For more information please visit: https://scgp.stonybrook.edu/archives/45985

featuring Tom Manuel, cornet; Chris Donohue, alto saxophone; John Marshall, tenor saxophone; Joe Devassy, trombone; Dean Johnson, bass; Jon Mele, drums; Alex Pastrano Sotto, piano. This show will feature a nonet performance of Miles Davis’ Birth of the Cool album.

Speaker:   Greta Panova
Title:   Asymptotic algebraic combinatorics Part 3
Abstract:   TBA
Hashtag: #Workshop

Speaker:   Tomohiro Sasamoto
Title:   Large deviations of interacting particle systems Part 1
Abstract:   We consider large deviations of interacting particle systems and their connections to integrable systems. For the most standard system of symmetric simple exclusion process(SEP), the large deviation principle (LDP) was established by Kipnis-Olla-Varadhan in 1990. A related but somewhat different physical approach, called the macroscopic fluctuation theory (MFT), was introduced and developed by Jona-Lasinio et al from 2001 for a larger class of interacting particle systems. In this formulation the large deviation is determined by solving the MFT equations, which is a coupled non-linear PDEs. As such they are in general difficult to solve, but the MFT equations for SEP was mapped to a classical integral system (AKNS system) and solved by inverse scattering method a few years ago.
In this lecture, we discuss these subjects by first giving an overview, explaining some basics about large deviation and then introducing a large deviation on a lattice for a class of interacting particle systems with spin. We also discuss connections to microscopic calculations by Bethe ansatz, ballistic version of MFT and so on.
Hashtag: #workshop

Speakers: TBA
Title:   Lightning Talks
Hashtag: #workshop

Speaker:   Tomohiro Sasamoto
Title:   Large deviations of interacting particle systems Part 2
Abstract:   We consider large deviations of interacting particle systems and their connections to integrable systems. For the most standard system of symmetric simple exclusion process(SEP), the large deviation principle (LDP) was established by Kipnis-Olla-Varadhan in 1990. A related but somewhat different physical approach, called the macroscopic fluctuation theory (MFT), was introduced and developed by Jona-Lasinio et al from 2001 for a larger class of interacting particle systems. In this formulation the large deviation is determined by solving the MFT equations, which is a coupled non-linear PDEs. As such they are in general difficult to solve, but the MFT equations for SEP was mapped to a classical integral system (AKNS system) and solved by inverse scattering method a few years ago.
In this lecture, we discuss these subjects by first giving an overview, explaining some basics about large deviation and then introducing a large deviation on a lattice for a class of interacting particle systems with spin. We also discuss connections to microscopic calculations by Bethe ansatz, ballistic version of MFT and so on.
Hashtag: #workshop

Speaker:   Lauren Williams
Title:   TBA

Speakers: TBA
Title:   Lightning Talks
Hashtag: #workshop

Speaker:   Tomohiro Sasamoto
Title:   Large deviations of interacting particle systems Part 3
Abstract:   We consider large deviations of interacting particle systems and their connections to integrable systems. For the most standard system of symmetric simple exclusion process(SEP), the large deviation principle (LDP) was established by Kipnis-Olla-Varadhan in 1990. A related but somewhat different physical approach, called the macroscopic fluctuation theory (MFT), was introduced and developed by Jona-Lasinio et al from 2001 for a larger class of interacting particle systems. In this formulation the large deviation is determined by solving the MFT equations, which is a coupled non-linear PDEs. As such they are in general difficult to solve, but the MFT equations for SEP was mapped to a classical integral system (AKNS system) and solved by inverse scattering method a few years ago.
In this lecture, we discuss these subjects by first giving an overview, explaining some basics about large deviation and then introducing a large deviation on a lattice for a class of interacting particle systems with spin. We also discuss connections to microscopic calculations by Bethe ansatz, ballistic version of MFT and so on.
Hashtag: #workshop

Speaker:   Rick Kenyon
Title:   Dimers and webs Part 1
Abstract:   We'll discuss connections between the dimer model and representation theory of SL_nand other Lie groups. The basic result is a generalization of Kasteleyn's theorem (which counts dimer covers of planar graphs using the Pfaffian of an adjacency-type matrix), to the setting of a graph with a matrix-valued connection on edges

Speaker:   Persi Diaconis
Title:   Markov Chains for Enumeration under Symmetry Lecture 1
Abstract:   Let G be a finite group acting on finite set X. This divides X into orbits and we ask ?how many orbits are there?, ?What is the typical size of an orbit?, ?do the orbits have 'nice names'?, ?do they fit together into some kind of 'moduli space'?. The Burnside process addresses the first two topics. It allows uniform sampling of a random orbit. It runs the following Markov chain on X: From x, choose g fixing x (uniformly) and then y fixed by this g (uniformly). The chain moves from x to y. This chain has stationary distribution pi(x) proportional to the size of the orbit containing x. Thus simply reporting the size of the current orbit gives a Markov chain with a uniform stationary distribution.
Problems abound: How do you actually carry out the two steps in problems of
interest? How can you convert the output into a useful estimate of the number and size of orbits? What is the running time and typical behaviour of this Markov chain? There is progress in special cases, but in general, all problems are open.

Lecture 1: An introduction to Polya Theory, the Burnside process and auxiliary variables algorithms
Lecture 2: Three real examples: Polya trees, contingency tables with fixed row and
column sums, The commuting graph process for generating random partitions of n (and counting the number of conjugacy classes in a finite group).
Lecture 3: Careful proof in a simple case: the binary Burnside process, from
Tchebychev polynomials to Schur-Weyl duality.
I expect the audience to know something about Markov chains (at the level of Levin-
Peres' book) and something about groups (a good undergraduate course should
suffice). These topics are based on my current research and you can look at papers on my homepage for the past year or two.
Hashtag: #workshop

Speakers: TBA
Title:   Lightning Talks
Hashtag: #workshop

Speaker:   Rick Kenyon
Title:   Dimers and webs Part 2
Abstract:   We'll discuss connections between the dimer model and representation theory of SL_nand other Lie groups. The basic result is a generalization of Kasteleyn's theorem (which counts dimer covers of planar graphs using the Pfaffian of an adjacency-type matrix), to the setting of a graph with a matrix-valued connection on edges

Speaker:   Persi Diaconis
Title:   Markov Chains for Enumeration under Symmetry Lecture 2
Abstract:   Let G be a finite group acting on finite set X. This divides X into orbits and we ask ?how many orbits are there?, ?What is the typical size of an orbit?, ?do the orbits have 'nice names'?, ?do they fit together into some kind of 'moduli space'?. The Burnside process addresses the first two topics. It allows uniform sampling of a random orbit. It runs the following Markov chain on X: From x, choose g fixing x (uniformly) and then y fixed by this g (uniformly). The chain moves from x to y. This chain has stationary distribution pi(x) proportional to the size of the orbit containing x. Thus simply reporting the size of the current orbit gives a Markov chain with a uniform stationary distribution.
Problems abound: How do you actually carry out the two steps in problems of
interest? How can you convert the output into a useful estimate of the number and size of orbits? What is the running time and typical behaviour of this Markov chain? There is progress in special cases, but in general, all problems are open.

Lecture 1: An introduction to Polya Theory, the Burnside process and auxiliary variables algorithms
Lecture 2: Three real examples: Polya trees, contingency tables with fixed row and
column sums, The commuting graph process for generating random partitions of n (and counting the number of conjugacy classes in a finite group).
Lecture 3: Careful proof in a simple case: the binary Burnside process, from
Tchebychev polynomials to Schur-Weyl duality.
I expect the audience to know something about Markov chains (at the level of Levin-
Peres' book) and something about groups (a good undergraduate course should
suffice). These topics are based on my current research and you can look at papers on my homepage for the past year or two.
Hashtag: #workshop

Speaker:   Amol Aggarwal
Title:   Asymptotics for the Toda Lattice Part 1
Abstract:   The Toda lattice prescribes the evolution of N particles interacting under certain Hamiltonian dynamics; it is an archetypal example of a completely integrable system. A question of interest is to understand how the model behaves, under random (or typical) initial data, when the number N of particles becomes large. In this course we describe several results explaining such asymptotics under certain
invariant initial data. The proofs proceed by finding a way to interpret the Toda lattice as a dense collection of "quasi-particles" that behave similarly to solitons, and providing a framework to study how these quasi-particles asymptotically evolve in time. In this analysis, arguments from random matrix theory, particularly the analysis of Lyapunov exponents governing the decay rates of eigenvectors of random tridiagonal matrices, play an important role.
Hashtag: #workshop

Join Long Island Chamber Music for a dynamic concert featuring some of the ‘Greatest Hits’ for string quartet. Experience legendary works by Beethoven, Borodin, Ravel, and Brahms and more.

Speaker:   Amol Aggarwal
Title:   Asymptotics for the Toda Lattice Part 2
Abstract:   The Toda lattice prescribes the evolution of N particles interacting under certain Hamiltonian dynamics; it is an archetypal example of a completely integrable system. A question of interest is to understand how the model behaves, under random (or typical) initial data, when the number N of particles becomes large. In this course we describe several results explaining such asymptotics under certain
invariant initial data. The proofs proceed by finding a way to interpret the Toda lattice as a dense collection of "quasi-particles" that behave similarly to solitons, and providing a framework to study how these quasi-particles asymptotically evolve in time. In this analysis, arguments from random matrix theory, particularly the analysis of Lyapunov exponents governing the decay rates of eigenvectors of random tridiagonal matrices, play an important role.
Hashtag: #workshop

Speaker:   Vadim Gorin
Title:   Random lozenge tilings via Nekrasov equations, Part 1
Abstract:   This course studies the macroscopic behavior and fluctuations of uniformly random lozenge tilings of polygonal domains. We identify tilings with their height functions and focus on central limit theorem-type results for these functions. Our main results establish Gaussian Free Field fluctuations for the associated two-dimensional fields, as well as discrete Gaussian asymptotics for height differences between boundary components of the tiled domain. The class of domains we consider consists of gluings of elementary building blocks, called trapezoids, along a single vertical line. Such gluings may have complicated topology and, in some cases, may even be non-orientable. Exact enumeration results and algebraic properties of lozenge tilings of trapezoids provide a rich set of tools for the analysis. A key ingredient is the identification of the model with a discrete log-gas, which allows for the use of Nekrasov equations (also known as discrete loop or discrete Dyson–Schwinger equations). The course is based on a research monograph written jointly with G. Borot and A. Guionnet.
Hashtag: #workshop