

Jesus De LoeraUniversity of California, Davisdeloera@math.ucdavis.edu

Matthias KoeppeDept. of MathematicsUniv. of California, Davismkoeppe@math.ucdavis.edu

MS50

Expected Complexity of Bisection Methods forReal Solving

We examine degree-d polynomials with iid coefficients un-der two zero-mean normal distributions, such that the i-thcoefficient has variance

(di

), or 1/i!. The expected bit com-

plexity of the Sturm solver is O∗(rd2τ ), where r bounds thenumber of real roots and τ the coefficient bitsize. For Bern-stein polynomials with iid coefficients of moderate standarddeviation, we show E[r] =

√2d ±O(1).

Ioannis Z. EmirisNational and Kapodistrian University of Athensemiris@di.uoa.gr

Andre’ GalligoUniversity of NiceFrancegalligo@math.unice.fr

Elias TsigaridasUniversity of Aarhuselias@cs.au.dk

MS50

Hybrid Method for Solving Bivariate PolynomialSystem

Many problems in science and engineering can be reducedto that of solving bivariate polynomial systems of equa-tions. Intensive research efforts have yielded reliable sym-bolic methods and efficient numeric methods. We com-bine the advantages of symbolic methods (in particularresultant) and numeric methods (in particular Hansen-Sengupta operator with slope form). The resulting methodcan reliably approximate all the solutions more efficientlythan the other (symbolic, numeric) methods in most cases.

Hoon HongNorth Carolina State Universityhong@math.ncsu.edu

MS50

Accurate Path Tracking

Tracking solution paths defined by a family of polynomialsystems is a basic operation for many algorithms in numer-ical algebraic geometry. Version 2.3.55 of PHCpack incor-porated the QD-2.3.9, a software library of Y. Hida, X.S.

76 AG11 Abstracts

Li, and D.H. Bailey for quad double arithmetic. In thistalk we will explain the adaptive use of double double andquad double arithmetic in the path trackers of PHCpack.

Jan VerscheldeDepartment of Mathematics, Statistics and ComputerScienceUniversity of Illinois at Chicagojan@math.uic.edu

Genady YoffeDept. of Mathematics, Statistics, and CSUniversity of Illinois, Chicagogyoffe2@uic.edu

MS50

An Efficient Exact Numerical Algorithm for Iso-topic Approximation of NonSingular Surfaces

We describe a subdivision algorithm for computing anisotopic ε-approximation of a nonsingular surface f−1(0)where f : R3 → R is a C1 function. Our algorithm (calledCxyz) is based on numerical predicates which are easy toimplement and does not suffer from implementation gaps.Cxyz combines the technique of non-local isotopy fromPlantinga-Vegter (2004) and parametrizability from Sny-der (1996). Our implementation shows that this is moreefficient than either approach separately. The correctnessof Cxyz is non-trivial.

Chee K. YapNew York UniversityCourant Instituteyap@cs.nyu.edu

Long LinCourant Institute, NYUllin@cs.nyu.edu

MS51

Polyhedral Methods For Positive Dimensional So-lution Sets

We present a polyhedral algorithm to manipulate positivedimensional solution sets. Using facet normals to Newtonpolytopes as pretropisms, we zero in at the first two termsof a Puiseux series expansion. We illustrate how this poly-hedral algorithm gives insight into the structure of a tropi-cal prevariety. This polyhedral algorithm is well suited forexploitation of symmetry, when it arises in systems of poly-nomials. Initial form systems with pretropisms in the samegroup orbit are solved only once, allowing for a systematicfiltration of redundant data. The computational resultswill be illustrated on cyclic n-roots polynomial systems.

Danko AdrovicDepartment of Mathematics, Statistics, and ComputerScienceUniversity of Illinois, Chicagoadrovic@math.uic.edu

Jan VerscheldeDepartment of Mathematics, Statistics and ComputerScienceUniversity of Illinois at Chicagojan@math.uic.edu

MS51

Hom4ps in Parallel

When solving systems of polynomial equations by the poly-hedral homotopy methods, while the path tracing part isnaturally parallel, the mixed cell computation which pro-vides the starting points for the homotopy paths is highlyserial. Recently, we have successfully reformulated themixed cell computation and achieved very high speed-upsin parallel. In this talk, we will report our most up-dated parallel implementation of both path tracing andmixed cell computation algorithms. The parallel imple-mentations are designed for a variety of parallel architec-tures such as multi-core processors, symmetric multipro-cessors, nonuniform memory access architectures, clusters,distributed computers, and the exciting new technology ofgeneral purpose GPUs.

Tien-Yien LiDepartment of MathematicsMichigan State Universityli@math.msu.edu

Tianran ChenMichigan State Universitychentia1@msu.edu

MS51

Real Solutions and Numerical Algebraic Geometry

In many applications, the real solutions to polynomial sys-tems are the ones of interest. This talk will consider usingclassical approaches with methods in numerical algebraicgeometry to compute real solutions of a given polynomialsystem. Examples arising from applications will be pre-sented to demonstrate the practicality of the methods.

Jonathan HauensteinTexas A&M Universityjhauenst@math.tamu.edu

Charles WamplerGeneral Motors Research Laboratories, EnterpriseSystems Lab30500 Mound Road, Warren, MI 48090-9055, USAcharles.w.wampler@gm.com

MS51

Multithreaded Newton’s Method and Path Track-ing

We have developed a multithreaded implementation ofNewton′s method on a multicore workstation in a staticmemory allocation multiprecision environment using thequad double arithmetic in the QD-2.3.9. library. Further-more, based on this implementation, we obtained a com-plete parallel version of a path tracker. In my talk I willreport on the details of work load distribution among thethreads and of synchronization in our parallel implemen-tation. I will also discuss to what extent our use of multi-threading compensates the overhead caused by employinghigher precision arithmetic, when tracking a solution path.

Genady YoffeDept. of Mathematics, Statistics, and CSUniversity of Illinois, Chicagogyoffe2@uic.edu

AG11 Abstracts 77

Jan VerscheldeDepartment of Mathematics, Statistics and ComputerScienceUniversity of Illinois at Chicagojan@math.uic.edu

MS52

AG Codes for Secure Multiparty Computation

Chen and Cramer (Crypto 2006) use algebraic curves toconstruct linear secret sharing schemes that are suitablefor secure multiparty computation. We describe the con-struction and we analyse its main properties in terms ofthe geometry of the curve.

Iwan DuursmaUniversity of Illinois at Urbana-ChampaignDepartment of Mathematicsduursma@math.uiuc.edu

MS52

Groebner Bases and Linear Codes

Many classes of good codes have been constructed fromalgebraic curves. These constructions, however, have re-strictions on the alphabet size (say q has to be a square)and on the block length (which depends on the number ofrational points). We are interested in a more general ap-proach that avoids these restrictions. More precisely, givenany prime power q and any block length n, Farr and Gao(2003) shows how Groebner basis theory can be used toconstruct linear codes of length n over GF(q) from any setof n points from an affine space. In this talk, we shall dis-cuss various computational problems related to this classof codes, including decoding via Groebner bases.

Shuhong GaoClemson Universitysgao@math.clemson.edu

MS52

Bounding the Number of Points on a Curve usinga Generalization of Weierstrass Semigroups and anApplication to Toric Codes

In this talk we use techniques from coding theory to deriveupper bounds for the number of rational places of an alge-braic curve defined over a finite field. The used techniquesyield upper bounds if the (generalized) Weierstrass semi-group for an n-tuple of places is known. This sometimesenables one to get an upper bound for the number of ra-tional places for families of function fields. We consider anapplication to toric codes.

Diego RuanoAalborg Universitydiego@math.aau.dk

Peter BeelenTechnical University of Denmarkp.beelen@mat.dtu.dk

MS52

Bent Functions on a Galois Ring and SystematicAuthentication Codes

Authentication codes with secrecy and without secrecy,produced by different techniques, have received consider-

able attention lately. A subclass of the latter is the Sys-tematic Authentication Codes (SACs). Several types ofSACs have appeared constructed by using highly nonlin-ear functions over finite fields or non-degenerated and ra-tional functions on a Galois ring. In this talk a class ofbent functions over a Galois ring of characteristic p2 is in-troduced and based on these functions a class of SACs isgiven. These SAC’s generalize some appearing in the liter-ature for finite fields.

Horacio Tapia-RecillasUAM Mexicohtr@xanum.uam.mx

Juan Carlos Ku-CauichUAM-Mexicojckc23@gmail.com

Claude CarletUniversity of Paris 8 and INRIAINRIA Projet CODESclaude.carlet@inria.fr

MS52

Complexity in Coding Theory

We will give an introduction to the complexity issues incoding theory, both for general codes and for specific codessuch as algebraic geometric codes of low genus.

Daqing WanUniversity of California, Irvinedwan@math.uci.edu

MS53

Some Examples of Non-finitely Generated Sym(N)-invariant Ideals

We are concerned with Sym(N)-invariant (or similar) ide-als of subalgebras of polynomial algebras in countably in-finitely many variables. These ideals arose recently fromapplications to statistics and chemistry. Aschenbrenner,Hillar, Sullivant and Draisma have proved that in certainsubalgebras all Sym(N)-invariant ideals are finitely gener-ated (as such). In this talk, we will present some examplesof non-finitely generated Sym(N)-invariant ideals in somesubalgebras closed to ones mentioned above.

Alexei KrasilnikovDepartamento de MatematicaUniversidade de Braslia, Braslia - DF, 70910-900, Brazilalexei@unb.br

MS53

Tensors of Bounded Rank are Defined in BoundedDegree

An element of a p-fold tensor product of vector spaces hasborder rank k if it may be approximated by a sum of k(pure) tensors. The variety of tensors of a given borderrank is a classical object of algebraic geometry (a highersecant variety) with important modern applications in var-ious diverse areas (algebraic geometry, statistics, complex-ity theory). Except for small k no system of defining equa-tions for this variety is known. We show that for each kthere exists a universal degree bound d such that for eachp there are defining equations of degree at most d for thevariety of p-dimensional tensors of border rank k. The cru-

78 AG11 Abstracts

cial ingredient is to remove the dependence on p by passingto the limit p → ∞ and using the action of some naturalgroup of symmetries. In fact, we show that up to thesesymmetries, there are finitely many equations defining thevariety for all p-dimensional tensors as long as p is largeenough.

Jan DraismaT.U.Eindhoven, The NetherlandsDepartment of Mathematicsj.draisma@tue.nl

Jochen KuttlerDepartment of Mathematical and Statistical SciencesUniversity of Albertajochen.kuttler@ualberta.ca

MS53

Associative Algebras and Letterplace Embedding

We introduce an algebra embedding ι : K〈X〉 → S fromthe free associative algebra K〈X〉 generated by a finite orcountable set X into the skew monoid ring S = P ∗ Σ,where P = K[X × N ] and Σ = 〈σ〉 is the monoid, gener-ated by a suitable endomorphism σ : P → P . We presenta general Grobner bases theory for graded two-sided idealsof the graded algebra S =

⊕iPσi, where P = K[Y ] is

any commutative polynomial ring and σ : P → P is anabstract endomorphism satisfying two explicit conditions.We obtain a bijective correspondence, preserving Grobnerbases, between graded Σ-invariant ideals of P and a classof graded two-sided ideals of S. As an application we show,that up to a given degree, Grobner bases of finitely gener-ated graded ordinary difference ideals can be computed bythe proposed methods in S in a finite number of steps.

Viktor LevandovskyyLehstuhl D fuer Mathematik, RWTH Aachenviktor.levandovskyy@math.rwth-aachen.de

Roberto La ScalaDipartimento di MathematicaUniversita degli studi di Barilascala@dm.uniba.it

MS53

Secant Varieties of Segre-Veronese Varieties

Secant varieties of Segre and Veronese varieties are classi-cal objects that go back to the Italian school in the nine-teen century. Surprisingly, very little is known about theirequations. Inspired by experiments related to algebraicstatistics, Garcia, Stillman and Sturmfels gave a conjec-tural description of the generators of the ideal of the secantline variety Sec(X) of a Segre variety X. This generalizesthe familiar result which states that matrices of rank twoare defined by the vanishing of their 3 × 3 minors. Fora Veronese variety X, it was known by work of Kanevthat the ideal of Sec(X) is generated in degree three byminors of catalecticant matrices. I will introduce a newtechnique for studying the equations of the secant varietiesof Segre-Veronese varieties, based on the usual representa-tion theoretic approach to this problem. I will explain howthis technique applies to show that for X a Segre-Veronesevariety, the ideal of Sec(X) is generated in degree threeby minors of matrices of flattenings, and to give a descrip-tion of the decomposition into irreducible representationsof the homogeneous coordinate ring of Sec(X). This willrecover as special cases the conjecture of Garcia, Stillman

and Sturmfels, and the result of Kanev.

Claudiu RaicuUC Berkeley /Princetonclaudiu@math.berkeley.edu

MS53

Twisted Commutative Algebras and Delta-modules

Twisted commutative algebras and Δ-modules are twokinds of algebraic structures that can be interpreted ashaving infinitely many variables. I will explain what theyare and state two finiteness theorems about them: the firstis a noetherianity result, the second a rationality result forHilbert series. If time permits, I will explain how these re-sults can be applied to study the syzygies of certain familiesof varieties.

Andrew SnowdenMITawsnowden@gmail.com

MS54

The Convex Geometry of Linear Inverse Problems

Building on the success of generalizing compressed sens-ing to low-rank matrix recovery, we extend the catalog ofstructures that can be recovered from partial information.We describe algorithms to decompose signals into sumsof atomic signals from a simple, not necessarily discrete,set. These algorithms are derived in a convex optimizationframework that generalizes previous methods based on l1-norm and nuclear norm minimization. We discuss generalrecovery guarantees for our approach and several exampleapplications.

Venkat ChandrasekaranMassachusetts Institute of Technologyvenkatc@mit.edu

Benjamin RechtUniversity of Wisconsin – Madisonbrecht@cs.wisc.edu

Pablo A. Parrilo, Alan WillskyMassachusetts Institute of Technologyparrilo@MIT.EDU, willsky@mit.edu

MS54

Extended BIC and Bayesian Marginal Likelihoodin Sparse Graphical Models

We consider the problem of recovering sparse graphicalstructure in the setting of Gaussian or discrete graphicalmodeling. While the total number of possible sparse mod-els is often too large for exhaustive search, information crit-era may be used to select a sparse model from a reducedset of candidate models (for instance, the “path” of modelsselected by a parameterized procedure such as the graphi-cal lasso). Modern applications of graphical models oftenhave a number of nodes that is comparable to or greaterthan the number of data points observed. In our treat-ment of an extended Bayesian information criterion (BIC)we thus consider asymptotic scenarios in which the num-ber of nodes p to grows as the sample size n increases. Ourwork examines the consistency properties of the extendedBIC as well as its relationship to the marginal likelihood

AG11 Abstracts 79

arising in fully Bayesian approaches.

Rina Foygel, Mathias DrtonDepartment of StatisticsUniversity of Chicagorina@uchicago.edu, drton@uchicago.edu

MS54

Diagonal/Low-Rank Decomposition and Correla-tion Matrices

Suppose an n×n matrix X is the sum of a diagonal and apsd low-rank matrix. Decomposing X into these unknownconstituents has applications in statistics, signal process-ing, and elsewhere. We give a simple condition on the col-umn space of the low rank matrix that ensures a tractibleconvex program can correctly decompose X. Our anal-ysis highlights connections between this problem and thestructure of the set of correlation matrices.

James SaundersonMassachusetts Institute of Technologyjamess@mit.edu

MS54

Chromosome Packing in Cell Nucleus

During most of the cell cycle each chromosome occupiesa roughly spherical domain called a chromosome territory.Chromosome territories can overlap and their radial andrelative positions are non-random and similar among simi-lar cell types. A chromosome arrangement can be viewed asa packing of overlapping spheres of various sizes inside anellipsoid, the cell nucleus. We present a non-convex modelfor chromosome arrangements together with an alternat-ing minimization algorithm. Using this model, we simulatechromosome arrangements and study the resulting numberand volume of internal ’holes’, which make chromosomesdeep inside accessible to regulatory factors.

Caroline UhlerDepartment of StatisticsUC Berkeleycuhler@stat.berkeley.edu

MS54

Finding Approximately Rank-One Submatriceswith the Nuclear Norm and l1-Norm

We propose a tractable method to find large approximatelyrank-one submatrices which can be used in nonnegativematrix factorization. We apply a proximal point algo-rithm to solve the resulting optimization problem whichis formulated with the nuclear norm and l1-norm. We re-port numerical results on realistic datasets of decomposablebitmap images and nearly separable corpus.

Xuan Vinh DoanC&O DepartmentUniversity of Waterloovanxuan@uwaterloo.ca

Kim-Chuan TohNational University of SingaporeDepartment of Mathematics and Singapore-MIT Alliancemattohkc@nus.edu.sg

Stephen A. Vavasis

University of WaterlooDept of Combinatorics & Optimizationvavasis@math.uwaterloo.ca

MS55

Continuous Amortization for the Complexity ofAdaptive Subdivision Methods

Adaptive subdivision algorithms subdivide more times nearchallenging features and fewer times elsewhere. Computa-tions of the size of the subdivision tree should consideradaptivity: e.g., bounds on the maximum depth do notreflect situations where the tree has a few deep paths, butremains modest in overall size. I use the new analysis tech-nique of continuous amortization to show that the tree sizeof a simple, evaluation-based root isolation algorithm isnearly optimal.

Michael A. BurrFordham Universitymburr1@fordham.edu

Felix KrahmerHausdorff Center for MathematicsUniversitat Bonnfelix.krahmer@hausdorff-center.uni-bonn.de

Chee K. YapNew York UniversityCourant Instituteyap@cs.nyu.edu

MS55

Patterns in Roots of the Derivatives of RandomUnivariate Polynomials

I will first recall old and recent results on distributions ofroots of random polynomials and eigenvalues of randommatrices. Then I will present original observations of pat-terns of roots of the derivatives of random polynomials. Iwill set some conjectures enforced by experiments and out-line proofs of some claims. This can be applied to designroot isolation algorithms.

Andre GalligoMathematics DepartmentlUniversite Nice Sophia Antipolisgalligo@math.unice.fr

MS55

Subadditivity in Polynomial Real Root Isolation

Isac Schoenberg’s theory of variation-diminishing lineartransformations, published in 1930, is used to prove the fol-lowing subadditivity property for all univariate real poly-nomials A,

var(T(A)) + var((T ◦ R)(A)) ≤ var(A).

Here, T, R and var are, respectively, the translation by 1,the reciprocal transformation, and the number of sign vari-ations. The subadditivity property is useful in the imple-mentation and analysis of the continued fractions methodfor polynomial real root isolation. This research is jointwork with G. E. Collins.

Werner KrandickDrexel Universitykrandick@cs.drexel.edu

80 AG11 Abstracts

MS55

Subdivision Methods for the Topology and Ar-rangement of Semi-algebraic Curves and Surfaces

Computing the topology and arrangement of algebraiccurves and surfaces appears in many geometric mod-elling problems, such as surface-surface intersection, self-intersection, ... In the presentation, we will describe subdi-vision methods which exploit information on the boundaryof regions and which only require the isolation of character-istic points. We will show how topological degree compu-tation can help analysing the number of branches at singu-lar points. Combining regularity criterion with subdivisionstrategies, this yields a complete algorithm for comput-ing the topology of (singular) algebraic curves. Extensionof this approach to arrangement computation of curvesor surfaces will be described. Experimentation with thealgebraic-geometric modeller AXEL will shortly be demon-strated.

Bernard MourrainINRIA Sophia AntipolisBernard.Mourrain@inria.fr

PP1

The Geometry of Tree Reconstruction

Tree agglomeration methods such as the Neighbor-Joiningand UPGMA algorithms continue to play a significant rolein phylogenetic research. We use polyhedral geometry toanalyze the natural subdivision of Euclidean space inducedby classifying the input vectors according to tree topologiesreturned by an algorithm. We also use lattice theory toanalyze disparities between observed and predicted shapestatistics on reconstructed trees.

Ruth E. Davidson, Seth SullivantNorth Carolina State Universityredavids@ncsu.edu, smsulli2@ncsu.edu

PP1

The Maximum Likelihood Degree for the RandomEffects Model

Given a statistical model, the maximum likelihood degree(ML degree) is the degree of the variety defined by thelikelihood equations. It measures the algebraic complexityof the computation to find model parameters that best ex-plain a given data point. In this poster, we give an explicitformula for the ML degree of the analysis of variance modelwith random effects. This is joint work with Mathias Drtonand Sonja Petrovic.

Elizabeth GrossUniversity of Illinois at Chicagoegross7@uic.edu

Mathias DrtonDepartment of StatisticsUniversity of Chicagodrton@uchicago.edu

Sonja PetrovicUniversity of Illinois, Chicagopetrovic@math.uic.edu

PP1

Unexpected Reality: the Monotone-Secant Conjec-

ture

The Monotone Secant Conjecture poses a rich class of poly-nomial systems, all of whose solutions are real. Thesesystems come from the Schubert calculus on ?ag mani-folds, and the Monotone Secant Conjecture is a compellinggeneralization of the Shapiro Conjecture for Grassmanni-ans (Theorem of Mukhin, Tarasov, and Varchenko). Wepresent the Monotone Secant Conjecture as a generaliza-tion of the Shapiro Conjecture and explain the massivecomputational evidence in its favor.

Nickolas J. HeinTexas A&M Universitynhein@math.tamu.edu

PP1

Dual Space Computations for Homotopy Continu-ation Using H-Bases

Using homotopy continuation methods, certain systemsend up with a number of paths tending toward infinity. Us-ing dual space computations through H-bases, our goal is topre-condition these systems. By moving to the dual spaceand using H-bases, we can cut out many of these extraneouspaths. When we come back from the dual space, we have asystem with fewer paths tracking to infinity thereby savingtime and computation. We examine how many times toefficiently implement this for optimal time and results.

Steven L. IhdeColorado State Universityihde@math.colostate.edu

PP1

Markov Bases for the Analysis of Partially RankedData

Suppose survey respondents have been asked to give a com-plete ranking of a collection of items. Diaconis and Sturm-fels, and Diaconis and Eriksson, have shown how Markovbases can be used to better understand relationships be-tween natural summary statistics associated with the re-sulting fully ranked data. In this talk, we show how theirideas can be fruitfully extended to situations in which re-spondents are asked to provide rankings of only a subsetof the items.

Michael E. Orrison, Ann JohnstonHarvey Mudd Collegeorrison@hmc.edu, ann k johnston@hmc.edu

Michael HansenN/Amhansen@gmail.com

PP1

Algebraic Ecological Inference

Ecological inference is the process of inferring individuallevel behavior from aggregate data. The problem of ecolog-ical inference occurs frequently in many applied sciences.In the present work, we present an algebraic solution to theproblem of ecological inference using tools from computa-tional algebra. This method can handle multi-dimensionalcontingency tables, deterministically respects bounds andcan incorporate information from many different sources.We illustrate the method by a few examples and provide

AG11 Abstracts 81

R code.

Vishesh KarwaPenn State Universityvishesh@psu.edu

PP1

The Convex Hull of Circles

We describe the boundaries of the convex hulls of perhapsthe simplest compact curves in three-dimensional space,namely pairs of circles. We show that the edge surfaceof such a convex hull is in general an irrational ruled sur-face whose ruling forms a (2,2)-curve, and conjecture thatany real (2,2)-curve can arise from the edge surface of theconvex hull of two circles.

Tina MaiDepartment of Mathematics, Texas A&M UniversityCollege Station, TX 77843-3368mai@math.tamu.edu

Frank SottileTexas A&M Universitysottile@math.tamu.edu

PP1

Developing Water Loss Prevention Robot SystemGeometrical Design

This research describes the geometrical design of an in-piperobot system able to redevelop fresh water pipes of differentdiameters. The robots geometry is calculated in conjunc-tion of its working environment, the size of the pipes, andits physical properties, such as weight and required forceto move and redevelop the pipe.

Luis A. Mateos, Markus VinczeAutomation and Control Institute (ACIN)Vienna University of Technologymateos@acin.tuwien.ac.at, vincze@acin.tuwien.ac.at

PP1

Perturbation Analysis to Design a Robust Decou-pling Geometric Technique in Linear Multi-InputMulti-Output Systems

The linear state-feedback control is one of the basic andfundamental part of the linear control theory. Eigenvectorsare used through the singular value decomposition (svd)in robust analysis and synthesis of robust controllers inmulti-input multi-output systems. In this paper the prob-lem of robustness of a decoupling controller is analyzed.In particular, it is shown that the robustness of a decou-pling controller is achieved using subspaces closer to theeigenvectors associated to the system. The analysis of therobustness is based on a perturbation method and a syn-thesis of a controller is shown. An application concerningan electrical motor is shown.

Paolo MercorelliOstfalia University of Applied Sciencesp.mercorelli@ostfalia.de

PP1

A Stochastic Framework for Discrete Models in

Systems Biology

We present a new modeling framework for gene regulatorynetworks that incorporates state dependent delays and thatis able to capture the cell-to-cell variability. We presentthis framework in the context of polynomial dynamical sys-tems where we can use computational algebra to analyzeour model. The state dependent delays represent the timedelays of activation and degradation. One of the new fea-tures of this framework is that it allows a finer analysis ofdiscrete models and the possibility to simulate cell popu-lations. We applied our methods to one of the best knownstochastic regulatory networks, that is involved in control-ling the outcome of lambda phage infection of bacteria.

David MurrugarraVirginia Bioinformatics InstituteVirginia Techdavidmur@vt.edu

Reinhard LaubenbacherVirginia Bioinformatics institute1880 Pratt Drivereinhard@vbi.vt.edu

PP1

An Implementation of Parameter Homotopies andtheir Applications

There are several implementations ofhomotopy continuation-based polynomial system solvers,some of which allow for so-called parameter homotopiesbetween polynomial systems that differ only in coefficients.Some applications from science and engineering require therepeated solution of polynomial systems with slightly dif-ferent coefficients. The new software package described inthis poster is a parallelized implementation of parameterhomotopies, making use of Bertini (D. Bates, J. Hauen-stein, A. Sommese, C. Wampler). An example consideringbistability in biochemical kinetic reactions is provided toshow the usefulness of these methods. This is joint workwith Dan Bates and Dan Brake.

Matt NiemergColorado State UniversityFort Collins, COmatthew.niemerg@gmail.com

PP1

Computing the Variety for K-Algebra Homomor-phisms

Let k be an algebraically closed field. Let A and B be ar-bitrary commutative (unitary) k-algebras. Assume V ⊂ Aand W ⊂ B are finite dimensional k-linear subspaces. De-note the subalgebras of A and B generated by V and Was A(V ) and B(W ). Then the set RHom(A,B, V,W ) ofk-algebra homomorphisms f : A(V ) → B(W ) such thatf(V ) ⊂ W is an affine k-variety in a natural way. Thestructure of the proof of this claim suggests an algorithmcould be developed to allow software to calculate the affinevariety Hom(A,B, V,W ). The goal of my research is to de-velop software capable of doing this calculation and use thissoftware to compute some classical algebras (e.g. group al-gebras, monomial algebras etc).

Jon YaggieUniversity of Illinois Chicagojyaggi2@uic.edu