## Project Leader

- Derek Krepski, University of Manitoba
- Steven Rayan, University of Saskatchewan

### Graduate Mentor

- Dinamo Djounvouna, University of Manitoba

### Undergraduate Team Members

- Pratham Lalwani, University of Northern British Columbia
- Zhikai Chen, University of Alberta

## Problem Statement

Topological Data Analysis (TDA) applies classical methods of algebraic topology to investigate and make inferences about the topology of (possibly large/complex) data sets. Still in its infancy, TDA has already been applied successfully to numerous areas of scientific research. In this project, participants will learn and apply current tools and techniques of TDA to objects of current interest at the interface of geometry, topology, and physics, such as the various moduli spaces investigated across several areas of mathematics research.

Understanding the geometry and topology of moduli spaces can be difficult; in some cases, even determining whether a moduli space is connected — a topological feature whereby the space consists of a single geometric/contiguous ‘piece’ — can be challenging. The methods of TDA, such as persistent homology, can be applied to a random sampling of data (points) from the moduli space (suitably embedded in Euclidean space) to give insights on the presence (or absence) of such topological features in the moduli space, helping guide further research in the subject.

Students working on this project will learn the fundamentals of topological data analysis while enhance their understanding of topology and geometry as mathematical subjects in general. Students will also encounter, for instance, aspects of linear algebra and other core courses that are less emphasized in traditional undergraduate course offerings.

### Final Report

This VXML project was completed by the participants listed above, as described in the following documents

### Details

**Expected team size**: 2**Student Experience Level**: Advanced: students who have taken multiple higher level mathematics courses

##### Prerequisites

- Elementary Topology

##### Skills

- Sage/Jupyter or similar programming tools (preferred but optional)