Research

My current research areas are primarily Topological Data Analysis (TDA) and Low-Dimensional Topology. In addition to these areas, I also find myself looking into areas of Statistics, Machine Learning (specifically Semi-Supervised Machine Learning), Topological Deep Learning (TDL), and Computer Science. I have included a link to my public GitHub repositories at the bottom of this page.

Papers

These are papers that I have written and/or helped write.

  1. M. Fairchild, M. Lemoine; "Topological Data Analysis of Mortality Patterns During the COVID-19 Pandemic". arXiv:2510.15066 (2025)
  2. B. Appiah, P. Dani, W. Ge, C. Hudson, S. Jain, Y. Lee, M. Lemoine, J. Murphy, A. Pandikkadan, K. Schreve; "The algebraic structure of hyperbolic graph braid groups". arXiv:2403.08623 (2024)
  3. M. Lemoine; "Factors that Relate to Academic Excellence". (completed in May 2022 in accordance with the degree requirements for B.S. in Mathematics at Louisiana Tech University) digitalcommons.latech.edu (2022)

Talks

Below are talks that I have given. I have included the titles, abstracts, and where these talks were given along with the context that these talks were given.

  1. "Introductory Talk for Topology (specifically Knot Theory)
    This talk was an introduction to knot theory given at an undergraduate workshop. At this workshop, I was a mentor helping undergraduate students learn about knot theory, and how to learn mathematics.

    At Louisiana State University for the RTG (Research Training Group) Undergraduate Workshop (May 18, 2026)

  2. "Sliding Window Embedding"
    Abstract: When we are looking at a dataset that has a time-dependency and is periodic (or quasi-periodic), we are interested in noticing differences in the period. Using sliding window embeddings (also known as time-delay reconstructions), we can look at the periods of a given dataset and analyze the persistent homology to detect changes in our periods. In this talk, we will be discussing the foundational paper in this area of Topological Data Analysis by Jose Perea and John Harer (arxiv:1307.6188v2).

    At Lousisiana State University for the Informal Geometry and Topology Seminar (February 4, 2026)

  3. "An Introduction to Topological Data Analysis with an Application"
    Abstract: In this talk, we will look at a thorough introduction to Topological Data Analysis and the tools that are used most frequently in this field. Then we will look at some work on a dataset from the Centers for Disease Control and Prevention analyzing mortality during the Covid-19 pandemic. We will look specifically at two regions during this time ('The Northeast' and 'The South') following "Topological Data Analysis of Mortality Patterns During the COVID-19 Pandemic". This paper is joint work with Megan Fairchild. This talk is aimed at being accessible to undergraduate students interested in Topology and Data Analysis.

    At Louisiana Tech University for Math Department Tea Time Talks (December 12, 2025)

  4. "Topological Data Analysis of Mortality Patterns During the COVID-19 Pandemic (joint work with M. Fairchild)"
    Abstract: This talk will be a brief overview of Topological Data Analysis and will go into some of the work that Megan and I have done. Topological Data Analysis is a relatively new field of study that uses topological invariants to study the shape of data. We analyze a dataset provided by the Centers for Disease Control and Prevention (CDC) using persistent homology and MAPPER. This dataset tracks mortality week-to-week from January 2020 to September 2023 in the United States during the COVID-19 pandemic. We examine the dataset as a whole and break the United States into geographic regions to analyze the overall shape of the data. Then, to explain this shape, we discuss events around the time of the pandemic and how they contribute to the observed patterns.

    At LSU for the Informal Geometry and Topology Seminar (November 12, 2025)

  5. "Topological Data Analysis"
    Abstract: In this talk, I will outline my research interests and highlight the big pieces in TDA. I will also be in a breakout room for people with questions afterward.

    At LSU for the GEAUX First-year Math Graduate Student Orientation (August 13, 2025)

  6. "The Classification Theorem of Vector Bundles over Compact Spaces"
    Abstract: In the summer for the Informal Seminar, we pick a book/paper and we all present over different aspects of the book/paper. In the summer of 2025, we picked a book on K-Theory by Inna Zakharevich.

    At LSU for the Informal Geometry and Topology Seminar (June 12, 2025)

  7. "Topological Data Analysis and the Persistent Laplacian"
    Abstract: In this talk, we will go through some basic information about Topological Data Analysis (TDA) such as Persistent Homology with the goal of getting to the Persistent Laplacian and how these tools are used to analyze data.

    At LSU for the Informal Geometry and Topology Seminar (April 16, 2025)

  8. "A Brief Introduction to Khovanov Homology through an example"
    Abstract: In this talk, we will discuss Khovanov Homology and how to compute this homology using an example with the trefoil knot. We will also discuss the relations between Khovanov Homology and the Jones Polynomial.

    At LSU for the Informal Geometry and Topology Seminar (October 30, 2024)

Conferences

Attended:
AMS Southeastern Sectional at Tulane University; New Orleans, Louisiana (October 2025)
Topology of Arrangements with an Eye to Applications at University of Pisa; Pisa, Italy (September 2025)
The Geometric Realization of AATRN at IMSI; Chicago, Illinois (attended over Zoom, August 2025)
The 68th Texas Geometry and Topology Conference at Texas A&M University; College Station, Texas (November 2024)

Funding and Grants

While at Louisiana State University, I have been primarily funded by Graduate Teaching Assistantships which include teaching, grading, and aiding different classes (See 'Teaching' for more details on what I have taught, and classes I have aided and graded.)
During the 2025-2026 school year at Louisiana State University, I have been on the Research Training Groups (RTG) Grant, which allowed me to work more on my research and not have teaching duties.

GitHub Link

I have worked on several different coding projects. (Including this website, which I coded using HTML and Javascript) Here is the link to my public GitHub repositories. GitHub Link