Georgia Institute of Technology College of Sciences

We will give an overview of results on the global existence of solutions to the initial value problem for nonlinear elastic and viscoelastic materials in 3d without boundary. Materials will be assumed to be isotropic, but both compressible and incompressible cases will be discussed. In the compressible case, a key null condition must be imposed to control nonlinear interactions of pressure waves. This necessary assumption is consistent with the physical model. Initial conditions are small perturbations of a stress free reference state. Existence is proven using a fixed point argument which combines energy estimates and with some new dispersive estimates.

We consider the the following fourth order degenerate parabolic equation h_t + (hh_xxx)_x = 0. The equation arises in the lubrication approximation regime, describing the spreading of a thin film liquid with height profile h >= 0 on a plate. We consider the equation as free boundary problem, defined on its positivity set. We address existence and regularity of classical solutions in weighted Hölder and Sobolev spaces.

The problem of understanding the parabolic hull of Brownian motion arises in two different fields. In mathematical physics this is the Burgers-Hopf caricature of turbulence (very interesting, even if not entirely turbulent). In statistics, the limit distribution we study was first considered by Chernoff, and forms the cornerstone of a large class of limit theorems that have now come to be called 'cube-root-asymptotics'. It was in the statistical context that the problem was first solved completely in the mid-80s by Groeneboom in a tour de force of hard analysis. We consider another approach to his solution motivated by recent work on stochastic coalescence (especially work of Duchon, Bertoin, and my joint work with Bob Pego). The virtues of this approach are simplicity, generality, and the appearance of a completely unexpected Lax pair. If time permits, I will also indicate some tantalizing links of this approach with random matrices. This work forms part of my student Ravi Srinivasan's dissertation.

The transportation problem can be formulated as the problem of finding the optimal way to transport a given measure into another with the same mass. In mathematics, there are at least two different but very important types of optimal transportation: Monge-Kantorovich problem and ramified transportation. In this talk, I will give a brief introduction to the theory of ramified optimal transportation. In terms of applied mathematics, optimal transport paths are used to model many "tree shaped" branching structures, which are commonly found in many living and nonliving systems. Trees, river channel networks, blood vessels, lungs, electrical power supply systems, draining and irrigation systems are just some examples. After briefly describing some basic properties (e.g. existence, regularity) as well as numerical simulation of optimal transport paths, I will use this theory to explain the dynamic formation of tree leaves. On the other hand, optimal transport paths provide excellent examples for studying geodesic problems in quasi-metric spaces, where the distance functions satisfied a relaxed triangle inequality: d(x,y) <= K(d(x,z)+d(z,y)). Then, I will introduce a new concept "dimensional distance" on the space of probability measures. With respect to this new metric, the dimension of a probability measure is just the distance of the measure to any atomic measure. In particular, measures concentrated on self-similar fractals (e.g. Cantor set, fat Cantor sets) will be of great interest to us.