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: TBA