Diophantine geometry is the study of integral solutions to polynomial equations. For instance, for integers *a,b,c* >1 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 all integegers *a,b,c* > 2 there are no non-trivial solutions.

I'll discuss various Diophantine problems, with an emphasis on the underlying intuition and conjectural framework. I will especially focus on the uniformity conjecture, and will explain new ideas from tropical geometry and recent partial progress toward the uniformity conjecture.

### Penn Mathematics Colloquium

Wednesday, September 27, 2023 - 3:45pm

#### David Zureick-Brown

Amherst College