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34th Summer Conference on Topology and its Applications University of the Witwatersrand Johannesburg, Gauteng, South Africa July 1–4, 2019

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Page 1: 34th Summer Conference on Topology and its Applications · 34th Summer Conference on Topology and its Applications University of the Witwatersrand Johannesburg, Gauteng, South Africa

34th Summer Conference on Topology and its

Applications

University of the WitwatersrandJohannesburg, Gauteng, South Africa

July 1–4, 2019

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Contents

Invited Speakers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1Daron Anderson . . . . . . . . . . . . . . . . . . . . . . . . . . . 1Taras Banakh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1Frederic Chazal . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2Szymon Dolecki . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2Mahmoud Filali . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3Osvaldo Guzman Gonzalez . . . . . . . . . . . . . . . . . . . . . 3Rodrigo Hernandez-Gutierrez . . . . . . . . . . . . . . . . . . . . 4Vladimir Itskov . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4Judy Kennedy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4Jan van Mill . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5Seithuti Moshokoa . . . . . . . . . . . . . . . . . . . . . . . . . . 5Dilip Raghavan . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6Francesco G. Russo . . . . . . . . . . . . . . . . . . . . . . . . . . 6Michel Schellekens . . . . . . . . . . . . . . . . . . . . . . . . . . 6Petra Staynova . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7Yusu Wang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

Topological Algebra and Analysis . . . . . . . . . . . . . . . . . . . . . 9T. M. G. Ahsanullah . . . . . . . . . . . . . . . . . . . . . . . . . 9Sergey Antonyan . . . . . . . . . . . . . . . . . . . . . . . . . . . 10Yanga Bavuma . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10Salvador Hernandez . . . . . . . . . . . . . . . . . . . . . . . . . 11Eniola Kazeem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11Noureen Khan . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11Atish J. Mitra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12Seid Kassaw Muhie . . . . . . . . . . . . . . . . . . . . . . . . . . 12Paul Antony Otieno . . . . . . . . . . . . . . . . . . . . . . . . . 12Yuli Rudyak . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13Menachem Shlossberg . . . . . . . . . . . . . . . . . . . . . . . . 13

Asymmetric Structures and Order . . . . . . . . . . . . . . . . . . . . 14Collins Amburo Agyingi . . . . . . . . . . . . . . . . . . . . . . . 14Fikreyohans Solomon Assfaw . . . . . . . . . . . . . . . . . . . . 14S.B.Nimse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15Yae Ulrich Gaba . . . . . . . . . . . . . . . . . . . . . . . . . . . 15Dimitris Georgiou . . . . . . . . . . . . . . . . . . . . . . . . . . 16

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Esteban Indurain . . . . . . . . . . . . . . . . . . . . . . . . . . . 17Minani Iragi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17Gunther Jager . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18Hans-Peter A Kunzi . . . . . . . . . . . . . . . . . . . . . . . . . 19Ralph Kopperman . . . . . . . . . . . . . . . . . . . . . . . . . . 20Sami Lazaar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20Danny Mukonda . . . . . . . . . . . . . . . . . . . . . . . . . . . 20Ando Razafindrakoto . . . . . . . . . . . . . . . . . . . . . . . . . 21Tom Richmond . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21Hope Sabao . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22Katlego Sebogodi . . . . . . . . . . . . . . . . . . . . . . . . . . . 22Josef Slapal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22Nicolo Zava . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

Dynamics and Continuum Theory . . . . . . . . . . . . . . . . . . . . 24Paul Bankston . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24David Herrera-Carrasco . . . . . . . . . . . . . . . . . . . . . . . 24Hisao, Kato . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24Robert Leek . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25Fernando Macas-Romero . . . . . . . . . . . . . . . . . . . . . . . 25Anima Nagar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25Patricia Pellicer-Covarrubias . . . . . . . . . . . . . . . . . . . . 26

Set-Theoretic Topology . . . . . . . . . . . . . . . . . . . . . . . . . . 27Hector Alonzo Barriga-Acosta . . . . . . . . . . . . . . . . . . . . 27Manoj Bhardwaj . . . . . . . . . . . . . . . . . . . . . . . . . . . 27Harsh Vardhan Singh Chauhan . . . . . . . . . . . . . . . . . . . 28Krzysztof Chris Ciesielski . . . . . . . . . . . . . . . . . . . . . . 28Pratulananda Das . . . . . . . . . . . . . . . . . . . . . . . . . . 29Tadeusz Dobrowolski . . . . . . . . . . . . . . . . . . . . . . . . . 29Ivan S. Gotchev . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29Joe Mashburn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30Jissy Nsonde Nsayi . . . . . . . . . . . . . . . . . . . . . . . . . . 30William Obeng-Denteh . . . . . . . . . . . . . . . . . . . . . . . . 31YUAN SUN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31Sumit Singh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31Jonathan Verner . . . . . . . . . . . . . . . . . . . . . . . . . . . 32Lynne Yengulalp . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

Topology in Data Science . . . . . . . . . . . . . . . . . . . . . . . . . 33Justin Curry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33Tamal Dey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33Davorin Lesnik . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33Facundo Memoli . . . . . . . . . . . . . . . . . . . . . . . . . . . 34Nina Otter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35Steve Oudot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35Jose Perea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35Gard Spreemann . . . . . . . . . . . . . . . . . . . . . . . . . . . 36Ziga Virk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

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Invited Speakers

New Indecomposable Continua with Exactly One ComposantDaron AndersonTrinity College [email protected]

Metric indecomposable continua have uncountably many composants, butthe same fails for non-metric continua. In particular there are ZFC examples ofindecomposable Hausdorff continua with exactly one composant. The first wasconstructed by Bellamy in 1978, and there have been no fundamentally differentexamples since. We adapt the methods of Bellamy and Smith to construct twonew examples, which illustrate the disparity between the metric and non-metriccases.

(1) There exists a Bellamy continuum with a strong non-cut point. Thatmeans the point-complement is continuumwise-connected. This contrasts withthe metric case where every point-complement has uncountably many continuum-components.

(2) There exists a Bellamy continuum which – despite being large in thesense of non-metrisable – is small in the sense of being separable.

Complete topologized posets and semilatticesTaras BanakhIvan Franko National University of Lviv (Ukraine) and Jan Kochanowski Uni-versity in Kielce (Poland)[email protected]

Coauthors: Serhii Bardyla

We shall discuss the notion of completeness of topologized posets, and sur-vey some recent results on closedness properties of complete topologized semi-lattices.

References[1] T.Banakh, S.Bardyla, Complete topologized posets and semilattices, preprint(https://arxiv.org/abs/1806.02869).

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Graph classification using a general neural network architecture toprocess persistence diagrams and their representations.Frederic ChazalInria Saclay - [email protected]

Coauthors: Mathieu Carrire (Columbia University), Tho Lacombe and MartinRoyer (Inria) and, Yiuchi Ike and Yuhei Umeda (Fujitsu Labs).

Graph classification is a difficult problem that has drawn a lot of attentionfrom the machine learning community over the past few years. This is mainlydue to the fact that, contrarily to Euclidean vectors, the inherent complexi-ty of graph structures can be quite hard to encode and handle for traditionalclassifiers. Even though kernels have been proposed in the literature, the in-crease in the dataset sizes has greatly limited the use of kernel methods sincecomputation and storage of kernel matrices has become impracticable. In thistalk, we will show that extended persistence diagrams of well-chosen functionsdefined on the vertices of graphs provide a relevant multiscale summary of thegraph structures. Then, building on the recent development of neural networksfor point clouds, we will introduce a neural network architecture for (extend-ed) persistence diagrams which is modular and easy-to-use. We will illustratethe usefulness of our approach by validating our architecture on several graphdatasets, on which the obtained results are comparable to the state-of-the-artfor graph classification. This is a joint work with Mathieu Carrire (ColumbiaUniversity), Tho Lacombe and Martin Royer (Inria) and, Yiuchi Ike and YuheiUmeda (Fujitsu Labs).

Why filter-convergences are indispensableSzymon DoleckiMathematical Institute of Burgundy, Dijon, [email protected]

Already in 1948 Gustave Choquet evidenced inadequacy of topologies in thestudy of hyperspaces. The language of filters enabled him to extend the class oftopologies to that of convergence spaces, which turned out to be closed underseveral important natural operations, for which the class of topologies is not.Probably the most important of these operations is that of exponentiation (theexistence of the coarsest object on the set of continuous maps, for which thenatural coupling is jointly continuous).

Therefore, it is justified to compare the relation between convergence spacesand topologies with that between complex numbers and the reals. Convergencetheory often makes complicated things simple, provides insight, unavailable inthe framework of topologies, and makes it possible to establish important facts,which go far beyond the topological universe.

These advantages will be illustrated here by duality theory of completeness.For example, the completeness number of a convergence space is equal to the

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pseudopaving number of the dual space. In topological framework, completenessnumbers can be arbitrarily large, while the pseudopaving number is always equalto 1.

The fact that non-adherent filters are dual to convergence covers, makes itpossible to grasp a mechanism of preservation of completeness numbers, alsothanks to the Stone transform.

Algebra in the Stone-Cech compactification versus algebra in Banachalgebras with an Arens productMahmoud FilaliUniversity of [email protected]

The Stone-Cech compactification βG of an infinite discrete group G is knownto have a rich algebraic structure. The topological centre of βG (the elementsin βG for which the map y 7→ xy : βG→ βG is continuous) is known to be G.

In the present talk, we explain how the same combinatorial reasons (on thegroup or its dual object) giving the rich algebraic structure to βG and makingthe topological centre of βG as G are the cause of non-Arens regularity (evenextreme) of Banach algebras in harmonic analysis and of the rich algebraicstructure in their second duals.

The talk is based partly on some recent joint work with Jorge Galindo.

The Katetov order on MAD familiesOsvaldo Guzman GonzalezUniversity of [email protected]

An infinite family A ⊆ ℘ (N) is almost disjoint (AD) if the intersectionof any two of its elements is finite and a MAD family is a maximal AD family.Almost disjoint families and MAD families have applications in many branches ofmathematics, for example in forcing theory (almost disjoint coding), in topology(Mrowka–Isbell spaces) and in functional analysis (in the study of masas in theCalkin algebra and constructions of interesting Banach spaces). MAD familiescan be studied using the Katetov order : Given A and B two MAD families,define A ≤K B if there is a function f : N −→ N such that f−1 (A) ∈ I (B) forevery A ∈ I (A) , where I (A) and I (B) are the ideals generated by A and B.The Katetov order is a powerful order for studying MAD families. In this talk,we will survey what is known and what are the open problems regarding thisorder.

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Countable dense homogeneity of functions spacesRodrigo Hernandez-GutierrezUniversidad Autonoma Metropolitana, [email protected]

In this talk we give the first (consistent) example of a countable dense ho-mogeneous space that is metrizable, non-Polish, arcwise connected and infinitedimensional. To be more specific: Let F be a free filter on ω and define XF bea countable space with a unique non-isolated point that has its neighborhoodsdefined by F . Then the function space Cp(XF ) is countable dense homogeneousif and only if F is a non-meager P-filter.

Directed complexes and convex sensing.Vladimir ItskovThe Pennsylvania State [email protected]

What is the embedding dimension, and more generally, the geometry of aset of sequences? This problem naturally arises in the context of neural codingand neural networks. Here one would like to infer the geometry of a space thatis measured (“sensed”) by unknown convex functions.

It turns out that the the information about the geometry, measured by con-vex functions, can be described via combinatorics and topology of directed com-plexes. In particular, the embedding dimension as well as some other geometricproperties of data can be estimated from the homology of an associated directedcomplex. Moreover, each such directed complex gives rise to a multi-parameterfiltration of Dowker complexes that provides a dual topological description ofthe underlying space. I will illustrate these methods in the neuroscience contextof understanding the “olfactory space”. This is a joint work with Philip Eggerand Alexandra Yarosh.

A construction of Mary ReesJudy KennedyLamar [email protected]

Coauthors: Jan Boronski, Piotr Oprocha, Xiaochuan Liu, Ivon Vidal Escobar

Mary Rees made a construction (also called the Denjoy-Rees construction)around 1980 that allowed her to modify a minimal homeomorphism on a torusto produce one with positive entropy. There has been much recent interest inher construction and quite a few new examples of homeomorphisms producedusing it. Quite recently, J. Boronki, J. Cinc, and P. Oprocha used it to producea homeomorphism on a pseudoarc with positive non-infinite entropy. We would

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like to generalize the Rees construction further and really figure out what makesit work, as well as use it to produce more examples.

Splitting Tychonoff cubes into homeomorphic and homogeneous partsJan van MillUniversity of [email protected]

Coauthors: A. V. Arhangelskii

Let τ be an infinite cardinal. We prove that [0, 1] and the Tychonoff cube[0, 1]τ can be split into two homeomorphic and homogeneous parts. If τ isuncountable, and if A is a cover of [01, ]τ such that |A| ≤ τ , then some elementA ∈ A is not homeomorphic to a topological group.

Distance spaces, asymmetric topology and some applicationsSeithuti MoshokoaDepartment of Mathematics and Statistics, Tshwane University of Technology,Arcadia Campus, Pretoria, [email protected]

Coauthors: Fanyana Ncongwane

Partial symmetric spaces were introduced by P. Waszkiewicz in 2003 andtopological properties of these spaces are discussed with applications to domaintheory. Recently in 2019 M. Asim et al presented non-topological properties ofpartial symmetric spaces with applications to fixed point theory. In the talk wepresent further properties of partial symmetric spaces with some applications.

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Order dimension of locally countable partial ordersDilip RaghavanNational University of [email protected]

Coauthors: Kojiro Higuchi, Steffen Lempp, Frank Stephan, Ashutosh Kumar

I will present several results on order dimension, focusing on the locallyfinite and locally countable cases. It will be shown that the order dimension ofall locally countable partial orders of size continuum is no more than the firstuncountable cardinal if the value of the continuum is bounded by the seconduncountable cardinal. On the other hand, it is consistent with the failure of thecontinuum hypothesis that there are locally countable partial orders of size anddimension equal to continuum. As an application, the axioms of ZFC do notdecide the order dimension of the Turing degrees.

Locally Compact Near Abelian GroupsFrancesco G. RussoUniversity of Cape [email protected]

The present talk will describe three theorems of structure for locally compactnear abelian groups. These groups have a fundamental role in the classificationof those locally compact groups, which are rich in pairs of closed commutingsubgroups.

Thermodynamic algorithms: modularity and entropy conservationMichel SchellekensUniversity College [email protected]

Timing-modularity, the capacity to determine the time of an algorithm fromthe times of its parts, is not guaranteed: a part’s time depends on inputs passedon by other parts. Modularity, when it holds, drastically simplifies timing, sup-porting recurrence equations for worst-case time, average case time and secondmoments. Little is known about the intrinsic properties of algorithms for whichthe analysis is guaranteed to to be modular.

Standard algorithm analysis assumes a uniform input distribution. Modularalgorithm analysis requires a similar analysis of each of the algorithm’s parts,for which input data once more need to be uniformly distributed. This provestoo strict a requirement. However, the analysis generalizes to inputs satisfying“local uniformity”, uniform distributions on parts of an input partition, referredto as a “global state”. Algorithms that preserve global states are amenable toan elegant modular analysis. Those that do not, lie at the root of deep open

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problems in algorithmic analysis, including Heapsort. The algorithm has beenredesigned to a global state preserving version, Percolating Heapsort, ensuringgreatly simplified and exact analysis. The redesign is based on the ModularQuantitative Analysis framework MOQA, which consists of: (a) modelling datastructures as partially-ordered finite sets; (b) modelling data on these by topo-logical sorts; (c) considering computation states as finite multisets of such data;(d) analysing algorithms by their induced transformations on states.

In this view, an abstract specification of a sorting algorithm has input s-tate given by any possible permutation of a finite set of elements (represented,according to (a) and (b), by a discrete partially-ordered set together with itstopological sorts given by all permutations) and output state a sorted list of el-ements (represented, again according to (a) and (b), by a linearly-ordered finiteset with its unique topological sort). Series-parallel (SP-)orders form an impor-tant, computationally tractable, class of data structures that help simplify ourmodel’s computations. SP-orders are constructed from a finite discrete orderby repeatedly applying a series and a parallel operation. The generalization ofthese operations to operate over topological sorts of SP-orders, yields a “diffu-sion” and a parallel operation. These form Mod-SP, the least MOQA-fragmentsufficient to construct all topological sorts on SP orders via computations guar-anteed to support modular timing. Mod-SPcomputations, when made reversiblethrough history-keeping, act as closed systems in which entropy is conserved.Thus modularity of timing is linked to entropy conservation of data flow, sharp-ening traditional entropy preservation guaranteed by the second law of thermo-dynamics for reversible systems. Conservation typically holds for the case ofenergy, but not for entropy. Physical systems typically do not obey this law forentropy. A salient point is that for Mod-SP-computations, supporting modularanalysis, entropy is neither created nor destroyed, merely transferred from oneform, i.e. quantitative entropy, to another, i.e. positional entropy. We refer tosuch algorithms as “thermodynamic algorithms”, provide basic examples anddiscuss entropy leakeage and potential connections with topology.

Ideals and DreamsPetra StaynovaUniversity of [email protected]

The Ellis semigroup was introduced by Robert Ellis in the 1960’s. Sincethen, it has been used in several settings throughout topological dynamics, fromthe proof of the Auslander-Ellis theorem to more recent elegant proofs of thetheorems of Malyutin and Ghys-Margulis. However, the relationship betweenclassical dynamical properties and the algebraic structure of the Ellis semigroupare evasive. In the present talk, we will explore some unexpected links betweenthe Ellis semigroup, its minimal ideals, and the dynamics of the underlying

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system. We give examples from substitution dynamics. We will pose some openquestions.

Topological and geometric methods for graph analysisYusu WangThe Ohio State [email protected]

Coauthors: This talk is based on several pieces of work, whose respective co-authors will be given during the talk.

In recent years, topological and geometric data analysis (TGDA) has e-merged as a new and promising field for processing, analyzing and understand-ing complex data. Indeed, geometry and topology form natural platforms fordata analysis, with geometry describing the “shape” and “structure” behinddata; and topology characterizing / summarizing both the domain where dataare sampled from, as well as functions and maps associated to them.

In this talk, I will show how topological and geometric ideas can be used toanalyze graph data, which occurs ubiquitously across science and engineering.Graphs could be geometric in nature, such as road networks in GIS, or relationaland abstract, such as protein-protein interaction networks. I will particularlyfocus on the reconstruction of hidden geometric graphs from noisy data, as wellas graph matching and classification. I will discuss the motivating applications,algorithm development, and theoretical guarantees for these methods. Throughthese topics, I aim to illustrate the important role that geometric and topologicalideas can play in data analysis.

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Topological Algebra and Analysis

On the categories of probabilistic convergence groupsT. M. G. AhsanullahDepartment of Mathematics, College of Science, King Saud University, Riyadh,Saudi [email protected]

Coauthors: Gunther Jager, University of Applied Sciences Stralsund, Stralsund,Germany. Email: [email protected]

Starting with Tardiff’s neighborhood system associated with probabilistic met-ric spaces under continuous triangle function [6-7], and following the notion-s of probabilistic convergence spaces [4] along with the probabilistic uniformconvergence spaces [1], we introduced and studied the notion of probabilisticconvergence groups, their uniformization and metrization [2], and also, proba-bilistic convergence transformation groups [3]. This work is a continuation ofour previous findings [1-4]; in so doing, at present, we are interested to: (a) thecategory of probabilistic neighborhood spaces, PNeigh as well as the categoryPTopNeigh of probabilistic topological neighborhood spaces, and the possibleprobabilistic metrization of probabilistic topological neighborhood spaces, (b)the category of probabilistic neighborhood groups, PNeighGrp, showing thatthis category is isomorphic to a subcategory PPrTopGrp of the category ofprobabilistic convergence groups, PConvGrp, and (c) the category of proba-bilistic Cauchy groups, PChyGrp, showing that the category RK-ChyGrp,herein called category of Richardson-Kent probabilistic Cauchy groups (cf. [5]),is a reflective subcategory of PChyGrp.References[1] T. M. G. Ahsanullah and G. Jager, Probabilistic uniform convergence spacesredefined, Acta Math. Hungar. 146(2)(2015), 376–390.[2] T. M. G. Ahsanullah and G. Jager, Probabilistic uniformization and proba-bilistic metrization of probabilistic convergence groups, Math. Slovaca 67(4)(2017),985–1000.[3] T. M. G. Ahsanullah and G. Jager, Probabilistic convergence transformationgroups, Math. Slovaca 68(6)(2018), 1447–1464.[4] G. Jager, A convergence theory for probabilistic metric spaces, Quaest.Math. 38(4)(2015), 587–599.[5] G. D. Richardson and D. C. Kent, Probabilistic convergence spaces, Austral.J. Math. Soc. (Series A) 61(1996), 400–442.[6] R. M. Tardiff, Topologies for probabilistic metric spaces, Pacific J. Math.65(1)(1976), 233–251.[7] B. Schweizer and A.Sklar, Probabilistic metric spaces, North Holland, NewYork, 1983.

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Simultaneous extension of equivariant mapsSergey AntonyanXATU, [email protected]

Coauthors: Lili Zhang

Let X be a metrizable space, A a closed subset of X, and L a locally convextopological vector space. Let C(X,L) denote the vector space of continuousfunctions from X into L, and similarly for C(A,L). We equip these functionspaces with the compact-open topology. The famuous Dugundji extension the-orem asserts that for every f ∈ C(A,L) there exists Λ(f) ∈ C(X,L such thatΛ(f)|A = f and ImΛ(f) ⊂ conv(Imf). In 1953, E. Michael and R. Arensindependently observed that the map Λ : C(A,L) → C(Z,L) constructed byDugundji is, in fact, a linear homeomorphic embedding. In this talk we willdiscuss how to extend this result to the category of G-spaces. We will provethat an analogous equivariant result is true when the acting group G is compactLie. By an example, we will show that the result may fail to be true when G isa compact non-Lie group. The speaker thanks the Department of Mathematicsof the Xi’an Technological University, where this joint research with Lili Zhangwas completed.

A note on the Hawaian groupsYanga BavumaUniversity of Cape [email protected]

Coauthors: My supervisor Dr Francesco G Russo.

We show an alternative argument for some of the main results of [A. Babaee,B. Mashayekhy, H. Mirebrahimi and H. Torabi, On a Van Kampen theorem forHawaiian groups, Topol. Appl. 241 (2018), 252–262]. This new approach is dueto the presence of notions of geometric group theory, which involve topologicallyfinitely presented groups.

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Subdirect products of finite abelian groupsSalvador HernandezUniversitat Jaume [email protected]

Coauthors: Maria V. Ferrer

A subgroup G of a product∏i∈N

Gi is rectangular if there are subgroups Hi of

Gi such that G =∏i∈N

Hi. We say that G is weakly rectangular if there are finite

subsets Fi ⊆ N and subgroups Hi of⊕j∈Fi

Gj that satisfy G =∏i∈N

Hi. In this

paper we discuss when a closed subgroup of a product is weakly rectangular.Some possible applications to the theory of group codes are also highlighted.

Subgroup commutativity degree of profinite groups.Eniola KazeemUniversity of Cape [email protected]

We find a probability measure which counts the pairs of closed commutingsubgroups in infinite groups. This measure turns out to be an extension of whatwas known in the finite case as subgroup commutativity degree. The extremalcase of probability one describes the so-called topologically quasihamiltoniangroups and is a useful tool in describing the distance of a profinite group fromthis special class. We have been inspired by an idea of Heyer in the context ofour problem.

Rational moves and p-coloring of virtual rational tanglesNoureen KhanUniversity of North Texas at [email protected]

Coauthors: NA

The theory of rational tangles was introduced by John Conway during hiswork on enumeration and classification of knots. We study virtual generalizationof rational tangles that form basis for equivalence classes of virtual knots andlinks. We introduce p-coloring and rational moves of virtual rational tangles,and use elementary combinatorics to classify virtual 3-braids.

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Geometric Groupoid Models for some C∗ AlgebrasAtish J. MitraMontana Tech, [email protected]

Coauthors: K. Austin

We use ideas from geometric topology to construct groupoid models forcertain C∗ algebras. The distinctive features of our method is that our groupoidmodeling is functorial, and easy to visualize geometrically. We demonstrate ourtechniques to provide groupoid models of some well known C∗ algebras builtfrom inductive systems.

The probability of commuting subgroups in arbitrary lattices of sub-groups.Seid Kassaw MuhieUniversity of Cape [email protected]

Coauthors: Dr.Francesco G. Russo

A finite groupG in which two randomly chosen subgroupsH andK commuteis very close to be abelian and its structure has been classified by Iwasawa in1943. It is possible to define a probabilistic notion, which measures how faris G from the groups studied by Iwasawa. Here we introduce the generalizedsubgroup commutativity degree gsd(G) for two arbitrary sublattices S(G) andT(G) of the lattice of subgroups L(G) of G. Upper and lower bounds for gsd(G)are shown and we study the behaviour of gsd(G) with respect to subgroups andquotients, discussing some numerical restrictions.

Homotopy type of mapping spaces between complex GrassmanniansPaul Antony OtienoStrathmore University, P.O Box 59857-00200, [email protected]

Coauthors: Prof. J-B Gatsinzi Prof. Onyango-Otieno Vitalis

The complex Grassmann Gr(k, n) is the space of k dimensional subspacesof Cn. It is a complex manifold of complex dimension k(n − k). There is anatural inclusion ik,n : Gr(k, n) ↪→ Gr(k, n+ r). In this paper, we use Sullivanmodels to compute the rational homotopy type of the component of the inclusionGr(2, n) ↪→ Gr(2, n+ r) in the space of mappings from Gr(2, n) to Gr(2, n+ r),r ≥ 1. We show in particular that

map(Gr(2, n), Gr(2, n+ 1); in

)has the rational homotopy type of a product of

odd spheres.

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Topological Complexity - History and Recent DevelopmentsYuli RudyakUniversity of [email protected]

Topological complexity TC(X) is a numerical topological invariant of a (topo-logical) space X that was invented in the beginning of 2000th by M. Farber. Thisinvariant is related (in fact, stimulated) with the motion planning problem ofrobotics. In greate detail, TC(X) is an integer reflecting the complexity of mo-tion planning algorithms for all systems (robots) having X as their configurationspace. Currently, methods of computation of TC are developed, and interestinggeneralizations of TC are presented. All these issues will be exposed in thissurvey.

Algebraic entropy on topologically quasihamiltonian groupsMenachem ShlossbergShamoon College of Engineering, Be’er Sheva, [email protected]

Coauthors: Wenfei Xi and Daniele Toller

We study the algebraic entropy of continuous endomorphisms of compactlycovered, locally compact, topologically quasihamiltonian groups. We provide aLimit-free formula which helps us to simplify the computations of this entropy.Moreover, several Addition Theorems are given. In particular, we prove thatthe Addition Theorem holds for endomorphisms of quasihamiltonian torsionFC-groups (e.g., Hamiltonian groups).

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Asymmetric Structures and Order

The Hausdorff measure of Isbell-convexity for a quasi-pseudometricspaceCollins Amburo AgyingiDepartment of Mathematics and Applied Mathematics, Nelson Mandela Uni-versity (Summerstrand Campus South), P.O. Box 77000, Port Elizabeth 6031,South [email protected]

Very recently, it has been shown that every T0-quasi-metric space X has aq-hyperconvex hull, QX which is join-compact whenever X is totally bounded.Moreoever, every q-hyperconvex space is bi-complete. In this note, a numericalparameter is introduced which enables us to determine when a bi-complete T0-quasi-metric space is q-hyperconvex.

Codenseness and openness with respect to an interior operatorFikreyohans Solomon AssfawDepartment of Mathematics and Applied Mathematics, University of the West-ern Cape, South [email protected]

Coauthors: David Holgate

Working in an arbitrary category endowed with a fixed (E, M)-factorizationsystem such that M is a fixed class of monomorphisms, we first define and studya concept of codense morphisms with respect to a given categorical interioroperator i in the sense of [1]. Some basic properties of these morphisms arediscussed. In particular, it is shown that i-codenseness is preserved under bothimages and dual images under morphisms in M and E, respectively. We thenintroduce and investigate a notion of quasi-open morphisms with respect to i.We prove that these morphisms are a generalization of the i-open morphismsthat are studied in [2]. We also show that morphisms which are both i-codenseand quasi i-open are i-open. Moreover, we obtain a characterization of quasii-open morphisms in terms of i-codense subobjects. Examples in topology andalgebra are provided.References[1] S.J.R. Vorster. Interior operators in general categories. Quaest. Math.,23(4):405-416, 2000.[2] G. Castellini.Interior operators, open morphisms and the preservation prop-erty. Appl. Categ. Structures., 23(3):311-322, 2015.

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Unified Approach to Topological StructuresS.B.NimseFormer Vice Chancellor, SRTM University Nanded and Lucknow University,Lucknow, [email protected]

Coauthors: Dr.S.V.Ingale Assistant Professor, New Arts, Commerce and Sci-ence College, Ahmednagar, Maharshtra , India

F.Hausdorff (1914), defined topology on a set, introducing neighbourhoodsystem of a point axiomatically. Notions such as uniform continuity, uniformconvergence and Cauchy sequences can not be generalized to arbitrary topologi-cal spaces. A. Weil (1937) introduced a new class of structures, the Uniformities,which constitute a generalization of metric structures and permit at the sametime of definition of the ideas mentioned above (uniform continuity etc.) V.A.Efremovic (1952) introduced the Proximity structure which is further developedby Smirnov and others. Proximity structure occupy an intermediate positionbetween Topologies and Uniformities. Axioms for Proximity structure are theabstraction of intuitive idea of Nearness between two sets. In 1974, H. Her-rrlich introduced the concept of Nearness, by abstracting nearness amongst thecollections of sets. Attempt to define a more generalized structure of whichTopological Uniformities and Proximity structures are particular cases, led tosearch of a common concept order (transitive relation). Using notion of order,A. Csaszar (1957), defined Syntopogeneous Structure. Topological, Proximityand Uniform structures can be derived as particular cases of SyntopogeneousStructures by putting suitable conditions on order Syntopogeneus Structure.In this paper, authors attempted to obtain Nearness Structures by introducingconditions on order in Syntopogeneuos like structures. This approach helps tounderstand Nearness spaces more vividly, establishing relations with other threestructures.References[1] A. Csaszar. Sur une classe de structures topologiques generals, Revue Math.Pures Appl. 2 (1957), 399 407.[2] A. Csaszar. Foundations of General Topology, Pergamon Press, 1963.[3] H. Herrlich. A Concept of Nearness , Topology Appl., 4 (1974), 191- 212.

Algebraic considerations in the m-splitting theory.Yae Ulrich GabaNorth West [email protected]

We continue our investigation on those (partially) ordered metric spaces(X,m,≤) for which there exists an m-splitting T0-quasi-metric d on X for whichx ≤ y if and only if d(x, y) = 0 for any x, y ∈ X. In the framework of the m-splitting-theory, we give an explicit simple algorithm to compute the largest m-splitting T0-quasi-metric D(m,≤), respectively in the case where the underlining

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ordered metric space (X,m,≤) is a lattice and the case when the underlining(X,m,≤) is a group (X, ∗).

A study of the small inductive dimension in the area of frames andfinite latticesDimitris GeorgiouDepartment of Mathematics, University of Patras, 265 04 Patras, [email protected]

Coauthors: A. Megaritis, G. Prinos, F. Sereti

Abstract. The Dimension Theory in the class of frames was the base of manyresearches. Especially, the large inductive dimension, the covering dimensionand the quasi covering dimension for frames and finite lattices have been studiedextensively (see [1-8]).

In this paper, we study the notion of the small inductive dimension, ind,for regular frames, as it is presented in [4], in combination with a differentmeaning of the small inductive dimension, frind, for the class of all frames, asit is presented in [9], comparing these two meanings.

Also, we insert and study the notion of the small inductive dimension, ind,in the class of all finite lattices.

References

[1] Banaschewski B., Gilmour G., Stone-Cech compactification and dimensiontheory for regular σ-frames, J. London Math. Soc. (2) 39 (1989), 1-8.

[2] Boyadzhiev D., Georgiou D. N., Megaritis A. C., Sereti F., A study of acovering dimension of finite lattices, Applied Mathematics and Computation333 (2018), 276-285.

[3] Boyadzhiev D., Georgiou D. N., Megaritis A. C., Sereti F., A study of thequasi covering dimension of finite lattices, accepted for publication in Compu-tational and Applied Mathematics, 2019.

[4] Brijlall D., Baboolal D., Some aspects of dimension theory of frames, IndianJ. Pure Appl. Math. 39 (5) (2008), 375-402.

[5] Brijlall D., Baboolal D., The Katetov-Morita theorem for the dimension ofmetric frames, Indian J. Pure Appl. Math. 41 (3) (2010), 535-553.

[6] Charalambous M. G., Dimension theory of σ-frames, J. London Math. Soc.(2) 8 (1974), 149-160.

[7] Dube T., Georgiou D. N., Megaritis A. C., Moshokoa S. P., A study ofcovering dimension for the class of finite lattices, Discrete Mathematics, 338(2015), 1096-1110.

[8] Georgiou D., Kougias I., Megaritis A., Prinos G., Sereti F., A study of a newdimension for frames, accepted for publication in Topology and its Applications,2019.

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[9] Georgiou D. N., Iliadis S. D., Megaritis A. C., Sereti F., Small inductivedimension and universality on frames, accepted for publication in Algebra Uni-versalis, 2019.

Open problems in the numerical representations of ordered struc-tures.Esteban IndurainPublic University of Navarre, Pamplona (SPAIN)[email protected]

Coauthors: Maria J. Campin

Different kinds of orderings, namely total preorders, interval orders andsemiorders are considered on an abstract nonempty set X. Each ordering actson X as a qualitative scale to compare elements. A classical framework studiesthe possibility of converting those qualitative scales into numerical or quanti-tative ones by means of one or more suitable real-valued functions, known asutility functions, isotonies, or, more generally, numerical representations. Theexistence of such representations is usually characterized in topological termsthat depend on topologies directly defined from the ordering stated a priori onX. Characterizations of the representability of total preorders, interval ordersand semiorders have already been given in that literature. However, many addi-tional problems (as, e.g., the existence of continuous numerical representations)remain still open. In our presentation we intend to furnish an updated accountof the state-of-art of this theory, pointing out the most important questions thathave not been solved yet.

Quasi-uniformities determined by closure and interior operatorsMinani IragiUniversity of the Western [email protected]

Coauthors: David Holgate

The introduction of a categorical quasi-uniform structure ([4]) and its re-lationship with idempotent closure operators ([1]) together with the fact thatevery topological space admits at least a quasi-uniformity, raise the questionof compatibility of a quasi-uniformity with a closure operator in a category.The purpose of this talk is to address this question. We shall prove that forany idempotent closure c ( interior i) operator on a category C with a proper(E, M)-factorization system there is a coarsest and a finest transitive quasi-uniformity on C compatible with c ( i). The result is used to characterize thoseidempotent closure and interior operators that admit a unique transitive quasi-uniformity on C. We also find a condition under which a topogenous order ([5])is compatible with a quasi-uniformity.

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References[1] D. Dikranjan and E. Giuli, Closure operators I, Topology and its Applica-tions, 27(2):129–143, 1987.[2] P. Fletcher and W. F. Lindgen, Quasi-uniform spaces, Lectures notes in PureApll. Math.77, Dekker, New York (1982).[3] D. Holgate and M. Iragi, Quasi-uniform structures and functors, QuaestionesMathematicae (under review), 2019.[4] D. Holgate and M. Iragi, Quasi-uniform and syntopogenous structures oncategories, Topology and its Applications (Accepted), 2018.[5] D. Holgate, M. Iragi and A. Razafindrakoto, Topogenous and nearness struc-tures on categories, Appl.Categor. Struct. 24 (2016), 447–455.

Wijsman convergence in the hyperspace of closed sets of an L-orderedsetGunther JagerSchool of Mechanical Engineering, University of Applied Sciences Stralsund,18435 Stralsund, [email protected]

For a commutative and integral quantale L = (L,≤, ∗) an L-ordered set (L-preordered set [7], L-category [2], L-metric [1]) is a set X with an L-relationd : X × X −→ L which is reflexive (d(x, x) = > for all x ∈ X) and transitive(d(x, y) ∗ d(y, z) ≤ d(x, z) for all x, y, z ∈ X). An L-ordered set has a naturalunderlying topology and thus we can define closed sets. In this talk we lookat a generalization of Wijsman convergence in the hyperspace of closed sets inthis context. To this end, we follow an approach of R. Lowen [5,6] and de-fine a quantale-valued convergence structure as initial structure w.r.t. “distancefunctionals” d(x,A). As in the “classical” metric case, for such initial construc-tions we have to use a suitable supercategory of the category of L-ordered sets.This supercategory of topological L-convergence spaces can be considered as anL-valued version of the category of Lowen’s approach spaces [5], see [4]. Wecharacterize the resulting L-Wijsman convergence structure in terms of neigh-bourhoods and in terms of systems of L-orders, show its quasi-uniformizabilityand discuss its relation to the L-Hausdorff structure [1].

References[1] A. Akhvlediani, M. M. Clementino and W, Tholen, On the categorical mean-ing of Hausdorff and Gromov distances, Topology and its Applications 157(2010) 1275 – 1295.[2] D. Hofmann, G.J. Seal and W. Tholen, Monoidal topology - A categoricalapproach to order, metric and topology, Cambridge University Press, Cambridge2014.

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[3] G. Jager and T.M.G. Ahsanullah, Characterization of quantale-valued metricspaces and quantale-valued partial metric spaces by convergence, Applied Gen.Topology 19(1), 129 – 144, 2018.[4] H. Lai and W. Tholen, Quantale-valued topological spaces via closure andconvergence, Topology Appl. 230 (2017), 599 – 620.[5] R. Lowen, Index Analysis, Springer-Verlag London 2015.[6] R. Lowen and M. Sioen,The Wijsman and Attouch-Wets topologies on hy-perspaces revisited, Top. Appl. 70 (1996), 179 – 197.[7] D. Zhang, An enriched category approach to many valued topology, FuzzySets and Systems 158 (2007), 349 – 366.

Symmetric connectedness in T0-quasi-metric spacesHans-Peter A KunziDept. Math. Appl. Math., University of Cape Town, Rondebosch 7701, [email protected]

Coauthors: Filiz Yildiz, Dept. Math., Hacettepe University, 06800 Beytepe,Ankara, Turkey

We introduce the property of symmetric connectedness for a T0-quasi-metricspace. We present some methods in order to find the symmetrically connectedpairs of a T0-quasi-metric space.

We also show that the problem to determine the symmetry components ofpoints turns out to be easier when formulated for the induced T0-quasi-metric ofan asymmetrically normed real vector space. In addition, as a kind of oppositeto the notion of a metric space, we define antisymmetric T0-quasi-metric spaces.Subsequently some useful results about antisymmetry can be emphasized bydescribing the property of antisymmetric connectedness for a T0-quasi-metricspace.

Finally, we observe that there are natural relations between the theory of(anti)symmetrically connected T0-quasi-metric spaces and the theory of con-nectedness in the sense of graph theory.

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Approximation using AspigoriesRalph KoppermanCity College of New York (retired)[email protected]

An aspigory is a category except that its objects need not have identitymaps.

We use this formalism to discuss known approximations of the reals by finitespaces of decimals and of normal topological spaces by finite topological spaces.

The topological group of autohomeomorphisms of homogeneous func-tionally Alexandroff spacesSami LazaarUniversity of Tunis El [email protected]

Coauthors: Houssem Sabri

In this paper, we study topological properties of the group of auto-homeomorphismsof functionally Alexandroff spaces. In particular, when the functionally Alexan-droff space (X,P (f)) is connected homogeneous space where the map f is k- toone with k > 1 is a finite cardinal number, then we prove that its autohomeomor-phism groupH(X) equipped with the permutation topology is second countable,totally disconnected, locally compact (s.c.t.d.l.c) non discrete Hausdorff group.Furthermore in this case or even when k is an infinite cardinal number, we provethat the topology of pointwise convergence, the compact open topology and thepermutation topology which equip the topological group H(X) are equivalent.Finally, one of the main results of this paper is to prove that the group H(X)is not locally compact when k is an infinite cardinal number. Consequently, wededuce that the group of all isometries of a regular tree is locally compact ifand only if the tree is locally finite.

Bornologies and asymmetric structuresDanny MukondaSchool of Mathematics-University of the [email protected]

Coauthors: O Olela Otafudu and W Toko

Over the last sixty years much progress has been made in the investigationsof bornology on a set. In particular, Hu, Galido, Levy, Costantini and Beer havepublished several articles concerning bornologies on metric spaces. For example,Hu in his paper investigated the concept of bornology on a metric space thatcorresponds to a bornology of the bounded set. Beer continued the work ofHu by studying the bornologies of extended metric spaces. Moreover, Piekosz

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and Wajch have shown that the Hu’s metrization theorm on bornologies can beadopted to quasi-metric spaces in terms of bitopological spaces. In this talk,we continue the work of Piekosz and Wajch and prove that most of the resultsof Beer can still be achieved without using the symmetric axiom of a metricspace. In short, with appropriate modifications, we obtain the Beer’s results inthe framework of quasi-metric spaces.

Ultrafilter spaces and reflective subcategoriesAndo RazafindrakotoUniversity of the Western [email protected]

Coauthors: MS Nxumalo

Ultrafilter spaces give rise to a monad (or triple) on the category of topo-logical spaces and continuous functions. Under mild conditions, this provides away to obtain certain standard compactifications.

The compactifications obtained are thus functorial and describe compactspaces (with suitable separation conditions) as algebras. We give certain con-ditions under which the compactification itself forms a monad, and list a fewexamples.References[1] S. Salbany, The completion monad and its algebra, Commentationes Math-ematicae Universitatis Carolinae 023.2 (1982), 301-311.

The lattice of functionally Alexandroff topologiesTom RichmondWestern Kentucky [email protected]

Coauthors: Jacob Menix

If f : X → X is a function, the associated functionally Alexandroff topologyPf on X is the topology whose closed sets are {A ⊆ X : f(A) ⊆ A}. If X isfinite, we present some cardinality considerations for | Pf | and show that thecollection FA(X) of all functionally Alexandroff topologies on X, ordered byinclusion, is a complemented lattice.

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Some aspects of quasi-uniform box productsHope SabaoUniversity of [email protected]

Coauthors: O Olela Otafudu

The quasi-uniform box product is a topology on a product of countablymany copies of a quasi-uniform space that is finer than the Tychonov producttopology but coarser than the uniform box product. In this talk, we presentsome properties of a quasi-uniform space that are preserved by its quasi-uniformbox product.

Isbell-convexity in modular quasi-metric spacesKatlego SebogodiSchool of Mathematics, University of the [email protected]

Coauthors: O Olela Otafudu

Few years ago, Kemajou, Kunzi and Olela Otafudu investigated the con-cept of hyperconvexity in quasi-pseudometric spaces which they called Isbell-convexity (or q-hyperconvexity). In this talk, we introduce the concept of Isbell-convexity in the setting of modular quasi-pseudometric spaces (which we callw-Isbell-convexity). We discuss the connections between Isbell-convexity andw-Isbell-convexity on a given modular set.

A digital Jordan surface theorem with respect to a closure operatoron Z3

Josef SlapalBrno University of [email protected]

We introduce closure operators on the digital line Z and show that they areassociated with n-ary relations on Z (n > 1 an integer). We focus on the closureoperator u associated with a ternary relation on Z. We equip the digital spaceZ3 with the closure operator obtained as a special product of three copies of u.The closure operator is shown to allow defining digital surfaces that satisfy adigital Jordan surface theorem.

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Asymptotic dimensions of preordered sets and quasi-coarse spacesNicolo ZavaUniversity of [email protected]

Coauthors: Ziga Virk

Coarse geometry is the study of global, large-scale properties of spaces. Thistheory found many applications in mathematics, for example in geometric grouptheory and in geometric topology. Coarse spaces were introduced by Roe to en-code and generalize the large-scale properties of metric spaces. However, coarsespaces are not suitable objects to describe interesting asymmetric spaces, for ex-ample, quasi-metric spaces and monoids. Quasi-coarse spaces have been recentlyintroduced to fill the gap, extending coarse spaces. In this talk, basic defini-tions and results concerning quasi-coarse spaces will be provided. Moreover, wewill introduce and discuss notions of asymptotic dimension that generalize theclassical corresponding definition for coarse spaces. In particular, we will focuson their applications to preordered sets.

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Dynamics and Continuum Theory

Replete continuaPaul BankstonMarquette University, [email protected]

A (Hausdorff) continuum is replete if it possesses as many composants aspoints. It is well known that each indecomposable metrizable continuum isreplete; here we discuss the existence of replete continua that are nonmetrizable.

Properties of (n,m)–fold hyperspace suspensionDavid Herrera-CarrascoBenemrita Universidad Autnoma de Puebla, Puebla, [email protected]

Coauthors: Fernando Macas-Romero, Antonio de J. Libreros-Lpez.

For a metric continuum X and n ∈ N, we consider the hyperspaces Cn(X)(respectively, Fn(X)) of all nonempty closed subsets of X with at most n compo-nents (respectively, n points). Given positive integers n and m such that m ≤ n,let HSnm(X) be the quotient space Cn(X)/Fm(X) with the quotient topology.In this talk we present some other property and we prove: if X is meshed con-tinua, Y is a continuum and HSnm(X) is homeomorphic to HSnm(Y ), then X ishomeomorphic to Y . We will also prove that, there exists X an almost meshedlocally connected continuum such that X not has unique hyperspace HSnm(X).

Topological entropy and IE-tuples of G-like continua for a graph GHisao, KatoUniversity of [email protected]

In this talk, for a graph G we define a new notion of ”free tracing propertyby free G-chains” on G-like continua and we prove that a positive topologicalentropy homeomorphism f of a G-like continuum X admits a Cantor set Z inX and an indecomposable subcontinuum H of X satisfying the following con-ditions (1)-(4);(1) Z has the free tracing property by free G-chains,(2) H is the unique minimal subcontinuum of X containing Z and no two pointsof Z belong the same composant of H,(3) any sequence (z1, z,..., zn) of points in Z is an IE-tuple of f , and(4) f is Li-Yorke chaotic on Z.Also we give a characterization theorem of G-like continua containing indecom-posable subcontinua.

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Revisiting the Auslander-Yorke dichotomyRobert LeekBirmingham, United [email protected]

Coauthors: Chris Good and Joel Mitchell

The Auslander-Yorke dichotomy states that a compact, metric, minimalsystem is either equicontinuous or sensitive. Transitivity plus sensitivity is oftenviewed as (parts of) one of the main definitions of chaos. In this talk, I willdiscuss recent work on equicontinuity and even-continuity pairs and local aspectsof stability.

Finite graphs have unique n-fold symmetric product suspensionFernando Macas-RomeroBenemrita Universidad Autnoma de Puebla, Puebla, [email protected]

Coauthors: David Herrera-Carrasco, German Montero-Rodrıguez

Let X be a continuum and let n ∈ N. We consider the hyperspace Fn(X) ofall nonempty closed subsets of X with at most n points. Given n ∈ N−{1}, then-fold symmetric product suspension of X is the quotient space Fn(X)/F1(X)and it is denoted by SFn(X). In this talk we prove: if X is a finite graph andY is a continuum such that SFn(X) is homeomorphic to SFn(Y ), then X ishomeomorphic to Y .

Strongly Transitive SystemsAnima NagarIndian Institute of Technology [email protected]

Strongly transitive systems are those that have dense backward orbits forevery point in the phase space. Some properties of such systems were recentlystudied. We add up to this theory.

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The hyperspace of nontrivial convergent sequencesPatricia [email protected]

Coauthors: D. Maya and R. Pichardo-Mendoza

Given a Hausdorff space X, we denote by Sc(X) the hyperspace of nontrivialconvergent sequences in X, equipped with the Vietoris Topology. In this talkwe present some general properties of this hyperspace, and we compare themwith those of some well known hyperspaces..

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Set-Theoretic Topology

Monotone normality on nabla spacesHector Alonzo Barriga-AcostaJoint Posgraduate in Mathematical Sciences [email protected]

Coauthors: Dr. Paul Gartside

We study the monotone normality on countable-nabla spaces. Monotonenormality is much stronger than normality. This is an aproach to understand alittle more the problem of normality on countable box products (Is the countablebox product of a compact metrizable space normal?) from a different point ofview. This problem has been studied since the 40’s and many concistent resultsconcerning to it’s paracompacness (and thus, it’s normality) have been proven.It looks like monotone normality plays an important role in this matter.

Relativization of star-Menger property in topological spacesManoj BhardwajUniversity of Delhi, Delhi, [email protected]

Coauthors: Dr. B. K. Tyagi, Sumit Singh

In this paper, we continued the study initiated in (Acta Math. Hungar.,2007) by M. Bonanzinga and B. A. Pansera. A subspace A of a topologicalspace X is said to have relatively star-Menger property (Acta Math. Hungar.,2007) in X (in short, RSM) or relatively star-Menger subspace of X if for eachsequence < Un : n ∈ ω > of open covers of X there is a sequence < Vn : n ∈ ω >such that for each n ∈ ω, Vn is a finite subset of Un and for each x ∈ A,belongs to St(

⋃Vn,Un) for some n. It is shown that relatively star-Menger

property is hereditary. We also studied preservation properties of star-Mengerproperty under two mappings. This property is studied in Alexandroff spaceand Isbell space. In this manner we obtained relationships of relatively stronglystar-Menger property with other existing Menger properties in literature.

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On connected and hyperconnected generalized topological spacesHarsh Vardhan Singh ChauhanUniversity of Delhi, Delhi, [email protected]

In this paper, I have studied non strong hyperconnected generalized topolog-ical spaces. Ekici (Acta Mathematica Hungarica, 2011) and Devi (Acta Math-ematica Hungarica, 2012) have provided the results of hyperconnectedness forstrong generalized topological spaces. I generalized these results for non stronggeneralized topological spaces. Through the notion of hyperconnectedness ofnon strong generalized topological spaces, I have constructed an example whichfails Hausdorff charactrization of topological spaces “A first countable spacesis Hausdorff if and only if every convergent sequence has unique limit”. Thisexample also serves the purpose of constructing anti Hausdorff Frechet space inwhich every convergent sequence has unique limit required by Novak in (Publ.Fac. Sci. Univ. Masaryk, 1968).

A century of Sierpiinski-Zygmund functionsKrzysztof Chris CiesielskiWest Virginia [email protected]

Coauthors: J.B. Seoane–Sepulveda

In the talk I will summarize the results presented in the survey “A century ofSierpiinski-Zygmund functions” which draft is available at https://math.wvu.edu/ k-cies/SubmittedPapers/SS36.SZsurvey.pdf

Sierpiinski–Zygmund, SZ, functions are the maps from R to R that have “aslittle continuity” as possible. In the survey we discuss the history behind theirdiscovery, their constructions, and generalizations. The presentation emphasizesthe algebraic properties of SZ maps and their relation to different classes ofgeneralized continuous-like functions. From the seminal work of H. Blumbergand the appearance of Sierpiinski–Zygmund’s result, we describe the currentstate of the art of this century-old class of functions and discuss the impact thatit has had on several different directions of research. Many typical proofs usedin the theory, often in a simplified format never published before, are included inthe presented material. Moreover, open problems and new directions of researchare indicated.

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Some observations on Hurewicz and I-Hurewicz propertyPratulananda DasDepartment of Mathematics, Jadavpur University, Kolkata - 700032, West Ben-gal, [email protected]

Coauthors: Upasana Samanta and Debraj Chandra

In this talk, we primarily consider the Hurewicz property and its ideal coun-terpart introduced by Das in (Houston J. Math., 2013). We present charac-terizations of Hurewicz and I-Hurewicz property regarding selection principlesterms involving large covers, I-γ and I-groupable open covers for certain typesof ideals. We present some preservation properties and make new observations.

On Countable Dense Homogeneity via InvolutionsTadeusz DobrowolskiPittsburg State [email protected]

Recall that a space is bihomogenous if for any x and y there exists a home-omorphism swapping x and y. A local version of bihomogeneity is defined sim-ilarly to that of local homogeneity. Similarly, counterparts of countably densehomogeneity (CDH) exist in the bihomogeneous framework. Recall that a s-pace is CDH if any two countable dense sets are like positioned with respect toa homeomorphism of the whole space. We will be interested in two variants ofthis property for bihomogeneity: countable dense bihomogeneity (CDB) and itsweak version. The CDB property means exactly that the positioning mentionabove must be obtained by an involution of the whole space. Preliminary facts,related to both finite and infinite dimensional cases, will be presented (includingthe smooth case).

More cardinal inequalities for Urysohn spacesIvan S. GotchevCentral Connecticut State [email protected]

Recall that for a topological space X, tθ1(X) is the smallest infinite cardinalκ such that for every A ⊂ X and every x ∈ cl(A) there exists a set B ⊂ Asuch that |B| ≤ κ and x ∈ clθ(B). Also, for every Urysohn space X the θ2-pseudocharacter of X, denoted by ψθ2(X), is the smallest infinite cardinal κsuch that for each x ∈ X, there is a collection Vx of open neighborhoods of xsuch that |Vx| ≤ κ and

⋂{clθ(cl(V )) : V ∈ Vx} = {x}.

Using these two cardinal functions, among other results, we show that ifX is a Urysohn space and A ⊂ X, then (1) |cl(A)| ≤ |A|tθ1 (X)ψθ2 (X); and (2)

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|X| ≤ 2tθ1 (X)ψθ2 (X)aLc(X), where aLc(X) is the almost Lindelof degree of Xwith respect to closed sets.

Since ψθ2(X) ≤ ψ(X)L(X), whenever X is a Urysohn space, as a corollaryof (2), we obtain that for every Urysohn space X we have |X| ≤ 2tθ1 (X)ψ(X)L(X).Therefore (2) sharpens, for the class of Urysohn spaces, the famous Arhangel’skiı–Sapirovskiı inequality |X| ≤ 2t(X)ψ(X)L(X), which is valid for every Hausdorffspace X.

Local homogeneity and cardinal boundsJoe MashburnUniversity of [email protected]

A space X is homogeneous if and only if for every p, q ∈ X there is ahomeomorphism h : X → X such that h(p) = q. It is power homogeneous ifand only if Xκ is homogeneous for some κ. It is locally homogeneous if andonly if for every p, q ∈ X there are neighborhoods U of p and V of q and ahomeomorphism f : U → V such that f(p) = q. A locally homogeneous spaceneed not be homogeneous or power homogeneous. We will explore the utility oflocal homogeneity in establishing cardinal bounds and show that several knowncardinal bounds for homogeneous or power homogeneous spaces also hold forlocally homogeneous spaces.

On Characterizations of EF -framesJissy Nsonde NsayiDepartment of Mathematics and Applied Mathematics, University of Limpopo,South [email protected]

In [1], Tahirefar defines a Tychonoff space X to be an EF -space if disjoint u-nions of clopen sets are completely separated. By first showing how this conceptcan be extended to locales in a conservative way, we will give characterizationsof EF -frames, mostly in terms of certain ring-theoretic properties of RL, thering of real-valued functions on L.

References[1] A. Taherifar, Some new classes of topological spaces and annihilator ideals,Topology Appl. 165 (2014), 84–97.

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Omega Limit Sets meet Alpha Limit SetsWilliam Obeng-DentehDepartment of Mathematics, Kwame Nkrumah University of Science and Tech-nology, Kumasi, [email protected]

The meeting point of Omega Limit Sets and Alpha Limit Sets are discussedand some observations depicted.

A study on symmetric products of generalized metric spacesYuan SunBeijing University of [email protected]

We study the relation between a space X satisfying certain generalizedmetric properties (for example, open (G), point-countable base, CollinsRoscoeproperty, semi-stratifiable, k-semistratifiable, semi-metrizable, scattered, point-countable cs-network, every compact set is metrizable) and its n-fold symmetricproduct Fn(X) satisfying the same properties. We also show that if X is anM1-space then F (X) is an M1-space, where F (X) is the hyperspace of finitesubsets of X. A space X is a paracompact p-space if and only if its 2-foldsymmetric product F2(X) is a paracompact p-space. A Tychonoff space X isa Lindelof 931;-space if and only if its 2-fold symmetric product F2(X) is aLindelof 931;-space.

Kocinac αi-properties and the idealsSumit SinghDepartment of Mathematics, University of Delhi, New Delhi-110007, [email protected]

Coauthors: Brij K. Tyagi, Atma Ram Sanatan Dharma College, University ofDelhi, New Delhi-110021, India and Manoj Bhardwaj, Department of Mathe-matics, University of Delhi, New Delhi-110007, India.

Kocinac introduced several αi-properties as a selection principles in (Tai-wanese Journal of Mathematics, 2008) and there were motivated by Arhangel’skiiαi-local properties (Soviet Math. Doklady, 1972). In this paper, we identifiedsome classesA and B of open covers in topological spaces, topological groups, hy-perspaces and abstract boundedness for which the Kocinac αi(A,B)-propertiesare closely related and often equivalent to S1(A,B), using the notion of an ideal.Further we introduced the ideal form of Hurewicz-bounded topological groupand characterize it using these αi-properties. Also the results in the paper ex-tend and improve some earlier results appeared in literature and present a moregeneral version with respect to ideal.

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Structure of the Rudin Keisler order of P-pointsJonathan VernerCharles [email protected]

We show that, under MA, the RK-ordering of P-points is downwards c-closed. This complements an earlier result showing that, under CH, the RK-ordering of rapid P-points is upwards c+-closed. The results presented are jointwork with Dilip Raghavan and Borisa Kuzeljevic.

Varieties of HomogeneityLynne YengulalpUniversity of [email protected]

We explore a local version of homogeneity: We say X is locally homoge-neous if for any points x and y of X there are neighborhoods U and V and ahomeomorphism from U to V taking x to y. We give examples of locally ho-mogeneous spaces that lack other homogeneity properties and generalize knowntechniques from the study of homogeneous and power homogeneous spaces tolocally homogeneous spaces.

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Topology in Data Science

Refining PersistenceJustin CurryUniversity at Albany [email protected]

In this talk I will provide a geometric and categorical introduction to someinverse problems in persistence. After reviewing the Elder Rule in persistenceI will provide a characterization of the fiber of the persistence map in two set-tings: functions defined on the interval and spheres embedded in R3. In thefirst setting, functions are identified up to orientation-preserving homeomor-phism and in the second setting, spheres are identified up to level-set preservingisotopy. Suggestions for topological summaries that enrich persistence and canbe metrized will be discussed.

Generalized Persistence Algorithm for Multi-parameter PersistenceModulesTamal DeyOhio State [email protected]

Coauthors: Cheng Xin

There is no known generalization of the classical matrix reduction basedpersistence algorithm for simplicial filtration in a multi-parameter setting. Wepresent for the first time such a generalization. It improves over the Meataxealgorithm commonly used for the purpose by several orders.

Sampling smooth manifolds using ellipsoidsDavorin LesnikUniversity of [email protected]

Coauthors: Sara Kalisnik

A common problem in data science is to determine properties of a space froma sample. For instance, under certain assumptions a subspace of a Euclideanspace may be homotopy equivalent to the union of balls around sample points,which is in turn homotopy equivalent to the 268; each complex of the sample.This enables us to determine the unknown space up to homotopy type, in par-ticular giving us the homology of the space. A seminal result by Niyogi, Smaleand Weinberger [1] states that if a sample of a closed smooth submanifold of aEuclidean space is dense enough (relative to the reach of the manifold), thereexists an interval of radii, for which the union of closed balls around sample

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points deformation retracts to the manifold. A tangent space is a good localapproximation of a manifold, so we can expect that an object, elongated in thetangent direction, will better approximate the manifold than a ball. We presentthe result that the union of closed ellipsoids of suitable size around sample pointsdeformation retracts to the manifold while requiring much smaller density thanin the case of union of balls. The proof requires new technics, as unlike thecase of balls, the normal projection of a union of ellipsoids is in general not adeformation retraction.References[1] Niyogi, P., Smale, S., Weinberger, S. (2008). Finding the homology of sub-manifolds with high confidence from random samples. Discrete ComputationalGeometry, 39(1-3), 419-441.

Metric Graph Approximations of Geodesic SpacesFacundo MemoliThe Ohio State University. Math and [email protected]

Coauthors: Osman Berat Okutan

A standard result in metric geometry is that every compact geodesic met-ric space can be approximated arbitrarily well by finite metric graphs in theGromov-Hausdorff sense. It is well known that the first Betti number of the ap-proximating graphs may blow up as the approximation gets finer. In our work,given a compact geodesic metric space X, we define a sequence (948;Xn)n8805; 0of non-negative real numbers by 948;Xn := inf{dGH(X,G) : G a finite metricgraph, 946; 1(G)8804;n}.

By construction, and the above result, this is a non-increasing sequencewith limit 0. We study this sequence and its rates of decay with n. We alsoidentify a precise relationship between the sequence and the first Vietoris-Ripspersistence barcode of X. Furthermore, we specifically analyze 948;X0 and findupper and lower bounds based on hyperbolicity and other metric invariants.As a consequence of the tools we develop, our work also provides a Gromov-Hausdorff stability result for the Reeb construction on geodesic metric spaceswith respect to the function given by distance to a reference point.

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The magnitude of a metric spaceNina [email protected]

The magnitude is an isometric invariant of metric spaces that was intro-duced by Tom Leinster in 2010, and is currently the object of intense research.Magnitude encodes many invariants of a metric space such as volume, dimen-sion, capacity, and so on. I will give an overview of existing results and currentresearch in this area; in particular, I will explain how magnitude is related tointrinsic volumes and persistent homology. This talk is partly based on thepreprint https://arxiv.org/abs/1807.01540.

Exact computation of the matching distance on 2-parameter persis-tence modulesSteve [email protected]

Coauthors: Michael Kerber, Michael Lesnick

The matching distance is a pseudometric on multi-parameter persistencemodules, defined in terms of the weighted bottleneck distance on the restrictionof the modules to affine lines. It is known that this distance is stable in areasonable sense, and can be approximated, which makes it a promising toolfor practical applications. In this talk, I will show that in the 2-parametersetting, the matching distance can be computed exactly in polynomial time. Theapproach subdivides the space of affine lines into regions, via a line arrangementin the dual space. In each region, the matching distance restricts to a simpleanalytic function, whose maximum is easily computed. As a byproduct, theanalysis establishes that the matching distance is a rational number, if thebigrades of the input modules are rational.

Persistence, Fiber Bundles and ApplicationsJose PereaMichigan State [email protected]

Many problems in data science can be articulated as learning functions be-tween structured spaces. The goal of this talk is to show how ideas from fiberbundle theory can be used in problems such as dimensionality reduction andmodel fusion.

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The simplicial Laplacian in topological data analysis - clustering con-volutional networks for simplicial complexesGard [email protected]

Coauthors: Stefania Ebli and Michal Defferrard

In topological data analysis, one typically works with (persistent) (co)homologyover finite fields, which certainly delivers some computational advantages. How-ever, by transitioning to working over the reals, one can define Laplacians thatgive access to different tools for data analysis. In this talk I will present twosuch recent developments. The first is a clustering scheme that can be seen asthe simplicial analog of spectral clustering for graphs. The second is a general-ization of convolutional neural networks to the case of simplicial cochains, whichallows a certain class of deep networks to work with learning on cochain data.

Nerve Theorem for Rips Complexes and Geometric Features Encodedin Higher-Dimensional PersistenceZiga VirkUniversity of [email protected]

Given a metric space and a scale, we will construct a cover whose nerve ishomotopy equivalent to the Rips complex at that scale. This will allow us toreconstruct the homotopy type of a large class of spaces using Rips complexesat appropriately small scales.

Going beyond small scales the homotopy structure of Rips complexes start-s getting more complicated. We will explain the emergence of some higher-dimensional persistent homology elements as footprints of geometric featuresof the space. In particular, we can detect (and locate) certain contractiblegeodesics using 2- and 3-dimensional persistent homology.

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