Abstract:  Given a permutation of size n, we can create a new permutation of size n+1 by appending a value k between 1 and n+1 to the end of the permutation, increasing previous values that are at least k, so the new sequence remains a permutation. This process is a natural way to generate a uniformly random permutation where at each step we chose k uniformly from {1,…,n+1}. If we restrict our choices of k at each step so that the new permutation avoids a particular pattern, we get interesting distributions on pattern-avoiding permutations that differ somewhat drastically from the uniform distribution on the same pattern-avoiding class. We will focus on the class of separable online permutations and describe a few different ways in which they are qualitatively different from other models of separable permutations.  We show a permuton limit which is deterministic (versus random in the other known models).  We also show that fluctuations about the deterministic limit converge to Brownian motion.

Abstract: In this talk, I will present recent progress on weighted Fourier restriction estimates. I will begin by introducing the problem and outlining its main motivations and applications. I will then discuss several sharp examples and highlight the key methods that have been developed to address the problem.

Abstract: Given N circles in the plane, no three tangent at a point, how many pairs of circles can be internally tangent? Tom Wolff posed this question in 1997 as a toy model for a problem about the wave equation. It has previously been shown that at most N^{3/2} pairs can be tangent.  Nobody has found an example with more than N^{4/3} tangent pairs (which can be accomplished with a grid of N evenly spaced circles), so there is a gap between N^{3/2} and N^{4/3}. I’ll explain how, using ideas from Fourier analysis—namely orthogonality and the uncertainty principle—we can prove that for every approximately gridlike set of N circles, no three tangent at a point, there are at most N^{25/18 + ε} many tangent pairs. Based on joint work with Dominique Maldague.

Abstract: We introduce a continuous policy-value iteration algorithm where the approximations of the value function of a stochastic control problem and the optimal control are simultaneously updated through Langevin-type dynamics. This framework applies to both the entropy-regularized relaxed control problems and the classical control problems, with infinite horizon. We establish policy improvement and demonstrate convergence to the optimal control under the monotonicity condition of the Hamiltonian. By utilizing Langevin-type stochastic differential equations for continuous updates along the policy iteration direction, our approach enables the use of distribution sampling and non-convex learning techniques in machine learning to optimize the value function and identify the optimal control simultaneously. Numerical examples will be presented for both concave and non-concave examples. This talk is based on a joint work with Gu Wang from WPI.