김동현 (서울대학교) 

Title: Schubert polynomials and the inhomogeneous TASEP on a ring

Abstract: The inhomogeneous totally asymmetric simple exclusion process (or TASEP) is a Markov chain on the set of permutations, in which adjacent numbers i and j swap places at rate if the larger number is clockwise of the smaller. Conjecturally, steady state probabilities can be written as a positive sum of (double) Schubert polynomials. We will start by giving some background on this model, including Cantini’s result showing that the inhomogeneous TASEP is a solvable lattice model. We will then use his result to show that a large number of states those corresponding to the "evil-avoiding" permutations (permutations avoiding patterns 2413, 4132, 4213, 3214) – have steady state probabilities which are proportional to a product of Schubert polynomials. Based on joint work with Lauren Williams.

김장수 (성균관대학교) 

Title: Lecture hall graphs and the Askey scheme
Abstract: We establish, for every family of orthogonal polynomials in the Askey scheme and the q-Askey scheme, a combinatorial model for mixed moments and coefficients in terms of paths on the lecture hall lattice. This generalizes to all families of orthogonal polynomials in the Askey scheme previous results of Corteel and Kim for the little q-Jacobi polynomials. This is joint work with Sylvie Corteel, Bhargavi Jonnadula, and Jon Keating.

남경식 (KAIST)

Title: Critical level set percolation of Gaussian Free Field

Abstract: Alexander and Orbach (AO) in 1982 conjectured that the simple random walk on critical percolation clusters exhibit mean field behavior; for instance, its spectral dimension is 4/3 regardless of the underlying dimension. In a breakthrough work, Kozma and Nachmias established the AO conjecture for critical bond percolation for the (spread out) lattice with dimension d>6. In this talk, I will talk about the AO conjecture for the critical level set of Gaussian Free Field, a canonical dependent percolation model of central importance.

변성수 (서울대학교)

Title: Spectral moments and superintegrability of the Gaussian beta ensemble and its (q,t) generalisation 

Abstract: The spectral moments of random matrices play a key role in understanding eigenvalue statistics. In this talk, I will discuss how the spectral moments of the Gaussian beta ensemble or its (q,t) generalization can be evaluated using the theory of symmetric polynomials, typically the Jack or Macdonald polynomials. In particular, I will explain the superintegrability conjecture, which relates to the average over a distinguished basis of symmetric functions, that Peter J. Forrester and I recently proved using a theory of multivariable Al-Salam and Carlitz polynomials based on Macdonald polynomials.

이지운 (KAIST) 

Title: Counting methods in random matrix theory
Abstract: Due to the canonical relation between random graphs and random matrices, many proofs in random matrix theory rely on counting. One of the most important example is the moment method, which is the technique for finding the properties of a random variable by the asymptotics of its moments. The moment method can be applied to prove various spectral properties of random matrices, most notably the Wigner semicircle law for the empirical spectral measure and the Tracy-Widom limit of the largest eigenvalues. Besides the moment method, counting is an important part of other tools including the Stieltjes transform method. In this talk, I will introduce several important ideas based on counting in random matrices and other related models.

장지혁 (성균관대학교)

Title: Combinatorics of the orthogonal polynomials on the unit circle
Abstract: Orthogonal polynomials on the unit circle (OPUC for short) are a family of polynomials whose orthogonality is given by integration over the unit circle in the complex plane. There have been combinatorial studies on the moments of various types of orthogonal polynomials, including classical orthogonal polynomials, Laurent biorthogonal polynomials, and orthogonal polynomials of type R1. In this talk, we study the moments of OPUC from a combinatorial perspective. We provide three path interpretations for them; L{}ukasiewicz paths, gentle Motzkin paths, and Schr"oder paths. Additionally, using these combinatorial interpretations, we derive explicit formulas for the generalized moments of some examples of OPUC, including the circular Jacobi polynomials and the Rogers--Szeg"o polynomials. Furthermore, we introduce several kinds of generalized linear coefficients and give combinatorial interpretations for each of them. This is joint work with Minho Song.