RESEARCH

A professor from undergrad once said to me: "Kaelyn, you're the kind of guy who could find MOST things in math interesting if they're presented to you in the right way." Looking back at it, I think that professor is correct, as I absolutely HATE pigeonholing myself mathematically. That being said, there are some branches of math I find myself gravitating to most naturally. Those branches are geometry, topology, combinatorics, and computational mathematics. These days, my favorite mathematical objects to think about are graphs, complexes, lattices, polytopes, hyperplanes, and matroids. As a result, I often think about problems in the following areas of math:

discrete & computational geometry

applied & computational topology

extremal & probabilistic combinatorics

applied & combinatorial algebraic geometry

I'm also interested in applications to machine learning, optimization theory, and topological data analysis. Below is a list of my projects, past and present.

## Current Projects

Here are the projects that consume the bulk of my research time these days.

Topology of Random Cubical Complexes: I am investigating various topological properties of random cubical complexes, in the spirit of Erdos', Renyi's, & Spencer's work on random graphs. In particular, I am interested in the Euler characteristic, low-order homology rank, homotopy type, and the existence of torsion in the homology & homotopy groups of these complexes. These complexes are constructed using different random processes assuming a uniform distribution of sampling. This is joint work with Jim Fowler & Matthew Kahle (Ohio State).

TDA on Random Graph Models: I am investigating the potential for application of tools from topological data analysis (such as persistent homology & barcodes) to study topological features of random graph models like Erdos-Renyi, Watts-Strogatz, small worlds, etc. I have recently begun studying the random dot product graph model in hopes of streamlining this process. This is joint work with Russell Funk & Tom Gebhart (UMN).

Tropical Geometry of Deep Neural Networks: I am investigating the use of tools in algebraic geometry for application in machine learning theory. In particular, I am interested in understanding how the "unreasonable effectiveness" of deep neural networks can be viewed through the lens of tropical geometry. A 2018 paper by Zhang, Naitzat, & Lim demonstrated that feedforward neural networks w/ ReLU activation can be viewed exactly as tropical rational functions, and I would like to see how the tropical geometric language can also be used to better understand features such as the linear regions of a neural network. This is joint work with Jeff Calder & Gregg Musiker (UMN).

## Past Projects

Here's a list of projects I've pursued before that I no longer actively work on but am still interested in. I'm willing to resume these projects at the right time, so feel free to ask about them anyway if you're interested as well.

d-Regular Cubulations of Genus-g Surfaces: I was attempting to answer the following question: " For which g is there a d-regular cubulation of a genus-g surface, for some given d?" This was joint work with Jim Fowler (Ohio State).

Cohen-Lenstra Heuristics of Random Cohomology Groups: I was interested in knowing if certain cohomology groups of random spanning 2-trees are Cohen-Lenstra distributed. This was joint work with Jim Fowler (Ohio State), with brief consultation from Melanie Matchett Wood (Harvard).

Random Tilings and the Arctic Circle Theorem: I was studying random tilings of the plane and attempting to determine if the Arctic Circle Theorem for Aztec Diamonds can be applied to other classes of tilings called "Aztec Dragons" and "Aztec Castles." This was joint work with Gregg Musiker (UMN).Â