Combinatorics & Topology Seminar at GIST

July 8th

 13:00~13:40 Sunhyuk Lim: Vietoris-Rips complex, hyperconvex space, and quantitative Borsuk-Ulam theorem

The Vietoris-Rips complex, originally introduced by Leopold Vietoris in the early 1900s to develop a homology theory for metric spaces, was later employed by Eliyahu Rips and Mikhail Gromov in their studies of hyperbolic groups. Recently, this complex has become pivotal in Topological Data Analysis (TDA) for its use in constructing persistent homology, a key tool in TDA. To thoroughly understand the strengths and limitations of persistent homology, it is essential to grasp the homotopy type of the Vietoris-Rips complex of manifolds.

In a recent paper by Lim, Memoli, and Okutan, the authors established a homotopy equivalence between the Vietoris-Rips complex and a tubular neighborhood in hyperconvex space, offering a geometric perspective on the complex and linking TDA, Quantitative Topology, and Metric Geometry. Additionally, papers by Lim, Memoli, Smith, and Lim et al. revealed that the infimum scale of the Vietoris-Rips complex of a lower-dimensional sphere, which "contains" a higher-dimensional sphere, can lower bound the discontinuity forced on an odd map from the higher-dimensional sphere to the lower-dimensional sphere.

This talk will explore these recent connections between the Vietoris-Rips complex, hyperconvex space, and the quantitative Borsuk-Ulam Theorem.

13:40~14:20 Woojin Kim: Persistence Diagrams at the Crossroads of Algebra and Combinatorics

Persistent Homology (PH) is a method used in Topological Data Analysis (TDA) to extract multiscale topological features from data. Via PH, the multiscale topological features of a given dataset are encoded into a persistence module and in turn, summarized by a persistence diagram. In order to extend PH so as to be able to study wider types of data (e.g. time-varying point clouds), variations of the indexing set of persistence modules must inevitably occur, leading for example to multiparameter persistence modules. It is however not always evident how to define a notion of persistence diagram for such variants. This talk will introduce a generalized notion of persistence diagram for such variants which arises through exploiting both the principle of inclusion and exclusion from combinatorics and the canonical map from the limit to the colimit of a diagram of vector spaces (these being notions from category theory). We also discuss (1) how the generalized persistence diagram subsumes some other well-known invariants of multiparameter persistence modules and (2) algorithmic considerations for computing the generalized persistence diagram. 

14:20~15:00 Jongbaek Song: Polyhedral product and persistence module

The polyhedral product is a mathematical platform to create a topological space associated with a simplicial complex. In homotopy theory, there are vast literatures about cohomological studies of polyhedral products. Hence, we may expect some additional topological information about data by applying cohomologies of certain polyhedral products to the usual persistence module. In this talk, we will discuss one of those applications, called the persistent Tor-algebra. It can be obtained by the cohomology of a particular type of the polyhedral product, called a moment-angle complex. This talk is based on the joint work with A. Bahri, I. Limonchenko, T. Panov and D. Stanley. 

15:00~15:30 Coffee Break

15:30~16:10 Jisu Kim: Homotopy Reconstruction via the Cech Complex and the Vietoris-Rips Complex 

A fundamental task in topological data analysis, geometric inference, and computational geometry is that of estimating the topology of a target space based on a collection of possibly noisy data points. I derive conditions under which the reconstruction of a target space is topologically correct via the Čech complex or the Vietoris-Rips complex obtained from data. To guarantee topological correctness, I developed two related theoretical results. First, I show that any intersection of balls of a target space with positive reach is contractible under some conditions, so that the Nerve theorem applies for the Čech complex. Second, I demonstrate the homotopy equivalence of a positive μ-reach set and its offsets. By applying these results, I formulate conditions guaranteeing that the target space is homotopy equivalent to the Čech complex or the Vietoris-Rips complex, in terms of the μ-reach. These results sharpen existing results. 

16:10~16:50 Kang-Ju Lee: Combinatorics and Optimization for Mapper in TDA

Mapper is a network-based visualization technique in Topological Data Analysis (TDA), which has shown applications in various fields. The Mapper algorithm generates the nerve of a refinement of the pullback cover induced by a lens function and a cover of its image. The algorithm requires tuning several parameters to generate a ``nice" Mapper graph. In fact, for a given data, the Mapper can generate any graph as output with a particular choice of parameters. Focusing on the cover, we present an algorithm that optimizes the cover of a Mapper graph by splitting a cover repeatedly according to a statistical test for normality.

16:50~17:30 Jinha Kim: Star clusters in independence complexes of hypergraphs

In 2013, Barmak introduced the concept of star clusters in independence complexes of graphs. We generalize Barmak's result, providing an analogous result for the independence complexes of hypergraphs. This implies that for a hypergraph H with a vertex v that is neither isolated nor contained in a Berge cycle of length 3, there exists a hypergraph K with fewer vertices than H such that the independence complex of H is homotopy equivalent to the suspension of the independence complex of K. As an application, we prove that if a hypergraph H has no Berge cycle of length divisible by 3, then the sum of all reduced Betti numbers of its independence complex is at most 1.

July 9th

10:00~10:40 Minki Kim: Some combinatorial properties of d-Leray complexes

Given a field F, an abstract simplicial complex is said to be d-Leray if all its induced subcomplexes have trivial reduced homology groups in dimension d or greater, with coefficients in F.  It is well-known in the theory of discrete geometry that Helly's theorem, along with its colorful and fractional generlizations, holds for d-Leray complexes. In this talk, we present extensions of variants of Helly's theorem for convex sets in d-dimensional Euclidean space to d-Leray complexes. This is based on joint research with Alan Lew.

This is a part of "Combinatorics and Topology Week at GIST" at Gwangju Institute of Science and Technology (GIST) in Gwangju, Republic of Korea from July 7th(Sun) to 13th(Sat) of 2024.