Course Descriptions Fall 2017

Department of Mathematics University of Illinois

Math 510- Riemann Surfaces and Algebraic Curves.

Instructor- Dodd

Course Description- This course will cover the basic theory of Riemann surfaces and algebraic

curves. In particular, we will define the notion of Riemann surface, study their basic analytic and topological properties - and prove the main theorems, including the Riemann existence theorem, the uniformization theorem, and the Riemann-Roch formula.

MATH 512: MODERN ALGEBRAIC GEOMETRY

WILLIAM J. HABOUSH

The coourse is devoted essentially to Chapter II of Hartshorne’s Al-gebraic Geometry. I will begin with a brief (about 3 lectures) reviewof chapter one. I will then discuss sheaves and related algebraic ideas.This will be followed by a discussion of the spectrum of a commu-tative ring and of general schemes. We will discuss such propertiesas separation and properness and we will consider sheaves of mod-ules with special attention to coherent and quasi-coherent sheaves. Weconsider projective and affine schemes and quasi-coherent sheaves onthem. Then we shall discuss projective embeddings, linear systems,divisors and invertible sheaves. We will consider certain special mor-phisms such as blowing up. We will conclude with Kahler differentialsnon-singularity and Bertini theorems.

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Math 518

Differentiable Manifolds I

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Instructor: Ely Kerman

Course Description

This graduate course is an introduction to the theory of smooth manifolds.We will define smooth manifolds, study many examples and constructions,and learn how to do calculus on them. Along the way we will study theintersection theory of submanifolds, the generalized Stokes’ Theorem andde Rham cohomology. We will also introduce the natural settings for manyfundamental theorems in Riemannian Geometry and Physics.

Grading Scheme

Weekly Homework 35%, Midterm 25%, Final Exam 40%.

Suggested Text

Introduction to Smooth Manifolds, Lee.

Prerequisites

Point set topology and linear algebra.

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MATH 526 Algebraic Topology IIFall 2017

Instructor: Vesna Stojanoska

Course description: This is the second semester of the algebraic topology sequence, and forthe most part will concentrate on studying singular cohomology, its structure and applications.In particular, we will talk about the cup product, Poincare duality, characteristic classes, andvarious applications. If time permits, we will study cohomology operations, spectral sequences,and/or generalized cohomology theories.

Prerequisites: MATH 525 or consent of instructor.

Textbooks: The main textbook will be

• Algebraic Topology, by Hatcher.

Other helpful references include:

• Algebraic Topology, by Switzer,

• A Concise Course in Algebraic Topology, by May,

• Geometry and Topology, by Bredon.

Assignments: There will be homework every 2-3 weeks, and a final project.

Mathematics 527 — Homotopy Theory(Fall 2017, 2-3:330 TuTh, 156 Henry)

Instructor: Charles RezkOffice: 242 Illini HallPhone: 5-6309Email: rezk@illinois.edu

Webpage: http://www.math.uiuc.edu/~rezk/

Prerequisites: Math 526, or instructor consent. It is possible to take Math 526 concurrently.

Topics: This course will be an introduction to the techniques of modern homotopy theory, fromthe point of view of model categories. After a brief review of the classical homotopy theory oftopological spaces (as covered in 526), we’ll discuss

• model structures in the sense of Quillen;

• the standard model structure on spaces;

• model structurs on categories of chain complexes, and derived categories;

• derived functors; homotopy limits and colimits;

• localization of model structures; Postnikov approximation;

• stable homotopy theory and spectra;

• examples such as bordism and K-theory.

Additional topics (time permitting) may include: equivariant homotopy theory, and algebraic struc-tures in homotopy theory (via operads).

Texts: There are no standard texts on all of this material. References for model categories include:

• Quillen, Homotopical algebra, Springer LNM 43, (1967).

• Hovey, Model categories, AMS Math Surveys 63, (1999).

• Dwyer & Spalinski, “Homotopy theories and model categories”.

References for stable homotopy theory include:

• Adams, Stable homotopy and generalised homology, U. of Chicago Press, (1974).

• Mandell & May, Orthogonal spectra and S-modules, AMS Memoirs 755, (2002).

• Schwede, Lectures on equivariant stable homotopy theory.

In addition, lecture notes will be posted.

FALL 2017, Math 531: Analytic Theory of Numbers I

Instructor: Alexandru ZaharescuOffice: 449 Altgeld HallPhone: 265-5439E-mail: zaharesc@illinois.edu

Lectures: TR 12:30-1:50pm

Course Description. Topics will include:

(1) Arithmetic functions.(2) Elementary theorems on the distribution of primes.(3) Dirichlet series and Euler products.(4) Properties of the Riemnann zeta function.(5) Analytic proof of the Prime Number Theorem.(6) Dirichlet’s theorem on primes in arithmetic progressions.

A more detailed description can be found at:http://www.math.uiuc.edu/Bourbaki/Syllabi/syllabus 531.pdf

Prerequisite: MATH 448 and either MATH 417 or MATH 453.

Grading Policy: Comprehensive final exam: 35%; Two midterm exams:2 × 25=50%; Homework: 15%.

Recommended Textbooks:

T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1sted. 1976.A. J. Hildebrand, Introduction to Analytic Number Theory, available on Pro-fessor Hildebrand’s webpage.

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Math 540: Real Analysis

Fall 2017

Instructor: Florin P. BocaOffice: 371 Altgeld HallE-mail: fboca@illinois.edu

Course description: This is the core graduate course in Real Analysis, covering thefollowing topics:

• Measures on the line.

Abstract measure theory, outer measure, Lebesgue measure on the real line, mea-surable sets, Borel sets, Cantor sets and functions, non-measurable sets, Baire’scategory theorem.

• Measurable functions.

Structure of measurable sets, approximation of measurable functions by simplefunctions, Littlewood’s three principles, Egorov and Lusin’s theorems.

• Integration.

Lebesgue theory of integration, convergence theorems (Monotone Convergence, Fa-tou’s Lemma, little Fubini, Dominated Convergence), comparison of the Riemannand Lebesgue integrals, modes of convergence, approximation of integrable func-tions by continuous functions, Fubini’s theorem for the plane, product measures,the general Fubini-Tonelli theorem, applications to probability, the convolutionproduct.

• Differentiability.

Functions of bounded variation (structure and differentiability), absolutely contin-uous functions, maximal functions, fundamental theorem of calculus, the Radon-Nikodym theorem.

• Lp spaces on intervals and ℓ

p spaces.

Jensen’s inequality, Holder and Minkowski’s inequalities, class of Lp functions,

completeness, duals of Lp; spaces, inclusions of Lp spaces.• Hilbert spaces and Fourier series

Elementary Hilbert space theory, orthogonal projections, Riesz representation the-orem, Bessel’s inequality, Riemann-Lebesgue lemma, Parseval’s identity, complete-ness of trigonometric spaces.

Prerequisite: MATH 447 or equivalent.

Textbook: There is no required textbook. Recommended textbooks include:

• Gerald B. Folland, Real Analysis. Modern Techniques and Their Applications.• Peter Loeb, Real Analysis.• Walter Rudin, Real and Complex Analysis.

Grading policy: Comprehensive final exam: 40%; Two midterm exams: 2x20 = 40%;Homework: 20%.

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Math 547 Banach Spaces Section F1, Fall 2017 Instructor: Zhong-Jin Ruan Classroom: 447 Altgeld Hall MWF 2:00-2:50 pm Office: 353 Altgeld Hall Email: z-ruan@illinois.edu In this course, we plan to cover the following topics.

1) Some properties on finite dimensional Banach spaces 2) Tensor norms and local properties of Banach spaces 3) Trace duality and certain norms on spaces of linear maps 4) Grothedieck approximation property 5) Type and cotype, application of probalistic techniques in Banach spaces 6) Grothendieck inequality 7) An introduction to operator spaces (, i.e. a natural quantization of Banach spaces).

Prerequisite: Math 541. Text: Summing and Nuclear Norms in Banach Spaces by G.J.O. Jameson London Mathematical Society, Study Texts 8, Cambridge University Press, 1987. References: 1) A course in functional analysis by J. Conway, Springer 2) Factorization of Linear Operators and Geometry of Banach Spaces by Pisier (Regional Conference Series in Math.). 3) Banach Spaces for Analysts, P. Wojtaszczyk, Cambridge studies in advanced mathematics

                                                            Math   550.   Dynamical   Systems   1.   Fall   2017.  

Instructor:   Vadim   Zharnitsky,   310   Altgeld   Hall.  

Meetings:   443   Altgeld,   MWF   12­1. 

Textbook   is   not   required   but   here   are   some   books   that   I   will   use:  

E. A. Coddington and N. Levinson,    Theory   of   Ordinary   Differential   Equations,    A. Katok and B. Hasselblatt ,   Introduction   to   the   Modern   theory   of   Dynamical   Systems, C. Chicone ,   Ordinary   Differential   Equations   and   Applications,    V. I. Arnold ,   Geometric methods   in   Ordinary   Differential   Equations. 

This      part   of   the   course      covers   comprehensive   exam   material: 

Introduction   to   continuous   dynamics   (6   lectures) 

●    Phase   space,   vector   fields,   flows ● Cauchy­Peano   existence   theorem,   uniqueness   theorem ● Dependence   on   initial   conditions   and   parameters ● Compact   differentiable   manifolds       where   local   flows   are   global   flows 

Introduction   to   discrete   dynamics   (3   lectures) 

● Iteration   of   maps,   fixed   points   and   stability ● Discrete   flows   on   manifolds       sphere,   torus,   surfaces   of   genus   g ● Chaotic   behavior:       Bernoulli   shift,   Cat   Map 

Linear   differential   equations   (3   lectures) 

●                            Real   and   Complex   Jordan   normal   forms ●                            Stability   of   linear   systems   with   constant   coefficients ●                         Variation   of   parameters ●                            Linearization   about   a   solution   and   the   equation   of   variation 

 

Other   material   will   include: 

Geometric   Methods   for   Nonlinear   Equations,    Nonlinear   Systems   near Equilibrium,   Structural   Stability,   Hamiltonian   Systems,   and   Bifurcation Theory. 

 

Course Description for Math 553Partial Differential Equations

Spring 2017

Lecturer: Eduard KirrOffice: 245 Illini HallE-mail: ekirr@illinois.edu

Time: 11:00 am - 12:20 pm

Texts: Course Packet for Math 553 available from Illini Union Bookstore;Partial Differential Equation by Lawrence C. Evans (not required);Equations of Mathematical Physics by V.S. Vladimirov (not required).

Syllabus: Basic introduction to the study of partial differential equations (PDE’s). Wewill cover all main classes of PDE’s: hyperbolic (such as wave equation), parabolic(such as heat equation) and elliptic equations (such as Laplace’s and Poisson’s equa-tions) from both the classical point of view, i.e. smooth solutions, and the modernpoint of view, i.e. weak solutions related to the integral form of the PDE.

1. Classical Theory includes: method of characteristics, power series method, en-ergy method, maximum principles and harmonic functions;

2. Modern Theory includes: weak solutions and formation of shock waves in con-servation laws, distribution theory, Fourier transforms, fundamental solutionsand Green’s functions, potential theory.

We will emphasize the applications of these methods to both initial value (Cauchy)problems and boundary value (Dirichlet, Neumann, Robin) problems for PDE’s .

Prerequisite: Consent of instructor.

Web page: http://www.math.uiuc.edu/~ekirr/page/teaching/math553.html

Homework: Assignments will be given approximately every two weeks. Homework is tobe written up independently. Solutions are posted, if necessary, after the due date,at TBA.

Grading Policy: Grades will be based on final exam 50%, midterm 30%, homework andclass activity 20%.

Math 562 (Probability II) Instructor: Renming Song Office: 338 Illini Hall Phone number: 217 244 6604 Text: Jean-Francois Le Gall : Brownian Motion, Martingales and Stochastic Calculus, 2016, Springer Course Topics: This is the second half of the basic graduate course in probability theory. This course will concentrate on stochastic calculus and its applications. In particular, we will cover, among other things, the following topics: Brownian motion, stochastic integrals, Ito's formula, martingale representation theorem, Girsanov's theorem, stochastic differential equations, connections to partial differential equations. If time allows, I will also present some applications to mathematical finance. Math 561 is a prerequisite for this course. However, if you have not taken Math 561, but are willing to invest some extra time to pick up the necessary materials from 561, you may register for this course. Grading Policy: Your grade will depend on homework assignment and a possible final exam.

Fall 2017 MATH 563Risk Modeling and Analysis

Dr. Runhuan Feng, FSA, CERA.

In our complex and interconnected world, dependence modeling has become an increasingly impor-

tant tool for quantifying, assessing and managing risks in all walks of life. This course provides an

introduction to the actuarial theory for modeling dependent risks.

Students will be able to learn various mathematical models and techniques to measure, compare

and model dependence. Specific topics include:

• Risk measures;

• Stochastic order relations;

• Bivariate and multivariate copulas;

• Concordance measures;

• Dependence order;

• Integral orderings and probability metrics;

• Stochastic bounds on functions of dependent risks.

Prerequisite: Math 409. (Students without prerequisite can register with instructor’s consent.)

Math 564- Applied Stochastic Processes

Professor Dey

Course Description- This is a graduate course on applied stochastic processes, designed for those graduate students who are going to need to use stochastic processes in their research but do not have the measure-theoretic background to take the Math 561-562 sequence. Measure theory is not a prerequisite for this course. However, a basic knowledge of probability theory (Math 461 or its equivalent) is expected, as well as some knowledge of linear algebra and analysis. The goal of this course is a good understanding of basic stochastic processes, in particular discrete-time and continuous-time Markov chains, and their applications. We will mainly follow James Norris' book "Markov Chains", augmenting the text when necessary. The materials covered in this course include the following: Fundamentals: background on probability, linear algebra, and set theory. Discrete-time Markov chains: classes, hitting times, absorption probabilities, recurrence and transience, invariant distribution, limiting distribution, reversibility, ergodic theorem; Continuous-time Markov chains: same topics as above, holding times, explosion, forward/backward Kolmogorov equations; Related topics: Discrete-time martingales, potential theory, Brownian motion; Applications: Queueing theory, population biology, Markov Chain Monte Carlo. This course can be tailored to the interests of the audience.

Math 567 Actuarial Models for Financial Economics

Instructor- Jing Wang

Course Description-

In this course, we will cover several important actuarial topics in the areas of stochastic

processes, risk, and financial economics. Our focus is on the mathematics and economics

underlying the pricing of financial options. Using Brownian motion as a modeling framework for

financial and economic processes, we explore the ideas and techniques for some advanced

actuarial and financial risk management material, such as the Black-Scholes option-pricing

model. We will also cover stochastic simulation of economic and financial variables, and the

modeling of interest rates. A thorough knowledge of calculus, probability, and interest theory is

required.

mathematical logic

math 570

Anush Tserunyan Fall 2017

The course will introduce the main ideas and basic results of mathematical logic from afairly modern prospective, providing a number of applications to other fields of mathematicssuch as algebra, algebraic geometry, and combinatorics. It will consist of three parts: basicmodel theory, basic recursion theory, and more.

Basic model theory. Model theory is a study of mathematical structures, examples ofwhich include groups, rings, fields, graphs, and partial orders. We will first abstractly studystructures and definability, theories, models and categoricity, as well as formal proofs, andthis will culminate in proofs of the Godel Completeness and Compactness Theorems—twoof the most useful tools of logic. Then, we’ll apply the developed techniques to concreteexamples such as the structure of natural numbers and algebraically closed fields; thelatter will yield a rigorous proof of the Lefschetz Principle (a first-order sentence is truein the field of complex numbers if and only if it is true in all algebraically closed fields ofsufficiently large characteristic) and an amusingly slick proof of Ax’s theorem (if a polynomialfunction Cn → Cn is injective, then it is surjective). We will also discuss applications of theCompactness theorem in deriving finitary analogues of the infinitary combinatorial statementssuch as the infinite Ramsey theorem, van der Waerden’s or Szemeredi’s theorems, graphcolorings, etc.

Basic recursion theory. This part will begin with a robust definition of computation(algorithm), followed by a rather short investigation of computable functions and sets. Theinvestigation will be short because we will quickly discover that many interesting functions andsets are not computable, as illustrated by the Godel Incompleteness Theorem and Church’stheorem on undecidability of first-order logic, both of which we will prove.

And more. Diving more into model theory, we will study quantifier elimination and modelcompleteness, and, as a quick application, give a transparent proof of Hilbert’s Nullstellensatz.If time permits, we will change gears and learn two completely different (set-theoretic andcombinatorial) constructions of structures from existing ones: ultraproducts and Fraısselimits. The former will involve a rather measure-theoretic introduction to ultrafilters, whilethe latter will touch base with probabilistic objects like the random graph.

Textbook: Logic lecture notes available on the instructor’s webpage 1.

Prerequisites: No background in mathematical logic is needed, but knowledge of under-graduate abstract algebra would be helpful.

Exams: One in-class midterm and an in-class final.

Homework: 6–8 problems every week to be submitted in both written and blackboardpresentation forms in problem sessions.

1http://www.math.uiuc.edu/~anush/

Math 580

Professor Balogh

Syllabus: This is a rigorous, graduate level introduction to combinatorics. It does not assume prior study, but requires mathematical maturity; it moves at a fast pace. The first half of the course is on enumeration. The second half covers graph theory. There are some topics that are treated more in depth in advanced graduate courses (Math 581, 582, 583, 584, 585): Ramsey theory, partially ordered sets, the probabilistic method and combinatorial designs (as time permits). Textbook: The FALL 2017 edition of the text COMBINATORIAL MATHEMATICS (by Douglas West) will available at TIS Bookstore. REQUIREMENTS: A raw score of 80% or higher guarantees an A while a score of 60% or higher guarantees a B- (grade drops by 5%). Additionally, for an A, in the final exam minimum 50%, for B+ (passing com) 40% required. (Near) weekly assignments. Each assignment will have 6 problems of your choice of 5/6 are graded. There are 10 homework assignments, each worth 6%, two tests, each 10%, and a final exam for 20%. The gradings: 80%− : A, 75%− : A−, 70%− : B+, 65%− : B, etc. Note that the writings of the solutions must have a high quality and typed, if the argument is messy or not typed then even if the solution is correct it could be returned without grading with 0 points. Late homework policy: In case the homework is not submitted on time, it could be submitted for the next class, with losing 10% of the score. If there is offical or medical reason then try to notify me in advance via e-mail. RESOURCES: Electronic mail is a medium for announcements and questions. PREREQUISITES: There are no official prerequisites, but students need the mathematical maturity and background for graduate-level mathematics.

Math 585- Probabilistic Combinatorics

Professor Balogh

Course Description- the Probabilistic Method is a powerful tool in tackling many problems in

discrete mathematics.It belongs to those areas of mathematics, which have experienced a most impressive growth in the past few decades. This course provides an extensive treatment of the Probabilistic Method, with emphasis on methodology. We will try to illustrate the main ideas by showing the application of probabilistic reasoning to various combinatorial problems. The topics covered in the class will include (but are not limited to):Linearity of expectation, the second moment method, the local lemma, correlation inequalities, martingales, large deviation inequalities, Janson and Talagrand inequalities, pseudorandomness, random graphs, random regular graphs, percolation, bootstrap percolation.

TEXTBOOKS: Most of the topics covered in the course appear in the following book: Other topics appear in recent papers. The Probabilistic Method, by N. Alon and J. H. Spencer, 4th Edition, Wiley, There will be a test during the semester, and several homework assignments. GRADING: 80%− : A, . . . , 50% − C−. OFFICE HOURS: After classes and by appointment. PREREQUISITES: There are no official prerequisites, but students need the mathematical maturity and background for graduate-level mathematics. For example, basics of linear algebra, probability and graph theory are assumed to be known.

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Graduate Course Description

Fall 2017 CRN: 64464Math 595: Anatomy of Integers and Random Permutations

Instructor: Kevin Ford

Time/place: MWF 1–1:50; Altgeld 347

Prerequisites: Basic analytic number theory will be very helpful (equivalent of the first half of Math 531; elementary

prime number estimates and multiplicative functions). Some knowledge of basic probability will be helpful but not

necessary.

Recommended Texts: Divisors, by R. R. Hall and G. Tenenbaum, Cambridge Tracts in Math. 90, paperback edition,

2008. (highly recommended for purchase).

Course Description.

Integers factor uniquely into a product of primes, and permutations factor uniquely into a product of cycles. Basic

questions one can ask about these structures are

• How many prime factors does a typical integer n ∈ [1, x] have? What is the distribution of those prime factors?

• How many cycles does a typical permutation of Sn have? How are the lengths of the cycles distributed?

Perhaps surprisingly, there is a close connection between these two problems, both distributions governed by the same

probabilistic law. In the first part of the course we will examine carefully, on many scales, the distribution of the prime

factors of typical integers and cycle decompositions of typical permutations. We will emphasize the connections

between these two structures, stressing probabilistic techniques and ideas. In the second part, we will apply this

knowledge to answer questions which about the distribution of divisors of integers, fixed sets of permutations (a

subset of {1, . . . , n} which is itself permuted by the permutation) and applications of these bounds. Some examples:

1. How likely is it that an integer has two divisors in a fixed dyadic interval (y, 2y]?

2. How likely is it that an integer has two divisors in some dyadic interval (y, 2y]?

3. How many distinct entries are there in and N ×N multiplication table?

4. How likely is it that a random permutation π ∈ Sn contains a fixed set of size equal to k (that is, contains cycles

with lengths summing to k)?

5. Given r random permutations of Sn, chosen independently, are there infinitely many k so that every permutation

fixes a set of size k? We show that when r = 3, the answer is yes with probability 1, and when r = 4 the answer

is no with probability 1.

Problems 1,2,3 are classical problems of Erdos. Problem 4 generalizes the classical derangement problem (k = 1).

Problem 5 has application to “invariable generation of Sn”.

Grades. The course grade will depend on homework assignments, which will be given periodically.

MATH 595 Higher Algebraic K-theory: classical results Fall 2017 CRN: 58570

Instructor: Jeremiah Heller (jbheller@illinois.edu)

Course description Algebraic K-theory was introduced in the 50’s by Grothendieck for hisRiemann-Roch theorem. Since then it has played an important role in many areas of mathe-matics: algebra, geometry, number theory, and topology. The goal of this class is to providean introduction to Quillen’s higher algebraic K-theory. A (tentative) list of topics we’ll coverincludes:

1. Classical results on K0, K1 of rings.

2. Quillen’s construction of higher algebraic K-theory and fundamental theorems.

3. Quillen’s computation of K-theory of finite fields.

4. Suslin’s computation of K-theory of algebraically closed fields; Suslin-Gabber rigidity.

Prerequisites: Some knowledge of basic algebra and basic homotopy theory.

Recommended text: “The K-book” by Chuck Weibel. (Available on his homepage.)

Math 595-Operator algebra methods in Quantum Information Theory

Instructor: Marius Junge

CRN: 62754/ Section: QITStarting date: October 23

Quantum Information Theory is situated somewhere between physics computer sci-

ence and math. In fact, the most mathematical aspect comes from Shannon’s infor-

mation theory, and the foundations of quantum mechanics. This course has three parts. In the first part we collect basic material from quantum mechanics and in-

formation theory, but little coding theory. In the second part we combine this in covering the basics about entanglement, superdense coding and teleportation. In the last part, we discuss some new aspects coming from functional analysis. However, the main focus here is on finite dimensional spaces and linear algebra, maybe basic finite dimensional Hilbert space theory, is the main prerequisite.

Literature: 1) Quantum Information Theory - Mark M. Wilde, Cambride Univer-

sity Press

2) Quantum Computation and Quantum Information 10 th Anniversary Edition, by

Nielsen and Chang.

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Math 595: Tube Formulas in Convex and Riemannian Geometry

CRN 48012

Fall 2017

Part B (Oct 23 – Dec 13, 2017)

TueThu 9:30-10:50 am

445 Altgeld Hall

Instructor: Jeremy Tyson

Office Location: 329 Altgeld Hall

tyson@illinois.edu

Course Description: Weyl’s tube formula computes the volume of a small tubular neighborhood of a smooth submanifold of Euclidean space. Remarkably, the volume of the scale r neighborhood is a polynomial in r, whose coefficients encode various curvatures of the submanifold. There are similar formulas in convex geometry for the volumes of neighborhoods of convex sets, or more generally, Minkowski sums of convex sets. These sometimes go under the name quermassintegrals or mixed volumes, and go back to the work of Jakob Steiner. In this course we will discuss various tube formulas for neighborhoods of convex sets and neighborhoods of smooth submanifolds in Riemannian manifolds. At the end of the course, time permitting, we will indicate some recent extensions of these ideas to sub-Riemannian manifolds, specifically, the sub-Riemannian Heisenberg group.

Textbook: There is no required textbook. In the first part of the course I will roughly follow the book

Tubes, by A. Gray (2nd edition), Progress in Mathematics, vol. 221, Birkhauser, 2004

Prerequisites: A working knowledge of real analysis and differential geometry (at the level of Math 540 and Math 518) will be assumed. Familiarity with Riemannian geometry (as in Math 519) will be helpful, but we will review all necessary tools of Riemannian geometry as needed. In the final part of the course some prior exposure to the theory of Lie groups may be helpful.

Grades: Grading will be based on attendance and participation. Each student will be asked to give a short presentation on a topic related to the subject of the course during the final week of the semester.