Event № 2006

I will discuss several recent results of (subsets of) Ben Antieau, Asher Auel, Ben Williams and myself where homotopy theory is being used to solve difficult questions in algebra, all concerning Azumaya algebras and their involutions.

Azumaya algebras --- which I will define during the talk --- are generalizations central simple algebras in which the base field is replaced with an arbitrary ring or scheme. Loosely speaking, they are sheaves of algebras looking "locally" like a matrix algebra. They are used in forming of the Brauer group of schemes and topological spaces, and feature in the construction of the Brauer-Manin obstruction for rational points on varieties. 

Azumaya algebras over a space X are also classified by PGL_n-principal homogeneous bundles on X, or equivalently maps from X to the classifying space of PGL_n. This feature allows one to translate various algebraic problems about Azumaya algebras to problems about maps between classifying spaces, which can then be attacked using various tools from algebraic topology. This technique has proved a powerful tool in constructing counterexamples.