Motivated by integrals of the Calculus of Variations considered in Nonlinear Elasticity, we study mathematical models which do not fit in the classical existence and regularity theory for elliptic and parabolic Partial Differential Equations. We consider general nonlinearities with non-standard p,q-growth, both in the elliptic and in the parabolic contexts. In particular, we introduce the notion of "variational solution/parabolic minimizer" for a class of Cauchy-Dirichlet problems related to systems of parabolic equations.

Using recent advances in the study of the mapping class group as our motivation, we introduce tools to study the outer automorphism group of a free group Out(F) via its action on free factors. As applications of this new approach, we describe a construction that embeds right-angled Artin groups into Out(F) and we provide a new method to produce fully irreducible automorphisms of free groups. Throughout the talk, we will focus on aspects of Out(F) that distinguish it from the mapping class group and how this affects the study of Out(F). This is a thesis defense.

The von Neumann-Day problem asks if every non-amenable group contains a free group. We discuss this problem in the context of measured equivalence relations and offer a solution in the case of hyperbolic equivalence relations.

Classical harmonic analysis is (among other things) concerned with the description of function spaces by exploiting underlying symmetries. In recent years an algebraic avatar of harmonic analysis has emerged, in which functions are replaced by the systems of linear PDEs they obey. I will survey this theory and some of its applications to representation theory (following joint work with David Nadler).

Inspired by the problem of sensory coding in neuroscience, we study the maximum entropy distribution on weighted graphs with a given expected degree sequence. This distribution is characterized by independent edge weights parameterized by vertex potentials at each node. Rather surprisingly, a single graph sample suffices to determine these parameters and thus the original distribution. We explain how we arrived at this result, first proved by Chatterjee, Diaconis, and Sly for the case of unweighted (binary) graphs, and how it relates to recent work of Sanyal, Sturmfels, and Vinzant on the entropic discriminant in algebraic geometry. Interestingly, our proofs require an intricate study of the inverses of diagonally dominant positive matrices and the combinatorics of bipartite graphs. (Joint work with Shaowei Lin and Andre Wibisono).

The phase space of the classical Chern-Simons theory is the character variety of a surface, i.e. the moduli space of representations of the fundamental group. Locality in field theory allows one to cut the surface into pieces, compute the character variety with its symplectic form on each piece separately and then glue everything back together. Derived symplectic structures appear naturally in the process. I will explain how the local computation of the symplectic structures is related to quasi-Hamiltonian reduction. If I have time, I will mention a parallel story with the classical Wess-Zumino-Witten model, moduli space of flat connections on a Riemann surface and Hamiltonian reduction. This is a thesis defense.

The talk discusses inverse homogenization problem which is a problem of deriving information about the microgeometry of a two-component composite media from given effective properties. The approach is based on reconstruction of the spectral measure of a self-adjoint operator that depends on the geometry of composite. Stieltjes analytic representation of the effective property relates the n-point correlation functions of the microstructure to the moments of the spectral measure, which contains all information about the microgeometry. I show that the problem of identification of the spectral function from effective measurements known in an interval of frequency, has a unique solution. In particular, the volume fractions of materials in the composite and an inclusion separation parameter, as well as the spectral gaps at the ends of the spectral interval, can be uniquely recovered. I will discuss reconstruction of microstructural parameters from electromagnetic and viscoelastic effective measurements, application to coupling of different effective properties, and show an extension to nonlinear composites.