# The geometric nature of Diophantine equations

## Document Type

Presentation Abstract

## Presentation Date

4-19-2021

## Abstract

Does there exist a box such that the distance between any two of its corners is a rational number? Which integers can be expressed as the sum of three cubes? These questions and many others can be reframed as *Diophantine problems*, that is, questions of existence of rational or integer solutions to polynomial equations. Each such Diophantine problem has a geometric manifestation called an *algebraic variety* whose properties often shed light on why these questions don't have elementary answers. In this talk I'll give an introduction to the guiding principle that geometry influences arithmetic, and describe work on the existence of (and obstructions to) rational solutions to equations that define algebraic surfaces.

## Recommended Citation

Berg, Jen, "The geometric nature of Diophantine equations" (2021). *Colloquia of the Department of Mathematical Sciences*. 604.

https://scholarworks.umt.edu/mathcolloquia/604

## Additional Details

April 19, 2021 at 3:00 p.m. via Zoom