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.

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.

We will start by looking at the basic setup of nonabelian Hodge theory using higher nonabelian cohomology stacks via simplicial presheaves on the site of schemes. The de Rham, Betti and Dolbeault cohomology stacks are defined and related. The de Rham to Dolbeault degeneration glues, via the Riemann-Hilbert correspondence, to create the twistor space. We then specialize to the character variety and look at how Hitchin's equations give rise to prefered sections and the hyperkahler structure. Parabolic structures for quasiprojective varieties fit into a weight yoga. The higher nonabelian cohomology stacks give structures relative to the character variety. Recent work includes an investigation of what happens near infinity in the character variety.

Mixed-norm $L^P$ spaces, generalizing the Lebesgue space, have been studied for over half a century, with various applications in pure and applied mathematics. For classical Lebesgue spaces, given exponents $p$ and $q$ and $\sigma$-finite measures $\mu$ and $\nu$ on the same measurable space, there are well-known conditions for when $L^p(\mu)$ is contained in $L^q(\nu)$. This talk presents a mostly complete solution describing when two (permuted) mixed-norm spaces, again with different exponents and measures, have such an inclusion.

The only non-trivial situation is when the mixed norms integrate over their variables in differing orders, as seen in Minkowski's integral inequality. J.J.F Fournier called these "permuted mixed norms" and developed a generalization of Minkowski's integral inequality which is key to this solution.

A full solution is given when no measure is purely atomic. This turns out to depend only on the necessary one-variable inclusions and a condition derived from Minkowski's integral inequality. When purely atomic measures are allowed, there are still partial solutions, but the situation is substantially more complicated. Solutions in some cases turn out to involve optimization problems in weighted $l^p$.

We will start by looking at the basic setup of nonabelian Hodge theory using higher nonabelian cohomology stacks via simplicial presheaves on the site of schemes. The de Rham, Betti and Dolbeault cohomology stacks are defined and related. The de Rham to Dolbeault degeneration glues, via the Riemann-Hilbert correspondence, to create the twistor space. We then specialize to the character variety and look at how Hitchin's equations give rise to prefered sections and the hyperkahler structure. Parabolic structures for quasiprojective varieties fit into a weight yoga. The higher nonabelian cohomology stacks give structures relative to the character variety. Recent work includes an investigation of what happens near infinity in the character variety.

We will start by looking at the basic setup of nonabelian Hodge theory using higher nonabelian cohomology stacks via simplicial presheaves on the site of schemes. The de Rham, Betti and Dolbeault cohomology stacks are defined and related. The de Rham to Dolbeault degeneration glues, via the Riemann-Hilbert correspondence, to create the twistor space. We then specialize to the character variety and look at how Hitchin's equations give rise to prefered sections and the hyperkahler structure. Parabolic structures for quasiprojective varieties fit into a weight yoga. The higher nonabelian cohomology stacks give structures relative to the character variety. Recent work includes an investigation of what happens near infinity in the character variety.