Siqi Li: Stabilizers of canonical matrices arising from matrix congruence. Poster
Abstract: My research is in noncommutative algebra and my project was to study the stabilizer groups of special matrices arising in the theory of congruent matrices as well as their applications. Our goal was to determine canonical forms under a certain type of congruence which corresponds to noncommutative quadratic polynomials in degree three. We extended results in the 2x2 case by Gaddis to the case of 3x3 matrices. Canonical forms under standard matrix congruence were developed by Horn-Sergeichuk. We first listed their 3x3 forms, and then computed the stabilizer groups for a representative of each form. Next, we proceeded to determine the canonical forms under standard form congruence for 4x4 matrices. This involved developing routines in Maple to aid in the computation. Additionally, we developed some general rules and conjectures for stabilizer groups.
Ke Liang: Four-Vertex quivers supporting graded Calabi-Yau algebras. Poster
Abstract: Our project focused on specific shapes arising from string theory, called Calabi–Yau manifolds, and the algebraic objects associated to them. The goal is to classify directed graphs (called quivers) on four vertices supporting graded Calabi-Yau algebras of global dimension 3 and finite GK dimension. The bulk of our study was to analyze the matrix-valued Hilbert series associated to each quiver. During our study, we found many `good’ matrices which support Calabi-Yau algebras, including certain circulant matrices, as well as matrices associated to McKay quivers and mutations. We also ruled out large classes of matrices which cannot support Calabi-Yau algebras. This extends work of Gaddis, Reyes, and Rogalski.