Talk 1. Janis Lazovskis, Pure Mathematics Department, University of Waterloo
“Sheaf cohomology and manifolds”
We will discuss the concept of sheaves of functions on manifolds and the related definitions, as well as the notion of an exact sequence of sheaves. After some basic examples, the process of going from a short exact sequence of sheaves to a long exact sequence on sheaf cohomology will be described, as well as the effect of the boundary map defining the first Chern class of a complex line bundle. This foreign differential geometric terminology will be meticulously defined in an algebraic geometry setting.
Talk 2. Mateusz Olechnowicz, Pure Mathematics Department, University of Waterloo
“The uniformization theorem”
Picking up where we left off, we will determine the differential equation for ℘ and use it to show that to each lattice we may assign an elliptic curve parametrized by elliptic functions. The converse of this assignment is known as the inversion problem for Eisenstein series. Its solution, given by the uniformization theorem, is the goal of our talk. Our main tools will be the modular group and the j-invariant.