Garrett Kepler

Garrett Kepler

Mathematician, Statistician, Physicist, and Artist

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publications

Paper Title Number 4

Published in GitHub Journal of Bugs, 2024

This paper is about fixing template issue #693.

Recommended citation: Your Name, You. (2024). "Paper Title Number 3." GitHub Journal of Bugs. 1(3).
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talks

Graphical Inginuety: Spectral Solutions to Combinatorial Conondrums

Published:

Abstract: Spectral graph theory (SGT) has various scientific applications, ranging from network optimization to quantum error correction. This presentation will explore the combinatorial realm of SGT by exploring its implications through several proofs of famous counting problems. By investigating methods, this talk aims to illustrate the intersection of combinatorics and spectral graph theory, offering insights into their synergistic relationship. 

Independent Sets and Graph Energy

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Abstract: The sum of the moduli of the adjacency spectra of a graph, the adjacency energy, has been shown to connect linear algebra and the electron energy of molecules. This talk will cover a conjectured lower bound relating the energy of a graph and the size of its largest independent set. A solution for graphs with proportionally large independent sets will be shown and the consequences explored.

Laplace, Fiedler, & Markov: A Graph Reconstruction Problem

Published:

Abstract: Many well established graph partitioning methods are determined by eigenvectors of matrices associated with the underlying graph. In this talk, we will explore the combinatorial intuition and computational methods involved in reconstructing a graph from subsets of such a partitioning. This will naturally lead us to characterize desirable attributes of graphs in this context and we will also uncover an application to information security.

Eigenspaces of Graphs and their Utility

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Abstract: Eigenspaces of linear operators associated with graphs have well known applications in the context of graph partitioning. In this talk, I will use combinatorial methods to evaluate the relationship between eigenvector entries and adjacency structure, particularly in constructing graphs with prescribed eigenvectors. As examples, I will characterize eigenspaces for well-studied graph families including Johnson graphs and perform a similar analysis for graph products. In addition to these combinatorial results, this approach also has applications to dimensionality reduction, a natural problem in data science.

A Probabilistic Excursion in Spectral Graph Theory

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Abstract: In this talk, we will explore the mutually beneficial relationship between probability theory and spectral graph theory. At their intersection lie lots of perplexing puzzles and fresh features. Through ample example, we will showcase their established significant symbiotic synergy. Moreover, we will meander through many musings and mysteries.

Graph Theory and Gerrymandering: Computationally Assessing Fairness

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Abstract: Per my former department’s request, we begin by reviewing my mathematical journey up to where I am today. We then explore the basics of Gerrymandering as a concept and redistricting as a mathematical application. Using graph thoeretic ideas, we define the redistricting methods used in practice. Then, we cover current research in spectral graph theory using eigenvectors of graphs as a tool for this application. Offering many paths for further exploration, we extend an invite to partake in math’s applications to redistricting, networks, and data.

A Look at Limits in Random Graphs

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Abstract: We will discuss the concept of Benjamini-Schramm convergence and its motivation. After discussion of properties of unimodularity in graphs, we discuss its importance in limits of random graphs. We will cover both pointed and rooted graphs and discuss the relationship between both through bias of random variables.

MCMC: An Introduction

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Abstract: As Math 420 reaches its close, I have been asked to give a talk on Markov Chain Monte Carlo (MCMC) and its applications. We will review some probability theory basics through an in-class excercise revealing the intuition of the Weak Law of Large Numebers. Afterward, we will have a demonstration in real time of MCMC’s approximation of a given distribution. After diving into the history of this revolutionary approach, we will end with applications to redistricting and crytography.

Applications in and Methods of Spectral Graph Theory

Published:

Abstract: As Math 453 comes to a close, I have been invited to give a special topics lecture on spectral graph theory. We will discuss relevant concepts of spectral graph theory and graph partitioning. Then, we will review some open research problems regarding pertubations of the eigensystem and spectral clustering. After a demonstration of Markov Chain Monte Carlo (MCMC), we will discuss applying the process to spectral partitioning problems and redistricting.

teaching

Teaching experience 1

Undergraduate course, University 1, Department, 2014

This is a description of a teaching experience. You can use markdown like any other post.

Teaching experience 2

Workshop, University 1, Department, 2015

This is a description of a teaching experience. You can use markdown like any other post.