# Topology, Geometry, and Applications - Graduate Student Seminar group

# Welcome to the TAGGS seminar group at Ohio State!

This is an informal seminar group comprising graduate students and postdocs studying a broad range of topics in topology, geometry, and their applications.

Organizers for 2020-2021: Mario Gomez and Francisco Martinez

Organizers for 2019-2020: Francisco Martinez and Kritika Singhal

Organizers for 2019-2020: Samir Chowdhury and Katie Ritchey

Time and place: Friday, 12:30 - 13:30 (ETS, OSU time) via Zoom

To subscribe to announcements and get the zoom links, join our Google Group! Send a blank email to osu-taggs+subscribe@googlegroups.com.

# Spring 2021 Schedule

## Upcoming Talk

**October 8, 2021**

Speaker: Francisco Martinez-Figueroa

## Previous Talks

**September 24, 2020**

Speaker: Mario Gomez

Title: A reinterpretation of the 4-point condition of metric trees as a limit of a generalized Ptolemy's inequality.

Abstract: In the study of persistence diagrams of 4-point subsets of the plane, Ptolemy's inequality is the key tool that provides a bound for the possible persistence diagrams. In 1970, J.E. Valentine published generalizations of Ptolemy's inequality to spherical and hyperbolic geometries, allowing for a similar characterization of the persistence diagrams of their 4-point subsets. On the other hand, tree metric spaces have no persistent homology in non-zero dimensions (as proven by Carlsson et al.). Since metric trees can be seen as spaces with constant curvature -infty, this prompts the question of whether our bounds can recover Carlsson's result, at least in the case of 4 points. The answer is not only positive, but it also leads to a reinterpretation of the 4-point condition of metric trees as the limit of Valentine's generalization of Ptolemy's inequality.

**December 4, 2020 **

Speaker: Johnathan Bush (CSU)

Title: Vietoris-Rips complexes of spheres and Borsuk-Ulam theorems

Abstract: Vietoris-Rips complexes defined on manifolds are generally not well understood, despite the fact that these complexes arise naturally in the context of persistent homology. I will summarize results about Vietoris-Rips complexes, and related spaces, defined on spheres. Then, I will outline an interesting connection between these spaces and generalizations of the Borsuk-Ulam theorem.

**November 20, 2020 **

Speaker: Ling Zhou

Title: Filtered chain complexes and verbose barcodes

Abstract: Persistent homology is a method to extract topological information of a data set at many different geometric scales. The lifetime of topological features obtained from persistent homology can be represented using a finite collection of intervals known as persistent barcodes. However, in the calculation of persistent barcodes, ephemeral features (features that are born and dead at the same time) are not taken into account. In this talk, I will present the notion of verbose barcodes by Usher and Zhang, and show how ephemeral features provide useful information of the data in some cases.

**October 30, 2020 **

Speaker: Jacob Hansen

Title: Dynamics on and of Cellular Sheaves .

Abstract: Cellular sheaves are a discrete, computable instantiation of sheaf theory. They can be seen as representing systems of linear constraints parameterized by a cell complex, and possess a robust cohomology theory. This sheaf cohomology admits a representation in terms of Hodge Laplacians, which are local operators on the spaces of sheaf-valued cochains. These Laplacians are a bountiful source of new ideas for network dynamics, providing novel equilibria and types of dynamics. Both the states of individual nodes and the sheaf structure may be affected by the dynamics, which we term dynamics on and of sheaves, respectively. This talk will explore the gamut of sheaf dynamics, with a running example of opinion dynamics in a social network. Here the sheaf describes a communication structure followed by the individuals in the network, and dynamics on the sheaf represent changes in opinions, while dynamics of the sheaf represent changes in the communication structure.

**October 16, 2020 **

Speaker: Jossiah Oh

Title: Geometry of Non-Transitive Graphs .

Abstract: In this talk we examine a class of non-transitive graphs and prove some results analogous to some classic theorems about the geometry of finitely generated groups. We also show that for each finitely generated group $G$, there are continuum many pairwise non-quasi-isometric regular graphs that share several geometric properties with $G$.

**October 3, 2020 **

Speaker: Zhengchao Wan

Title: Urysohn universal ultrametric space

Abstract: A Urysohn universal ultrametric space is an ultrametric space that satisfies both the universality and the ultra-homogeneity conditions. In this talk, I will present a novel construction of a Urysohn universal ultrametric space via the Gromov-Hausdorff ultrametric.

**February 28, 2020 **

Speaker: Mario Gomez

Title: Curvature sets and persistence diagrams

Abstract: In order to simplify the computation of the Vietoris-Rips complex of a metric space X, we can take an n point sample from the space. We can then reinterpret the Vietoris-Rips as a functor from the curvature sets of X. I will show a stability result and some computational examples.

**February 14, 2020 **

Speaker: Francisco Martinez

Title: The Chromatic Number of Random Borsuk Graphs

Abstract: We study a model of random graph where vertices are n i.i.d. uniform random points on the unit sphere S^d in R^(d+1), and a pair of vertices is connected if the Euclidean distance between them is at least 2−ϵ. We are interested in the chromatic number of this graph as n tends to infinity. Motivated by Lovász's result from 1978 relating the topology of a graph with its chromatic number, we show that for this model topological lower bounds are tight. This contrasts with the Erdős–Rényi random graph studied by Kahle in 2007, where these bounds are not efficient. Our proof depends on combining topological methods (the Lyusternik–Schnirelman–Borsuk theorem) with geometric probability arguments. This is joint work with Matthew Kahle.

**November 1, 2019 **

Speaker: Hyeran Cho

Title: Derivation of Schubert normal forms of 2-bridge knots from (1,1)-diagrams

Abstract: A genus one 1-bridge knot (simply called a (1, 1)-knot) is a knot that can be decomposed into two trivial arcs embed in two solid tori in a genus one Heegaard splitting of a lens space. A (1,1)-knot can be described by a (1,1)-diagram D(a, b, c, r) determined by four integers a, b, c, and r. It is known that every 2-bride knot is a (1, 1)-knot and has a (1, 1)-diagram of the form D(a, 0, 1, r). In this talk, we give the dual diagram of D(a, 0, 1, r) explicitly and present how to derive a Schubert normal form of a 2-bridge knot from the dual diagram. This gives an alternative proof of the Grasselli and Mulazzani’s result asserting that D(a, 0, 1, r) is a (1, 1)-diagram of 2-bridge knot with a Schubert normal form b(2a+1, 2r).

**October 25, 2019**

Speaker: Yubin Shin

Title: Image segmentation using persistence and watershed algorithm

Abstract: Watershed algorithm is one of the segmentation methods, which has a tendency to overdo the segmentation. We correct for this excessive segmentation by distinguishing the desired features from noise by persistent homology from topological data analysis. We will discuss properties of the method including some advantages, disadvantages and algorithmic efficiency.

**October 18, 2019**

Speaker: Paul Duncan

Title: Cocycle Counting in Random Simplicial Complexes

Abstract: A rich theory of random graphs has developed since the introduction of the Erdos-Renyi model. Similar questions can be asked of random simplicial complexes, which are much less well understood. I will talk about the technique used in determining thresholds in these complexes for the vanishing of homology, which is analogous to graph connectivity.

**October 4, 2020 (Location - MW 154)**

Speaker: Josiah Oh

Title: An Introduction to Large-Scale Geometry

Abstract: I plan to give a general survey of some important concepts in geometric group theory, and more specifically, large-scale geometry. I plan to introduce quasi-isometries and their invariants (Gromov hyperbolicity, growth of groups, ends of groups, etc.), and I will state some of the significant theorems that concern these notions. The talk will focus more on breadth than depth, and thus is geared more towards those who are new to (but interested in!) geometric group theory.

**September 20,2019**

Speaker: Hao Xing

Title: Johnson–Lindenstrauss lemma

Abstract: The Johnson–Lindenstrauss lemma is a result concerning low-distortion embeddings of points from high-dimensional into low-dimensional Euclidean space. In this short talk, I will give a brief introduction to the Johnson–Lindenstrauss lemma and sketch the idea of the proof.

**September 13, 2019**

Speaker: Zhengchao Wan

Title: Gromov-Hausdorff distance on Ultrametric spaces

Abstract: I will introduce a generalized Gromov-Hausdorff distance uGH on the collection U of ultrametric spaces which makes (U,uGH) itself an ultrametric space. This distance function can be characterized by distortion of correspondence as in the case of dGH. Moreover, I will show a structural theorem of uGH that allows us to devise a poly time algorithm to compute uGH.

**April 19, 2019 **

Speaker: Anastasios Stefanou

Title: Applied topology for phylogenetic networks

Abstract: In mathematical phylogenetics we are interested in developing mathematical methods that model the structure of *phylogenetic networks, *which in turn, can be used for developing metrics so that any pair of phylogenetic networks can be compared*.* In practice we often encounter tree-like networks called *phylogenetic trees* whose structure is already very well understood: one can show (Sokal and Rohlf, 1962) that, via the so called cophenetic map (which will be defined and explained during the lecture), phylogenetic trees with n leaves embed as vectors in the [n(n+1)/2]-dimensional Euclidean space. There we pull back l^p norms to define metrics on these trees. However very few are known for the general case of phylogenetic networks. In this talk I will explain why Reeb graphs is a natural model for phylogenetic networks and I will discuss a canonical way to decompose a phylogenetic network with n-labelled leaves and s cycles into a set of phylogenetic trees with (n + s)-labelled leaves. By combining this tree-decomposition and the cophenetic map, any such network embed as a finite set of points in the [(n+s)(n+s+1)/2]-dimensional Euclidean space. Hence, we can utilize the Hausdorff metric as a metric for comparison of phylogenetic networks.

**March 29, 2019**

Speaker: Alexander Wagner

Title: Discrete Morse Theory for Approximating Persistence Diagrams

Abstract: In this talk, I will briefly introduce discrete Morse theory and its extension to the filtered setting before sketching some work-in-progress that aims to use a discrete Morse function on a complex X to approximate the persistent homology of a filtration on X.

**March 22, 2019**

Speaker: Kathryn Hess

Title: Topological characterization of neuron morphologies

Abstract: The Topological Morphology Descriptor (TMD) encodes complex branching patterns, such as those of neuronal trees, as barcodes. It provides an unbiased benchmark test for the categorization of neuronal morphologies, enabling us to quantify and characterize the structural differences between distinct morphological classes and thus to increase our understanding of the anatomy and diversity of branching morphologies. We showed that applying the TMD to dendritic arbors of rat PCs provides an objective, reliable classification into 17 type. Our topological classification does not require expert input, is stable, and helps settle the long-standing debate on whether cell-types are discrete or form a morphological continuum. Applying the TMD to the apical dendrites of 60 3D reconstructed pyramidal neurons from layers 2 and 3 in the human temporal cortex revealed the existence of two morphologically distinct classes that also had distinct electrical behavior.

Work on this project has been led by Lida Kanari.

**March 1, 2019**

Speaker: Tatsuya Mikami

Title: Percolation on homology generators in codimension one

Abstract: Percolation theory is a branch of probability theory which mainly studies the behavior of clusters in a random graph. Recently, craze formation in polymer materials is gaining attention as a new type of percolation phenomenon in the sense that a large void corresponding to a craze of the polymer starts to appear by the process of coalescence of many small voids. In this talk, I will introduce a new percolation model motivated from the craze formation. For the sake of modeling the coalescence of nanovoids, this model focuses on clusters of "holes" of a random figure, which are formulated as homology generators in codimension one, while the classical percolation theory mainly studies clusters of vertices (i.e., 0-dimensional objects). This is a joint work with Yasuaki Hiraoka. (Reference: https://arxiv.org/abs/1809.07490)

Speaker: Shu Kanazawa

Title: Local structures of random simplicial complexes

Abstract: The notion of local weak convergence of finite graphs, introduced by Benjamini and Schramm in 2001, provided new asymptotic results for random graphs. In recent years, many tools in this framework have turned out to be extremely powerful even for random simplicial complexes. In this talk we will briefly describe the notion of local weak convergence of random simplicial complexes and how to read off the asymptotic behavior of the Betti numbers (nonlocal parameters) from its limiting local structures. Finally, we will report a recent asymptotic result of Betti numbers for a class of random simplicial complexes.

**February 22, 2019**

Speaker: Woojin Kim

Title: Multiparameter Persistent Homology for Time-Varying Metric Data

Abstract: Characterizing the dynamics of time-evolving data within the framework of topological data analysis has been attracting increasingly more attention. Popular instances of time-evolving data include flocking/swarming behaviors in animals and social networks in the human sphere. A natural mathematical model for such collective behaviors is a dynamic point cloud, or more generally a dynamic metric space (DMS). We will discuss (1) how to induce a multiparameter persistent homology as an invariant of a DMS, and (2) stability of these invariants. In order to address the stability, we extend the Gromov-Hausdorff distance on metric spaces to the setting of DMSs. In our framework, the celebrated Bottleneck-GH stability theorem for (static) metric spaces becomes a special case. This is a joint work with Facundo Mémoli. (Reference: https://arxiv.org/abs/1812.00949)

**February 15, 2019**

Speaker: Fedor Manin

Title: Maps between spheres are hard to optimize

Abstract: It's not too hard to imagine most possible metrics on the 2-sphere. But higher-dimensional spheres are much weirder. There are a number of different ways of measuring this, but I will focus on a measure of "roundness" called the hypersphericity. It turns out that in dimensions 3 and up, the hypersphericity of a given metric sphere is computationally hard to figure out, even approximately. In other words, it's hard to tell a pretty round sphere from a very skinny one "just by looking". This is joint work with Zarathustra Brady and Larry Guth.

**November 30, 2018**

Speaker: Jimin Kim

Title: Configuration space of thick particles on a metric graph

Abstract: Configuration space of thick particles on a metric graph is an interesting topic that lies in the intersection of topology, geometry, and combinatorics. Still, it has been little known since Deeley first studied this space in 2011. We will see an example of a configuration space of thick particles on a Y- shaped graph and discuss some of future research problems.

**November 16, 2018**

Speaker: Woojin Kim

Title: Rank for arbitrary diagrams

Abstract: The rank of a linear map f is the dimension of the image of f. Hence, the rank of f measures the "nondegenerateness" of f. We extend this notion and define the rank of a diagram of vector spaces and linear maps. We do this, based on a category theoretical perspective. If time permits, we also explore how this generalization can be used in the realm of topological data analysis. This is a joint work with Facundo Memoli.

**November 9, 2018**

Speaker: Nate Clause

Title: An Approach to Constructing Stable Filtration Functors via Basepoint Filtrations

Abstract: Persistent Homology is the primary tool used in Topological Data Analysis. Persistent Homology takes a dataset viewed as a finite metric space and converts it into persistence diagrams or barcodes which convey topological information of the underlying dataset. The first step in this process is to use a map called a filtration on the finite metric space, which builds up a simplicial complex on the metric space over time. Currently, a small number of filtrations such as Vietoris-Rips are well studied and put in practice, due to possessing some desirable theoretical properties and having clear geometric and topological intuition. However, there has been no generalized approach for constructing new filtrations which might provide novel insights for datasets. We will define a new method for constructing filtrations via curvature sets and valuations, and then generalize this approach to a method for constructing what we call basepoint filtrations. We outline desirable properties of filtrations constructed through this process, and provide computational examples.

**October 26, 2018**

Speaker: Francisco Martinez Figueroa

Title: Kneser-Lovasz Theorem

Abstract: In 1978 Lovasz proved Kneser’s conjecture about the chromatic number of Kneser Graphs. His proof is among the firsts and most prominent examples of an application of algebraic topology tools to a problem about finite combinatorial objects. Many more proofs of this result are now known, most of them involve some topological tools. One of the simplest known proofs is due to Greene (2002) which follows from Borsuk-Ulam’s theorem. We will discuss some properties of Kneser’s graphs and Greene’s proof.

**October 19, 2018**

Speaker: Paul Duncan

Title: Percolation

Abstract: Percolation theory is a study of natural physical models that has drawn considerable attention in the past sixty years. In two dimensions the classical model on the integer lattice is well understood, but there are many possible generalizations to higher dimensions. We will review known results and discuss some topologically motivated ways forward.

### September 21, 2018

Speaker: Samir Chowdhury

Title: Computational Optimal Transport

Abstract: In the past two decades, computational optimal transport has generated significant interest in the computer vision and machine learning communities. We will outline the OT approach to shape matching tasks as described by Rubner, Tomasi, and Guibas (2000), and explain the associated computational problem. Time permitting, we will briefly describe the notion of entropy regularized optimal transport that has been popularized over the past five years.