## Department Colloquium

### Friday, February 9th, 2018

**Time:** 3:30 p.m. **Place:** Jeffery Hall 234

**Speaker:** Daniel Le, University of Toronto

**Title:** The geometry of Galois representations

**Abstract: **The arithmetic of number fields can be profitably studied through the representation theory of their absolute Galois groups. These representations exhibit a number of elegant and surprising phenomena, most famously the quadratic reciprocity law. Many of these phenomena are explained by the modularity conjecture of Langlands that all Galois representations come from modular forms. Startling progress towards this conjecture began with Taylor and Wiles's study of Galois deformation spaces. We give a construction of local models for some Galois deformation spaces coming from geometric representation theory, and describe some applications to modularity conjectures and congruences between modular forms. Much of what we discuss is joint work with Bao Le Hung, Brandon Levin, and Stefano Morra.