Title: Energy concentration and dispersion in large data wave maps

Abstract: While global existence and scattering is known for small data wave maps, the large data problem is still open. In this talk I will describe ongoing work, joint with Jacob Sterbenz, whose aim is to establish a dichotomy between dispersive behaviour and soliton-like concentration for large data problems.

Title: Existence of Solutions to Boundary Value Problems for Smectic Liquid Crystals

Abstract: We investigate variational problems for an energy developed by the physicists Chen and Lubensky, that was used to to describe stable equilibrium states for liquid crystal materials. This energy and some modifications of it have been used since the 1970's to analyze liquid crystal materials that include two co-existing phases, namely nematic and smectic states. It has also been used by physicists and mathematicians to explain phase transitions from nematic to smectic states under the application of an exterior magnetic field.
Although physically, smectic phases often occur as a result of a phase melt at the boundary, existence to minimizers of these energies subject to a general class of boundary values was unknown. We present recent results on existence and regularity of minimizers subject to a robust class of bounary values for a Chen-Lubensky energy.

Title: Water waves over a random topography

Abstract: We consider the problem of nonlinear wave motion of the free surface of a body of fluid over a variable bottom. The object is to describe the character of wave propagation in a long wave asymptotic regime, under the assumption that the bottom of the fluid region is described by a stationary random process whose variations take place on short length scales. Our principal result is the derivation of effective equations and a consistency analysis. We compute the effects of random modulation of solutions, and give an explicit expression for the scattered component of the solution due to waves interacting with the random bottom.

Title: Energy Estimates for Degenerate Hyperbolic Equations

Abstract: Energy estimates are important in discussions of various classes of differential equations. It is well known that they hold for Cauchy problems for hyperbolic equations. However, it is not known whether they hold for Cauchy problems for general degenerate hyperbolic equations. In this talk, we will discuss some recent progress on this subject. The geometry of zero sets of degenerate coefficients plays a crucial role.

Title: A One Parameter Family of Expanding Wave Solutions of the Einstein Equations that Induces an Anomalous Acceleration into The Standard Model of Cosmology

Abstract: Following up on an idea of Blake Temple that the anomalous acceleration of the universe might be due to a secondary expansion wave reflected back from the shock wave in our shock wave cosmology model (and numerical computations of his student, Zeke Vogler, using a locally inertial numerical method that Temple and Groah derived), Temple invited me to join him on this project when he held the Gehring Chair at U of M. Together we discovered a surprising new family of expansion waves which perturb The Standard Model of Cosmology, and lead to a mathematically rigorous, non-ad-hoc possible explanation of the accelerating universe based only on Einstein's equations of General Relativity.

Title: Classical solutions to purely integro-differential Hamilton-Jacobi-Bellman equations.

Abstract: Hamilton-Jacobi-Bellman equations arise from stochastic control problems. When the stochastic processes involved are purely jump Levy processes, the corresponding equation is a supremum of linear integro-differential ones. We prove that their viscosity solutions are regular enough to consider the solutions to be classical. For the case of second order elliptic PDEs, this corresponds to a celebrated theorem by Evans and Krylov.
This is a joint work with Luis Caffarelli.