Abstract: The Duistermaat-Heckman exact stationary phase approximation was shown by Berline-Vergne (and independently by Atiyah-Bott) to be a consequence of a localization theorem for equivariant differential forms. This result, although stated in the setting of a Hamiltonian action of a compact Lie group on a symplectic manifold, is really a result about circle actions, since it relies on first choosing a Hamiltonian vector field with isolated zeros and periodic flow. In his paper "Two dimensional gauge theories revisited", Witten proposed a "not necessarily abelian" version of localization for compact Lie groups, and investigated some of the properties of such a localization. His results were motivated by physics, and not fully rigorous from a mathematical point of view. His ideas were subsequently explored by Jeffrey and Kirwan, and later by Paradan, and put on a sound mathematical footing. I will explain Witten's ideas, and briefly explore the approach of Jeffrey and Kirwan before outlining Paradan's approach to the problem, and how his results can be applied to obtain a cohomological formula for the index of transversally elliptic operators.

Abstract: A monad is a device for encoding algebraic structure. Conversely, if an adjunction is monadic (i.e., encodes a category of algebras), this implies several useful categorical properties.

This talk describes joint work with Dominic Verity to develop this theory for quasi-categories (aka infinity-categories). We introduce the free homotopy coherent adjunction, demonstrate that any adjunction of quasi-categories gives rise to such, and give a formal proof of the monadicity theorem that is directly applicable in other contexts.