Georgia Institute of Technology College of Sciences

We will study one and two weight inequalities for several different operators from harmonic analysis, with an emphasis on vector-valued operators. A large portion of current research in the area of one weight inequalities is devoted to estimating a given operators' norm in terms of a weight's A_p characteristic; we consider some related problems and the extension of several results to the vector-valued setting. In the two weight setting we consider some of the difficulties of characterizing a two weight inequality through Sawyer-type testing conditions.

This is an expository talk on the arc complex and translation distance of open book decompositions. We will discuss curve complexes, arc complex, open books, and finally the application to contact manifolds.

For many evolutionary dynamics, within a population there are finitely many types that compete with each other. If we think of a type as a strategy, we may consider this dynamic from a game theoretic perspective. This evolution is frequency dependent, where the fitness of each type is given by the expected payoff for an individual in that subpopulation. Considering the frequencies of the population, the logarithmic growth is given by the difference of the respective fitness and the average fitness of the population as a whole. This dynamic is Darwinian in nature, where Nash Equilibria are fixed points, and Evolutionary Stable Strategies are asymptotically stable. Fudenberg and Harris modified this deterministic dynamic by assuming the fitness of each type are subject to population level shocks, which they model by Brownian motion. The authors characterize the two strategy case, while various other authors considered the arbitrary finite strategy case, as well as different variations of this model. Considering how ecological and social anomalies affect fitness, I expand upon the Fudenberg and Harris model by adding a compensated Poisson term. This type of stochastic differential equation is no longer continuous, which complicates the analysis of the model. We will discuss the approximation of the 2 strategy case, stability of Evolutionary Stable Strategies and extinction of dominated strategies for the arbitrary finite strategy case. Examples of applications are given. Prior knowledge of game theory is not needed for this talk.

We study the Dirichlet and Neumann type initial-boundary value problems for strongly degenerate parabolic-hyperbolic equations. We suggest the notions of entropy solutions for these problems and establish the uniqueness of entropy solutions. The existence of entropy solutions is also discussed（joint work with Yuxi Hu and Qin Wang).