Algebra/Topology seminar

Speaker: David Zureick-Brown (Emory University)

Title: Diophantine and tropical geometry

Abstract: Diophantine geometry is the study of integral solutions to a polynomial equation. For instance, for integers $a,b,c \geq 2$ satisfying $1/a + 1/b + 1/c > 1$, Darmon and Granville proved that the individual generalized Fermat equation $x^a + y^b = z^c$ has only finitely many coprime integer solutions. Conjecturally something stronger is true: for $a,b,c \geq 3$ there are no non-trivial solutions.

I'll discuss various other Diophantine problems, with a focus on the underlying intuition and conjectural framework. I will especially focus on the uniformity conjecture, and will explain new ideas from tropical geometry and our recent partial proof of the uniformity conjecture.