Diophantine equations and modular curves

A Diophantine equation is an equation for which one seeks whole number solutions. Although the study of Diophantine equations dates back to antiquity, they still play a central role in modern number theory. Since Wiles' celebrated proof of Fermat's Last Theorem at the end of the 20th century, a strategy for solving Diophantine equations known as the "modular method" has seen significant development. The broad aim of this project is to further develop this method by introducing new techniques, both abstract and computational, and to then use these to study various families of Diophantine equations, such as the "generalised Fermat" and "Lebesgue-Nagell" equations. A key focus will be on the role played by modular curves, complex geometric objects often associated with Diophantine equations. In particular, new geometric techniques will be developed to study these curves by exploiting their symmetries.