Welcome to the Rutgers Symmetric Functions & Probability Theory seminar page.
This seminar runs at 10:45 AM on Wednesdays in Hill 705.
The organizers are Swee Hong Chan , Konstantin Matveev, Siddhartha Sahi and Hong Chen.
This page is currently maintained by Hong Chen. If you wish to join our mailing list, please click here or email hc813@math.rutgers.edu.
For past talks, click here.
Speaker: Michael Chapman (IAS)
Title: Subgroup Tests and the Aldous--Lyons Conjecture
Abstract: The Aldous-Lyons Conjecture states that every (unimodular random rooted) infinite graph can be (Benjamini-Schramm) approximated by finite graphs. This conjecture is an analogue of other influential conjectures in mathematics concerning how well certain infinite objects can be approximated by finite ones; examples include Connes' embedding problem (CEP) in functional analysis and the soficity problem of Gromov-Weiss in group theory. These became major open problems in their respective fields, as many other long standing open problems, that seem unrelated to any approximation property, were shown to be true for the class of finitely-approximated objects. For example, Gottschalk's conjecture and Kaplansky's direct finiteness conjecture are known to be true for sofic groups, but are still wide open for general groups.
In 2019, Ji, Natarajan, Vidick, Wright and Yuen resolved CEP in the negative. Quite remarkably, their result is deduced from complexity theory, and specifically from undecidability in certain quantum interactive proof systems. Inspired by their work, we suggest a novel interactive proof system which is related to the Aldous-Lyons conjecture in the following way: If the Aldous-Lyons conjecture was true, then every language in this interactive proof system is decidable. A key concept we introduce for this purpose is that of a Subgroup Test, which is our analogue of a Non-local Game. By providing a reduction from the Halting Problem to this new proof system, we refute the Aldous-Lyons Conjecture.
This talk is based on joint work with Lewis Bowen, Alex Lubotzky, and Thomas Vidick.
No special background in probability theory, group theory, complexity theory or quantum information theory will be assumed.
Speaker: Siddhartha Sahi (Rutgers)
Title: A power series with positive coefficients
Abstract: Consider the $n$-variable polynomial $P(x)=P(x_1,..,x_n)= (1-x_1-...-x_n) \prod_{i=1}^n (1-x_i)$. We show that the power series expansion of $1-P(x)^a$ has positive coefficients for $0\le a\le 1/4$.
A similar positivity result was conjectured by H. Lewy and K. Friedrichs in the course of their work on the discretized wave equation, and proved by G. Szego. This result has been extended considerably over the years by Askey-Gasper, Gillis-Resnick-Zeilberger, Scott-Sokal and others, but these techniques do not seem to apply to our problem.
We will explain the genesis of this problem, which has to do with the speaker's conjectural generalization of the FKG inequality; and give a self-contained proof, which involves a couple of new ideas.
Speaker: Hong Chen (Rutgers)
Title: Characterization of Macdonald's hypergeometric series ${}_r\Phi_s$ via $q$-difference equations
Abstract: Macdonald introduced the hypergeometric series ${}_r\Phi_s$ associated with Macdonald polynomials, extending both the $q$-hypergeometric and Jack hypergeometric series. In this talk, I will present $q$-difference equations that characterize Macdonald's hypergeometric series.
Speaker: Eyob Tsegaye (Princeton)
Title: Limit Profile for the Mixing Time of the TASEP in the High and Low Density Phase
Abstract: The totally asymmetric simple exclusion process is a widely studied interacting particle system where particles hop to the right on a line according to independent rate 1 Poisson clocks, with the constraint that two particles cannot occupy the same space. We consider this process on a finite line segment of length N with open boundaries, so that particles can also enter from the left boundary and exit from the right. It was previously shown by Elboim and Schmid that under high and low density conditions, this process rapidly changes from unmixed to mixed in a critical time window around CN, where C is an explicit constant - the so-called cutoff phenomenon. We refine this statement to show the exact decay of the total variation distance within the critical time window. Along the way, we obtain an enlightening picture of how the process gradually mixes. This is based on ongoing work with Dominik Schmid.
Speaker: Charles Bordenave (Institut de Mathématiques de Marseille, IAS)
Title: Cutoff for geodesic paths on hyperbolic manifolds
Abstract: This is a joint work with Joffrey Mathien. We establish new instances of the cutoff phenomenon for geodesic paths and for the Brownian motion on compact hyperbolic manifolds. We prove that for any fixed compact hyperbolic manifold, the geodesic path started on a spatially localized initial condition exhibits cutoff. Our work also extends results obtained by Golubev and Kamber on hyperbolic surfaces of large volume to any dimension. More generally, we will discuss ongoing works on the cutoff phenomenon in mixing dynamical systems.
Speaker: Michael Chapman (IAS)
Title: Subgroup Tests and the Aldous--Lyons Conjecture
Abstract: The Aldous-Lyons Conjecture states that every (unimodular random rooted) infinite graph can be (Benjamini-Schramm) approximated by finite graphs. This conjecture is an analogue of other influential conjectures in mathematics concerning how well certain infinite objects can be approximated by finite ones; examples include Connes' embedding problem (CEP) in functional analysis and the soficity problem of Gromov-Weiss in group theory. These became major open problems in their respective fields, as many other long standing open problems, that seem unrelated to any approximation property, were shown to be true for the class of finitely-approximated objects. For example, Gottschalk's conjecture and Kaplansky's direct finiteness conjecture are known to be true for sofic groups, but are still wide open for general groups.
In 2019, Ji, Natarajan, Vidick, Wright and Yuen resolved CEP in the negative. Quite remarkably, their result is deduced from complexity theory, and specifically from undecidability in certain quantum interactive proof systems. Inspired by their work, we suggest a novel interactive proof system which is related to the Aldous-Lyons conjecture in the following way: If the Aldous-Lyons conjecture was true, then every language in this interactive proof system is decidable. A key concept we introduce for this purpose is that of a Subgroup Test, which is our analogue of a Non-local Game. By providing a reduction from the Halting Problem to this new proof system, we refute the Aldous-Lyons Conjecture.
This talk is based on joint work with Lewis Bowen, Alex Lubotzky, and Thomas Vidick.
No special background in probability theory, group theory, complexity theory or quantum information theory will be assumed.
Speaker: Marcelo Sales (UC Irvine)
Title: On the edge expansion of random polytopes
Abstract: A $0/1$-polytope in $\mathbb{R}^n$ is the convex hull of a subset of $\{0,1\}^n$. The graph of a polytope $P$ is the graph whose vertices are the zero-dimensional faces of $P$ and whose edges are the one-dimensional faces of $P$. A conjecture of Mihail and Vazirani states that the edge expansion of the graph of every $0/1$-polytope is at least one. In this talk, we study a random version of the problem, where the polytope is generated by selecting vertices of $\{0,1\}^n$ independently at random with probability $p\in (0,1)$. Improving earlier results, we show that, for any $p\in (0,1)$, with high probability the edge expansion of the random $0/1$-polytope is bounded from below by an absolute constant.
This is joint work with Asaf Ferber, Michael Krivelevich, and Wojciech Samotij.
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Special time: 2:00 pm
Speaker: Cesar Cuenca (Ohio State)
Title: Discrete $N$-particle ensembles at high temperature through Jack symmetric functions
Abstract: I will speak about random discrete N-particle systems with the inverse temperature parameter theta. We find necessary and sufficient conditions for the Law of Large Numbers as the size N of the system tends to infinity simultaneously with the inverse temperature going to zero. We obtain the LLN for multiparameter families of Markov chains of N nonintersecting particles and the LLN for the multiplication of Jack symmetric functions, as the inverse temperature tends to zero. We express the answer in terms of novel one-parameter deformations of cumulants and discuss their relation to (quantized) free probability. Finally, we discuss a crystallization phenomenon and describe it in terms of the countable real roots of certain special functions. The talk is based on joint work with Maciej Dolega.
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