Hello there, my name is Andrew Michael Thomas. I am a postdoctoral associate in the Center for Applied Mathematics at Cornell University. You can find my CV here. I spend most of my research time working on the probability and statistics side of topological data analysis (TDA). Envision raindrops hitting pavement: what is the number of dry regions surrounded by wet ones? The questions I'm interested in range from investigating the limiting behavior of topological functionals treated as stochastic processes to how to integrate TDA with statistics and data science and then deploying that in scientific applications. I am also intrigued by the interaction between extreme value theory and stochastic topology/geometry, wanting to understand the statistical properties of various summaries of shape when they arise from low-density regions. Please check out the publications section to see my most recent work. In June 2022, I presented a poster on some of my latest work at the intersection of the extreme value theory and stochastic geometry at the 2022 Stochastic Networks Conference. In September 2022, I presented another poster on my work combining statistics, TDA, and image processing in a catalysis/microscopy application, at the AATRN/APATG Fall 2022 poster session. The **detecTDA** package implementing the procedures described in the work described on the aforementioned poster (Thomas, Crozier, Xu and Matteson, 2022) can be found on its github repository.

The work you see below, and which comprises my PhD dissertation, falls under the purview of random geometric complexes, the study of which was initiated in earnest by Kahle (2011). For introductory resources on stochastic process limits (Donsker's theorem), random geometric complexes and topological data analysis in general, please look in the miscellany section for some of my favorite introductions to TDA below.

** Central limit theorems and asymptotic independence for local U-statistics on diverging halfspaces (2022). **

Andrew M. Thomas

To appear in

** Dynamic atomic column detection in transmission electron microscopy videos via ridge estimation (2023). **

Yuchen Xu, Andrew M. Thomas, Peter A. Crozier, David S. Matteson.

*arXiv Preprint*, arXiv:2302.00816.

** Feature detection and hypothesis testing for extremely noisy nanoparticle images using topological data analysis (2022). **

Andrew M. Thomas, Peter A. Crozier, Yuchen Xu, and David S. Matteson

** Fractal dimension, approximation and data sets (2022). **

Livia Betti, Ivan Chio, Julie Fleischman, Alex Iosevich, Filippo Iulianelli, Scott Kirila, Michele Martino, Azita Mayeli, Svetlana Pack,

Zhelin Sheng, Conor Taliancic, Andrew [M.] Thomas, Nathan
Whybra, Emmett Wyman, Ustun Yildirim, and Kaiyuan Zhao.

*arXiv Preprint*, arXiv:2209.12079.

** The VC-dimension of a class of multiples of the primes, and a connection to AdaBoost (2022). **

Andrew M. Thomas

*arXiv Preprint*, arXiv:2208.06442.

** Functional strong laws of large numbers for Euler characteristic processes of extreme sample clouds (2021). **

Andrew M. Thomas and Takashi Owada

*Extremes*, DOI: https://doi.org/10.1007/s10687-021-00419-1. Publisher Link.

** Functional limit theorems for the Euler characteristic process in the critical regime (2021). **

Andrew M. Thomas and Takashi Owada

*Advances in Applied Probability* **53(1)**, 57–80. Publisher Link. See also: arXiv:1910.00751.

** Limit theorems for process-level Betti numbers for sparse and critical regimes (2020). **

Takashi Owada and Andrew M. Thomas

*Advances in Applied Probability* **52(1)**, 1-31. Publisher Link.

First off, here's a link to my PhD dissertation, which I think provides a pretty good history of random geometric complexes. Here's a video of an online talk that I gave on July 1, 2020 on functional limit theorems for the Euler characteristic process (or, Euler curves) at the Applied Algebraic Topology Research Network (AATRN).
I also contributed to AATRN & WinCompTop's Tutorial-a-Thon in the Spring of 2021. The tutorial I submitted is entitled "Let’s talk about random Cech and Vietoris-Rips complexes" and can be seen here:
On August 3, 2020 I gave a talk at the virtual Joint Statistical Meetings (JSM) held in virtual Philadelphia about the history of StatCom and how to keep student volunteers and clients engaged to complete projects and ensure that the volunteers have ample opportunities for service-learning. My slide deck can be accesssed here. I've also appeared twice in the IMS bulletin, for solving problems in the Student Puzzle Corner. You can see here, I sent in a solution to an interesting problem on random distances on hyperspheres. Be sure to read your IMS Bulletin!

Finally, I have compiled a list of some soothing and cool articles that pertain to my research. They will soothe your mathritis pain: Mathematical IcyHot®.

- A short introduction to Donsker's theorem, written by me, in R Markdown.
- A great early introduction and survey of TDA from a topological perspective—Barcodes: the persistent topology of data by Robert Ghrist.
- A concise yet intuitive first step in Betti numbers and random complexes, among many other more advanced excursions—Persistent homology for random fields and complexes by Adler et al.
- Foundational work on random geometric complexes— Random geometric complexes by Matthew Kahle.

Here is a link to my CV. Please feel free to email me at amt269@cornell.edu. Follow @amtprob Last updated: 2/14/2023.