Georgia Institute of Technology College of Sciences

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.

In this talk we examine the typical behavior of a trajectory of a piecewise smooth system in the neighborhood of a co-dimension 2 discontinuity manifold $\Sigma$. It is well known that (in the class of Filippov vector fields, and under commonly occurring conditions) one may anticipate sliding motion on $\Sigma$. However, this motion itself is not in general uniquely defined, and recent contributions in the literature have been trying to resolve this ambiguity either by justifying a particular selection of a Filippov vector field or by substituting the original discontinuous problem with a regularized one. However, in this talk, our concern is different: we look at what we should expect of a typical solution of the given discontinuous system in a neighborhood of $\Sigma$. Our ultimate goal is to detect properties that are satisfied by a sufficiently wide class of discontinuous systems and that (we believe) should be preserved by any technique employed to define a sliding solution on $\Sigma$.