Congratulations to Math major Ethan Hurst for being awarded first place in the 2022 Undergraduate Research & Creative Activity Conference. Ethan was honored at the April 18th URCA awards luncheon.
Ethan is a junior studying mathematics with minors in English and the Randall Research Scholars Program. He and Dr. Kyungyong Lee have been working on this project since January 2021.
Project abstract: The Jacobian Conjecture is a famous unsolved problem in mathematics lying at the intersection of algebraic geometry and combinatorics. It deals with polynomial mappings in arbitrary dimensional space and their inverses. The problem was originally stated in 1939 by Keller, and has been the subject of a number of subtly incorrect proofs since its original publication. Our project seeks to analyze the conjecture using Magnus’s formula, which provides a method for relating homogenous degree pieces of two functions by their determinant. Magnus published his formula in order to analyze homogenous degree pieces along a typical grading, or only in a single direction. Our project was able to generalize this formula to work in an arbitrary direction. Additionally, we were able to apply the generalized form of Magnus’s formula to the generic case of the Jacobian conjecture in two-dimensions, which has allowed us to draw significant results about the conjecture. Specifically, we were able to prove a special case of the conjecture in two-dimensions, which bodes well for future results. Solving this problem will have implications in many fields of mathematics, such as commutative algebras and dynamical systems. Solving this problem will also prove the Dixmier Conjecture, as that problem is stably equivalent to the Jacobian Conjecture. Such a solution would also have ramifications in quantum field theory, representation theory, and in many other areas.