In this course, we will learn Margulis' argument for counting lattice points in balls of large radius in negative curvature. We will then focus on various counting problems, and see how one can attack them using some tricks or by running a modification of Margulis' argument.

The course syllabus is as follows:

Geometry and dynamics in negative curvature. I will explain how in negative curvature, the picture is essentially the same as Poincare disk. I will introduce notions like Horospheres, Busemann cocycle, comparison triangles, and I’ll give a crash course on coarse geometry.

Margulis’ argument. With the stage set, I will then explain this beautiful and influential argument. This is the heart of the course, and I’ll mostly follow Roblin’s thesis here.

Various recent applications. If time allows, I’ll move to more recent developments, ex., the works of Oh-Shah and Paulin-Parkkonen. I might go over some of my recent works as well.

The goal is that, at the end of the course, students will be able to read (and understand) the recent literature on the topic.