Title: Adventures in Arithmetic Geometry

Abstract: In this talk, we will discuss some of the big ideas in arithmetic geometry, the area of mathematics that brings together number theory and geometry. We will begin by seeing how the Pythagorean theorem opens the door to one of the field's central questions: finding rational or integer solutions to equations defining curves. This theme runs through centuries of mathematics, from Diophantus's Arithmetica to Wiles's proof of Fermat's Last Theorem, and, even more recently, to the work of Balakrishnan et al on the so-called Cursed Curve.