## UIC Graduate Student Seminar

Friday October 5

3:00 pm

636 SEO

Speaker: Henri Gillet

Title: Counting solutions of equations three variables

Abstract:
In this lecture we shall discuss the following question:
Given a homogeneous polynomial equation f(x,y,z)=0 with integer
coefficients does it have infinitely many solutions in which x,y,z are
all integers?

The solutions of this equation in which x,y,z are allowed to be complex
numbers form a topological surface, i.e., a many holed donut. It has
been know for a long time that if this donut had at most one hole then
the equations have infinitely many integer solutions.

It was conjectured by Mordell in 1922 that if the surface has genus (the
number of holes) at least two, then the number of solutions is finite.
This was proved in 1983 by Gerd Faltings.