# Course Descriptions

**Uhlenbeck Lecture Course**: A glimpse into the Langlands program

Lecturer: Ana Caraiani, Imperial College London

Teaching Assistant: Alice Pozzi, University of Bristol

The goal of this lecture series is to give you a glimpse into the Langlands program, a central topic at the intersection of algebraic number theory, algebraic geometry and representation theory. In the first lecture, we will look at a celebrated instance of the Langlands correspondence, namely the modularity of elliptic curves. I will try to give you a sense of the different meanings of modularity and of the multitude of ingredients that go into establishing such a result. In the following lectures, I will focus on the more geometric ingredients, first in the special case of the modular curve and then for higher-dimensional Shimura varieties.

**Terng Lecture Course**: Deligne-Lusztig theory: examples and applications

Lecturer: Charlotte Chan, University of Michigan

Teaching Assistant: Si Ying Lee, Harvard University

Geometry and representation theory are intertwined in deep and foundational ways. One of the most important instances of this relationship was uncovered in the 1970s by Deligne and Lusztig: the representation theory of matrix groups over finite fields is encoded in the geometry of a natural "partition" of flag varieties. Recent developments have revealed rich connections between Deligne-Lusztig varieties and geometry studied in number-theoretic contexts. In this lecture series, we give an example-based tour of these ideas, focusing on how to extract concrete information from theory.

**Prerequisites for the courses**:

Uhlenbeck Course: One course in algebraic number theory and one course in algebraic geometry.

Terng Course: Two courses in algebra.