Talk 1. Mohamed El Alami - 1:00pm
Pure Mathematics Department, University of Waterloo
“Introduction to schemes”
In the previous century, Grothendieck revolutionized geometry by introducing his theory of schemes which has become one of the most important tools in the study of Algebraic Geometry. In this talk, I will cover some of the tools required to even define what a scheme is; I will briefly introduce presheaves on topological spaces, sheaves, stalks, morphisms of presheaves etc. I will then talk about ringed spaces and some of their properties which will help us define schemes. If time permits, I will also discuss some important properties of schemes and maybe some examples as well.
Talk 2. Ehsaan Hossain - 2:30pm
Pure Mathematics Department, University of Waterloo
“Projective modules”
A theorem of Serre-Swan states that the algebraic counterpart of a vector bundle is a finitely-generated projective module. One can think of projective modules as generalizing free modules - instead of free, they are only required to be "locally free", similar to how vector bundles generalize Cartesian products. The goal of this talk is to demystify the connection between these two objects.