Georgia Institute of Technology College of Sciences

I present a formalism and an computational scheme to quantify the dynamics of grain boundary migration in polycrystalline materials, applicable to three-dimensional microstructure data obtained from non-destructive coarsening experiments. I will describe a geometric technique of interface tracking using well-established optimization algorithms and demonstrate how, when coupled with very basic physical assumptions, one can effectively measure grain boundary energy density and mobility of a given misorientation type in the two-parameter subspace of boundary inclinations. By doing away with any specific model or parameterization for the energetics, I seek to have my analysis applicable to general anisotropies in energy and mobility. I present results in two proof-of-concept test cases, one first described in closed form by J. von Neumann more than half a century ago, and the other which assumes analytic but anisotropic energy and mobility known in advance.

In this talk, we will consider semiconcavity of viscosity solutions for a class of degenerate elliptic integro-differential equations in R^n. This class of equations includes Bellman equations containing operators of Levy-Ito type. Holder and Lipschitz continuity of viscosity solutions for a more general class of degenerate elliptic integro-differential equations are also provided.

Let f be a rational self-map of the complex projective plane. A central problem when analyzing the dynamics of f is to understand the sequence of degrees deg(f^n) of the iterates of f. Knowing the growth rate and structure of this sequence in many cases enables one to construct invariant currents/measures for dynamical system as well as bound its topological entropy. Unfortunately, the structure of this sequence remains mysterious for general rational maps. Over the last ten years, however, an approach to the problem through studying dynamics on spaces of valuations has proved fruitful. In this talk, I aim to discuss the link between dynamics on valuation spaces and problems of degree/order growth in complex dynamics, and discuss some of the positive results that have come from its exploration.

We investigate the existence of quasi-periodic solutions for state-dependent delay differential equationsusing the parameterization method, which is different from the usual way-working on the solution manifold. Under the assumption of finite-time differentiability of functions and exponential dichotomy, the existence and smoothness of quasi-periodic solutions are investigated by using contraction arguments We also develop a KAM theory to seek analytic quasi-periodic solutions. In contrast with the finite differentonable theory, this requires adjusting parameters. We prove that the set of parameters which guarantee the existence of analytic quasi-periodic solutions is of positive measure. All of these results are given in an a-posterior form. Namely, given a approximate solution satisfying some non-degeneracy conditions, there is a true solution nearby.