Certain classes of algebraic structures, such as Galois extensions, central division algebras and quadratic forms have played an outsized role in understanding the arithmetic of fields, and find rich and unexpected connections to other branches of mathematics. The task of describing these and similar objects rests on the interplay of three interacting parts: their structure theory, the geometry of the parameters which describe them, and the arithmetic of fields. Despite the range of contexts of such problems, there are some basic ideas which are fundamental in how we approach them. In this talk I will discuss various algebraic avatars of the notion of classifying spaces, and how aspects of field arithmetic may be encoded in combinatorial ways.

### Penn Mathematics Colloquium

Wednesday, April 7, 2021 - 3:30pm

#### Danny Krashen

Rutgers