We invite to the seminar of the Department of Applied Mathematics
Monday, January 22, 11:15am room 3/19
Our speaker is Amitesh Datta, a PhD student of Princeton University
Title: Polynomials, Fixed Points, and Stability.
I will introduce and motivate recent work of Church, Farb and others on the stability of certain counts of polynomials over finite fields. More precisely, the number of polynomials in one variable of degree $d$ over a finite field $F_q$ satisfying a chosen property (e.g., one of the following properties: with no square factors, with exactly one root/linear factor, with exactly one irreducible quadratic factor etc.) can be expressed as a polynomial in $q$ for each $d$, and the (suitably normalized) coefficients of these polynomials stabilize in the limit as $d\to \infty$. An important feature of this work is the use of ideas from algebraic topology pertaining to counting the number of fixed points of functions, to count the number of solutions of Diophantine equations arising in number theory. I will focus on illustrating these ideas through several elementary and explicit examples, such as that of the space of all configurations of points in the plane.