Georgia Institute of Technology College of Sciences

We use Fourier analysis to establish $L^p$ bounds for Stein's spherical maximal theorem in the setting of compactly supported Borel measures $\mu, \nu$ satisfying natural local size assumptions $\mu(B(x,r)) \leq Cr^{s_{\mu}}, \nu(B(x,r)) \leq Cr^{s_{\nu}}$. As an application, we address the following geometric problem: Suppose that $E\subset \mathbb{R}^d$ is a union of translations of the unit circle, $\{z \in \mathbb{R}^d: |z|=1\}$, by points in a set $U\subset \mathbb{R}^d$. What are the minimal assumptions on the set $U$ which guarantee that the $d-$dimensional Lebesgue measure of $E$ is positive?

Abstract: If P(z) is a polynomial, then log|P(z)| is a potential. We discuss some facets of this observation, and some gems in classical potential theory. A special topics course on potential theory will be offered in the fall.

Fokker-Planck equations, along with stochastic differential equations, play vital roles in physics, population modeling, game theory and optimization (finite or infinite dimensional). In this thesis, we study three topics, both theoretically and computationally, centered around them.In part one, we consider the optimal transport for finite discrete states, which are on a finite but arbitrary graph. By defining a discrete 2-Wasserstein metric, we derive Fokker-Planck equations on finite graphs as gradient flows of free energies. By using dynamical viewpoint, we obtain an exponential convergence result to equilibrium. This derivation provides tools for many applications, including numerics for nonlinear partial differential equations and evolutionary game theory.In part two, we introduce a new stochastic differential equation based framework for optimal control with constraints. The framework can efficiently solve several real world problems in differential games and Robotics, including the path-planning problem.In part three, we introduce a new noise model for stochastic oscillators. With this model, we prove global boundedness of trajectories. In addition, we derive a pair of associated Fokker-Planck equations.

In Watchareepan Atiponrat's thesis the properties of decomposable exact Lagrangian codordisms betweenLegendrian links in R^3 with the standard contact structure were studied. In particular, for any decomposableexact Lagrangian filling L of a Legendrian link K, one may obtain a normal ruling of K associated with L.Atiponrat's main result is that the associated normal rulings must have an even number of clasps. As a result, there exists a Legendrian (4,-(2n +5))-torus knot, for each n >= 0, which does not have a decomposable exact Lagrangian filling because it has only 1 normal ruling and this normal rolling has odd number of clasps.