The Atiyah-Bott-Shapiro map and the interacting classification of free fermions

Speaker: Arun Debray – Purdue University

Abstract: The Atiyah-Bott-Shapiro map $\widehat{A}\colon MSpin \longrightarrow KO$ is the preeminent example of an orientation in stable homotopy theory. In the first part of this talk, I will describe the data of $\widehat{A}$: what are $MSpin$ and $KO$, and what is the map between them? In the second half of the talk, I will describe how $\widehat{A}$ and some of its variants provide a mathematical model for the classification of topological phases of matter with free fermion Hamiltonians. This second half will include work of Freed-Hopkins and work in progress of myself, Krulewski, Pacheco-Tallaj, and Stehouwer.