Student Algebraic Geometry

The goal of this talk is to introduce how to use some fine techniques in algebraic geometry to obtain concrete combinatorial results. I will assume almost no background.

Roughly, "arithmetic statistics" refers to problems of counting a massive amount of arithmetic objects. For example, consider the smooth affine plane curves of the form y^2 = f(x) with deg(f) = d over a finite field of p elements. A natural question to ask is: what is the probability of a degree d random curve to have many (i.e., 2p) points? An explicit answer to this question is known when d goes to infinity, but one may ask what happens when d is some large finite integer.

In this talk, I will explain how to approach such a question using explicit Betti numbers of manifolds called "configuration spaces". The connection of point-counting and Betti numbers can be established by a machine called "\'etale cohomology", especially via Grothedieck trace formula, an analog of Lefschetz fixed point theorem in algebraic topology. In particular, I will sketch how the convergence in d of the distribution of points on curves of the form y^2 = f(x) is a consequence of homological stability. Speaker(s): GilYoung Cheong (UM)