# 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 2018-2019: Katie Ritchey and Samir Chowdhury

Time and place: Variable in Spring 2019; see announcements below

Frequency: See below for our upcoming schedule

Lunch is provided, so bring an appetite!

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

**Next meeting: **

Time and location: TBD

Speaker: TBD

Title: TBD

Abstract: TBD

# Spring 2019 Schedule

February 15: Fedor Manin

February 22: Woojin Kim

March 1: Shu Kanazawa and Tatsuya Mikami

March 22: Kathryn Hess (special faculty lecture)

March 29: Alex Wagner

April 19: Anastasios Stefanou

# Autumn 2018 Schedule

September 7: Organizational Meeting

September 21: Samir Chowdhury

October 19: Paul Duncan

October 26: Francisco Martinez Figueroa

November 9: Nate Clause

November 16: Woojin Kim

November 30: Jimin Kim

## Previous talks

**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.