Abstract: 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 $l^p$.

Abstract: 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.