Geometric Techniques in Number Theory

The study of Diophantine equations is the study of equations where we insist that the solutions be integers. There are many fascinating topics in this area, most recently Wiles' proof of Fermat's Last Theorem. In this talk we will introduce some elementary geometric methods and show how they can be used to generate solutions to certain Diophantine equations.

1. Quadratic Curves and Surfaces
2. Singular Curves and Surfaces
3. Elliptic Curves
4. Ramanujan's Observation
5. Hardy's Question
6. A Variant on Waring's Problem
7. Two Challenges