|Date: Monday, March 14, 2022
Location: Virtual (1:00 PM to 3:00 PM)
Title: Dissertation Defense: Topics on Polynomial Equations in Noncommutative Rings and Motivic Aspects of Moduli Spaces
ID: 962 5612 7474
We discuss three topics stemming from the study of polynomial equations. In the beginning part of the talk, we give a historical and example-driven account on several rich research areas motivated by polynomial equations. This part should be accessible to the general audience.
Next, as the first topic, we start with a classical "unit equation theorem" which relates addition and multiplication of complex numbers in a fundamental way. We give an analogue of this result, but with the commutative multiplication on complex numbers replaced by the noncommutative multiplication on quaternions. The flavor of this result is number-theoretic, and we will explain how this result naturally arises from and applies to the study of iterations of self-maps on abelian varieties.
As the second topic, we discuss several counting problems on the solution sets of certain polynomial equations on the noncommutative algebra of n by n matrices. The flavor of these problems turns out to be combinatorial and algebro-geometric, and we will explain how these problems are essentially point-counting problems on certain moduli spaces (spaces that parametrize all possible structures).
As the third topic, we focus on several combinatorial behaviors of certain geometric invariants of configuration spaces, which are an infinite sequence of moduli spaces determined by a fixed base space. Our results involve discovering and explaining their unexpected resemblances with the known behavior of point counts of configuration spaces. Normally, an invariant that behaves like the point count is called "motivic", but the geometric invariants in question are not motivic, so a separate explanation of the unexpected resemblances is needed.
Yifeng's advisor is Michael Zieve.
Speaker: Yifeng Huang