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Texas A&M University
Mathematics

Events for 10/04/2019 from all calendars

Probability Seminar

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Time: 11:30AM - 12:30PM

Location: BLOC 628

Speaker: Axel Sáenz, Tulane

Title: Stationary dynamics in finite time for the totally asymmetric simple exclusion process

Abstract: The totally asymmetric simple exclusion process (TASEP) is a Markov process that is the prototypical model for transport phenomena in non-equilibrium statistical mechanics. It was first introduced by Spitzer in 1970, and in the last 20 years, it has gained a strong resurgence in the emerging field of "Integrable Probability" due to asymptotic results from Johansson in 2000 and Tracy and Widom in 2007 (among other related results). In particular, these formulas led to great insights regarding fluctuations related to the Tracy-Widom distribution and scalings to the Kardar-Parisi-Zhang (KPZ) stochastic differential equation. In this joint work with Leonid Petrov (University of Virginia), we introduce a new and simple Markov process that maps the distribution of the TASEP at time t>0 , given step initial time data, to the distribution of the TASEP at some earlier time t−s>0. This process "back in time" is closely related to the Hammersley process introduced by Hammersley in 1972, which later found a resurgence in the longest increasing subsequence problem in the work of Aldous and Diaconis in 1995. Hence, we call our process the backwards Hammersley-type process (BHP). As a fun application of our results, we have a new proof of the limit shape for the TASEP. The central objects in our constructions and proofs are the Schur point processes and the Yang-Baxter equation for the sl_2 quantum affine Lie algebra. In this talk, we will discuss the background in more detail and will explain the main ideas behind the constructions and proof.


Mathematical Physics and Harmonic Analysis Seminar

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Time: 1:50PM - 2:50PM

Location: BLOC 628

Speaker: Wencai Liu, Texas A&M University

Title: Anderson localization for multi-frequency quasi-periodic operators on higher dimensional latices

Abstract: The first part of the talk, based on a joint work with S. Jitomirskaya and Y. Shi, is devoted to study multi-frequency quasi-periodic operators on higher dimensional lattices. We establish the Anderson localization for general analytic $k$-frequency quasi-periodic operators on $\Z^d$ for arbitrary $k, d$. This is a generalization of Bourgain-Goldstein-Schlag's result $b=d=2$ and Bourgain's result $b=d\geq 3$. Our proof works for Toeplitz operators as well. In the second part of the talk, I will discuss several closely related topics. For example, 1. Use the quantitative unique continuation to establish the Anderson localization of random Schr\"odinger operators with singular distributions. 2. Use rotation $C^{\star}$ algebra to tackle the dry ten Martini problem (gap labelling theorem). 3. Use the machinery of proof of Anderson localization to construct KAM (Kolmogorov-Arnold-Moser) tori for NLS and NLW equations.


Algebra and Combinatorics Seminar

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Time: 3:00PM - 3:00PM

Location: BLOC 628

Speaker: Daniel Eman, Wisconsin

Title: Asymptotic syzygies

Abstract: Asymptotic syzygies refers to the study of the syzygies of a variety under increasingly ample embeddings; the canonical example is to study the syzygies of projective space under the d-uple embedding as d goes to infinity. I’ll discuss some open questions related to asymptotic syzygies, and some recent work of myself and Jay Yang which uses a combinatorial model to produce new heuristics about this topic.


Geometry Seminar

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Time: 4:00PM - 5:00PM

Location: BLOC 628

Speaker: Ursula Whitcher, University of Michigan

Title: Zeta functions of alternate mirror Calabi-Yau families

Abstract: Mirror symmetry predicts surprising geometric correspondences between distinct families of algebraic varieties. In some cases, these correspondences have arithmetic consequences. For example, one can use mirror symmetry to explore the structure of the zeta function, which encapsulates information about the number of points on a variety over a finite field. We prove that if two Calabi-Yau invertible pencils in projective space have the same dual weights, then they share a common polynomial factor in their zeta functions related to a hypergeometric Picard-Fuchs differential equation. The polynomial factor is defined over the rational numbers and has degree greater than or equal to the order of the Picard-Fuchs equation. This talk describes joint work with Charles Doran, Tyler Kelly, Adriana Salerno, Steven Sperber, and John Voight.


Linear Analysis Seminar

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Time: 4:00PM - 5:00PM

Location: BLOC 220

Speaker: Jurij Volcic, TAMU

Title: Noncommutative polynomials describing convex sets

Abstract: The free semialgebraic set D determined by a hermitian noncommutative polynomial f is the set of tuples of hermitian matrices X such that f(X) is positive semidefinite. When f is a hermitian monic linear pencil, D is called a free spectrahedron. Since it is the feasible set of a linear matrix inequality (LMI), it is evidently convex. Conversely, it is well-known that every convex free semialgebraic set is a free spectrahedron. This talk presents a solution to the basic problem of determining those noncommutative polynomials f for which D is convex. A consequence is an effective probabilistic algorithm that not only determines if D is convex, but also produces its LMI representation. Further results address 1x1 noncommutative polynomials describing convex sets.

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