Titles and Abstracts:

Emily Riehl: Applications of Functoriality

This talk will introduce the concept of a functor from category theory assuming no prior acquaintance and highlight some applications of this notion. In the first part we will conceive of a functor as a large mapping that builds a bridge between two different mathematical theories. As a prototypical example we will consider the fundamental group construction from algebraic topology and explain how its functoriality leads to a proof of the Brouwer fixed point theorem. In the second part, we will meet much smaller functors which can be used to define combinatorial models of topological data. We will explain how the search for a functorial clustering algorithm lead to a breakthrough in topological data analysis and speculate how a similar functor might be used to define a convenient category of metric spaces.

Chris Hall: The Weil Conjectures (Basic Notions)

Abstact: Andre Weil posed his famous conjectures in 1949. He knew them to be true for curves having contributed a large part of the proof. They inspired Alexander Grothendieck to completely rewrite the foundations of algebraic geometry and forever change the subject. He envisioned they would following by resolving his Standard Conjectures, but those conjectures remain unresolved and his vision unfulfilled. Pierre Deligne, his brightest student, saw a way around them though, and in 1974 he presented a proof of Weil's conjectures.

In 1980, Deligne published a new proof of the conjectures and a lot more. In this talk we will recall the statement of the conjectures and discuss aspects of Deligne's proofs. If time permits, we will also highlight the significant new results in his second proof. This talk is meant for non-experts, so interested graduate students are encouraged to attend.