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Theresa Anderson, Purdue University
Spheres, primes, and triangles: tales from the interface of harmonic analysis and number theory
Pioneered by Bourgain, the fusion of Fourier analytic and number theoretic techniques in novel ways have led to a variety of discrete operator bounds where continuous techniques fail. Moreover, many distributional questions can be answered in a quantitatively strong way by knowing such bounds. We discuss recent work pertaining to distribution of primes on spheres, higher degree spherical maximal functions and three-point configurations.
Erin Beckman, Duke University
Phase transitions in chase-escape with death
Chase-escape with death is a competitive random growth model taking place on a graph occupied by red and blue particles. Red particles can spread to adjacent, unexplored sites, and blue particles can expand to adjacent, red sites. Red particles can also die, leaving the site uninhabitable. We study the process on the d-ary, rooted tree and are interested in the impact of varying two parameters: the expansion speed of red particles and the death rate of red particles. Our recent work shows the existence of two phase transitions and gives connections between this process and several combinatorial objects, including weighted Catalan numbers and continued fractions.
Jintao Deng, Texas A&M University
Novikov conjecture and group extension
An important problem in higher dimensional topology is the Novikov conjecture on the homotopy invariance of higher signature. The Novikov conjecture is a consequence of the Strong Novikov conjecture in the K-theory of group C*-algebras. In 2000, G. Yu proved that the Novikov conjecture holds for the group which admits a coarsely embedding into Hilbert space. The coarse embedding is not preserved by the group extension. In this talk, I will talk about the Novikov conjecture for the extension of the coarse embeddable group.
David Kerr, Texas A&M University
Dynamical alternating groups, property Gamma, and inner amenability
I will show that the alternating group of a topologically free action of a countably infinite amenable group on the Cantor set has property Gamma (and in particular is inner amenable) and that there are large classes of such groups which are simple, finitely generated, and nonamenable. This is joint work with Robin Tucker-Drob.
Jeffrey Kuan, Texas A&M University
Systematic constructions of Markov duality functions
Markov duality in spin chains and exclusion processes has found a wide variety of applications throughout probability theory. We review the duality of the asymmetric simple exclusion process (ASEP) and its underlying algebraic symmetry. We then explain how the algebraic structure leads to a wide generalization of models with duality, such as higher spin exclusion processes, zero range processes, stochastic vertex models, dynamic models, and their multi-species analogues.
Dominique Maldague, UC Berkeley
Problems related to the Riesz-Sobolev and Brascamp-Lieb inequalities
The Riesz-Sobolev inequality and Brascamp-Lieb inequality both involve multilinear integrals (without any oscillatory factor). We compare the different types of questions that have been asked about these inequalities. In particular, I will describe recent work studying maximizing configurations for a generalization of the Riesz-Sobolev inequality. I will also describe a recent weak Brascamp-Lieb inequality.
Elizabeth Meckes, Case Western Reserve University
Random matrices with prescribed eigenvalues
Much of classical random matrix theory deals with prescribing the joint distribution of the matrix entries, and then asking about the distribution of the resulting eigenvalues. In joint work with M. Meckes, we consider the opposite question: given a matrix whose eigenvalues are specified, but is otherwise random, what do the entries typically look like? More specifically, if a random matrix is chosen according to the canonical probability distribution on the set of real symmetric or complex Hermitian matrices having eigenvalues λ1,…,λn, then under mild conditions, when n is large linear functionals of the entries of such random matrices have approximately Gaussian joint distributions. In the context of quantum mechanics, these results can be viewed as describing the joint probability distribution of the expectation values of a family of observables on a quantum system in a random mixed state. I will also discuss other applications, in particular to the spectral distributions of submatrices, the classical invariant ensembles, and to a probabilistic counterpart of the Schur–Horn theorem, relating eigenvalues and diagonal entries of Hermitian matrices.
Jelani Nelson, UC Berkeley
Sampling from sketching, an optimal lower bound
The work of Jowhari et al. 2011 showed how to use sketches for sparse recovery, popular in the compressed sensing literature, to develop low-memory schemes for sampling from a set being dynamically updated in a data stream. Ahn, Guha, and McGregor in a sequence of two papers then showed how to use this sampling primitive to solve a number of dynamic graph problems using low memory, such as connectivity, and computing a spanning forest, max matching, minimum spanning tree, and other such graph problems. Much follow-up work has extended the AGM approach to several other graph problems as well. In this talk, we show optimality of the sampling primitive developed by Jowhari et al.
This talk is based on joint work with Michael Kapralov, Jakub Pachocki, Zhengyu Wang, David P. Woodruff, and Mobin Yahyazadeh.
Lauren Ruth, Vanderbilt University
Properly proximal groups and measure equivalence
Boutonnet, Ioana, and Peterson introduced the class of properly proximal groups and studied their von Neumann algebras in 2018. In this talk, we will review definitions and examples to illustrate the wide scope of this class of groups; then, we will show that proper proximality is a measure-equivalence invariant: If two groups are measure equivalent, then one group is properly proximal if and only if the other group is properly proximal. This is joint work with Ishan Ishan and Jesse Peterson.