Colloquium - Prasit Bhattacharya

Date: February 7, 2022

Time: 4:00PM - 5:00PM

Location: BLOC 117

Speaker: Prasit Bhattacharya, University of Notre Dame

Description:
Title: The Atiyah Real stable Adams conjecture
Abstract: The Adams conjecture, which was famously proved by Quillen and Sullivan independently in the 70s, remains one of the most influential results in homotopy theory. Many geometric results follow, as the statement predicts the behavior of the spherical fibration associated with any vector bundle. The stable Adams conjecture, which is an enhancement of the Adams conjecture in terms of cohomology theories, has its own set of important consequences.
The recent new proof of the stable Adams conjecture (joint with N. Kitchloo) can be modified to prove a version of the Adams conjecture which involves Atiyah Real vector bundles (complex vector bundles with involutions). This talk will present an Atiyah real analog of the Adams conjecture, its stable enhancement in terms of genuine $C_2$-cohomology theories, as well as the following applications to $C_2$-equivariant geometry. Our first application identifies a new infinite family in the $C_2$-equivariant stable homotopy of spheres, and the second, calculates the James periodicity number of Atiyah real projective spaces.