# 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 Autumn 2018: Katie Ritchey and Samir Chowdhury

Time and place: MW 724, 12:40 pm to 1:35 pm

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.

**Winter break! We plan to start again in Spring 2019.**

# Spring 2019 Schedule (in progress)

February 22: Woojin Kim

# 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

**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.