# Research

Broadly speaking, my research interests are in algebra, algebraic geometry, and number theory. I study abstract questions about algebraic groups that are of interest to mathematicians on the cutting edge by reducing to concrete problems that are accessible to undergraduates. Specifically, I study the representation theory of algebraic tori using techniques from algebraic groups, integral lattices, and the classification of finite simple groups.

Dissertation

Here is my doctoral dissertation, which studies degrees of matrix representations of algebraic via G-lattices. I expect to soon have a (much shorter) pre-print with these results as well.

SPARC Grant

I was awarded a Support to Promote Advancement of Research and Creativity (SPARC) research grant through the University of South Carolina for May 2023 - July 2024. Here is the narrative proposal portion of my application, which summarizes my project, including its significance and some partial results, for the non-mathematician.

Prepared Materials

Below are some materials I've produced for presentations and for other purposes, listed in reverse chronological order.

Representation Dimensions of Algebraic Tori, an invited talk at the University of Georgia Algebraic Geometry Seminar (October 2023). Intended for a graduate audience.

G-Lattices at G-Burg: Reflections, Research, and Representations, an invited Gettysburg College Homecoming Alumni Mathematics Colloquium (October 2023). Intended for undergraduates, talks about both my experience as a graduate student and my research.

An Invitation to Algebraic Geometry (February 2023). These slides were presented at the spring 2023 Pi Mu Epsilon/Gamecock Math Club induction ceremony at the University of South Carolina. They are intended to give undergraduates (with no more than a Calculus I background) an introduction to the field of algebraic geometry.

Symmetric Ranks of Matrix Groups at the Carolina Math Seminar at Columbia College (November 2022). These slides introduce my research from the perspective of matrix groups and share my results as of November 11, 2022. Intended for an undergraduate audience with background in linear algebra but not in abstract algebra.

Lattice Talk About Root Systems! at the Graduate Colloquium in the mathematics department at the University of South Carolina (April 2022). These slides provide a fairly detailed introduction to root systems, their Weyl groups, and their associated lattices. Assumes graduate algebra, but an undergraduate abstract algebra course would likely be sufficient.

Written comprehensive exam at the University of South Carolina (September 2021). This document provides a thorough, mostly self-contained introduction to the subject of algebraic tori through the lens of affine group schemes. Assumes graduate algebra.

Psychology Research

As an undergraduate at Gettysburg College, I did social psychological research under Dr. Chris Barlett (now at Kansas State University). Below are publications and posters which resulted from that work.

Barlett, C.P., DeWitt, C.C., Madison, C.S., Heath, J.B., Maronna, B., & Kirkpatrick, S.M. (2020). Hot temperatures and even hotter tempers: Sociological mediators in the relationship between global climate change and homicide. Psychology of Violence, 10, 1-7.

Barlett, C.P., Heath, J.B., Madison, C.S., DeWitt, C.C., & Kirkpatrick, S.M. (2020). You're not anonymous online: The development and validation of a new cyberbullying intervention curriculum. Psychology of Popular Media, 9, 135-144.

Barlett, C.P., Madison, C., DeWitt, C.C., & Heath, J.B. (2019). The moderating effect of dispositional fear of retaliation on the relationship between provocation and aggressive behavior. Personality and Individual Differences, 138, 257-265.

Barlett, C.P., Madison, C.S., Heath, J.B., & DeWitt, C.C. (2019). Please browse responsibly: A correlational examination of technology access and time spent online in the Barlett Gentile Cyberbullying Model. Computers in Human Behavior, 92, 250-255.

I Think, Therefore I Aggress? The Relationship Between Need for Cognition and Aggression (poster), Cross Disciplinary Science Institute at Gettysburg (X-SIG) Showcase 2018, Gettysburg College.

A Temporal Correlation Examination of Global Warming: Consequences on Violence (poster with Caroline "Ellie" DeWitt), Celebration 2018, Gettysburg College.

As an undergraduate, I conducted a study with Celeste Campbell and Ellie DeWitt on embodied cognition as part of a cognition course taught by Dr. Becca Fincher-Kiefer. Below are a resulting poster and paper.

Heath, Jason B. "The Heart Wants What it Wants: Effects of Desirability and Body Part Salience on Distance Perceptions (Heath)" (2018). Student Publications. 640. https://cupola.gettysburg.edu/student_scholarship/640/

The Heart Wants What it Wants: Effects of Desirability and Body Part Salience on Distance Perceptions (poster with Celeste Campbell and Ellie DeWitt), Celebration 2018, Gettysburg College.