CURRENT PROJECTS

• • 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.  Recent work by Montufar et al derived combinatorial results for maxout networks and I'm interested in learning if these results can be tropicalized. I'm also interested in deriving tropical representations of other deep learning architectures like the convolutional neural network and the transformer model. This is joint work with Jeff Calder & Gregg Musiker (UMN).

  • Spectral Properties of the Algebraic Path Problem: The algebraic path problem unifies a number of optimal routing algorithms on graphs under a single generic, algorithmic umbrella. While these path problems have been well-studied computationally, their relationship to spectral graph theory is still relatively under-explored which, in turn, prompts a number of interesting topological questions. For example, to what extent can the shortest paths in the graph be computed from the spectrum of the graph itself, and which graph topologies interfere in this relationship? Does the algebraic path problem generalize to higher dimensions? If so, can this generalized problem be expressed in the language of cellular (co)sheaves, and what aspects of (co)sheaf (co)homology may be utilized to study these generalized algebraic path problems? How might we apply this relationship between the algebraic path problem and algebraic topology? I am investigating answers to these very questions, and am also investigating a possible connection to tropical geometry. This is joint work with Russell Funk & Tom Gebhart (UMN).

  • 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).

  • Chip-Firing Models of Infectious Disease Dynamics: In this project, I am constructing a novel model of infectious disease spread using the theory of chip-firing on graphs pioneered by Caroline Klivans. The model promotes a concept of "probabilistic chip-firing" where nodes on a graph fire toward their neighbors with weighted probability, and I am interested in the nature of SIR dynamics under such a concept. This is joint work with Param Thakkar (high school student). 

PAST PROJECTS

• • TDA on Random Graph Models: I was 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. In an effort to revive this project, I have recently begun studying the random dot product graph model in hopes of streamlining this process. This was joint work with Russell Funk & Tom Gebhart (UMN).

  • 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). 

PREPRINTS

1. J. Fowler, K. Willingham. "Homological Properties of Random Cubical Complexes." (in preparation)

2. P. Thakkar, K. Willingham. "A Chip-Firing Model of Infectious Disease Dynamics." (in preparation)

3. R. Funk, T. Gebhart, K. Willingham. "Spectral Properties of the Algebraic Path Problem." (in preparation)

PUBLICATIONS

1. TO BE ANNOUNCED

POSTERS

1. Spectral Properties of the Algebraic Path Problem  (Mid-Atlantic Topology Conference 2024 - March 2024)

2. Neural Network Theory through the Lens of Tropical Geometry (WAGS 2023 - December 2023)

3. Topological Properties of Random Cubical Subcomplexes of the n-Cube (Ohio State TDAI Research Forum - November 2019)

TALK SLIDES

1. Tropical Geometry and Applications to Path Problems (MAA NCS Fall 2024 Meeting - October 2024)

2. A Tropical Geometric Study of Deep Learning Architectures (UMN Preliminary Oral Exam - August 2024)

3. Spectral Properties of the Algebraic Path Problem (Young Topologists Meeting 2024 - August 2024)

4. Tropical Geometry and the Algebraic Path Problem (GSCC 2024 - April 2024)

5. Quiver Representations of Neural Networks (UMN Student Summer Representation Theory Seminar - August 2023)

6. Tropical Geometry in Deep Learning (GTDAML 2023 - June 2023)

7. Topological Combinatorics of Random Graphs & Complexes (2023 AMS Spring Central Sectional Meeting - April 2023)

8. How Deep Learning Algorithms can help answer Combinatorial Questions (UMN Student Combinatorics & Algebra Seminar - September 2022)

9. An Extension of the Arctic Circle Theorem to Aztec Castles & Dragons via Domino Shuffling & Perfect Matchings (UMN GSDO Summer Institute 2021 - August 2021)

TALK NOTES

1. Tropical Geometry and the Connection to Graph Theory (UMN Student Combinatorics & Algebra Seminar - February 2024)

EXPOSITORY WRITING

1. A Tropical Geometric Study of Deep Learning Architectures (UMN Preliminary Oral Exam - August 2024)