Wojciech J. Gajda

Abstract:

One of the most astonishing success stories in recent mathematics is arithmetic geo- metry, which unites methods from classical number theory with algebraic geometry ("schemes"). In this context an extremely important role is played by the Galois groups of base schemes like rings of integers of number fields or rings of holomor- phic functions of curves over finite fields. These groups are the algebraic analogues of topological fundamental groups, and their representations induced by the action on divisor class groups of varieties over these domains yield spectacular results like Serre's Conjecture for two-dimensional representations of the Galois group of Q, which implies for example the modularity of elliptic curves over Q and so Fermat's Last Theorem (and much more). Expanded version of the abstract is available here. Lecture notes are available here.