Title and Abstract
■ Ivan Corwin (Columbia/Clay Math Inst/Inst Henri Poincare/MIT)
Stochastic quantum integrable systems and the KPZ class
Abstract: We describe how by leveraging certain structures in the theory of quantum integrable systems we can create an umbrella theory which encompasses all known integrable models in the Kardar-Parisi-Zhang universality class. Concepts which will feature in this talk includes Bethe ansatz, symmetric functions, Plancherel theory and Markov dualities. This is based on works with various collaborators including Barraquand, Borodin, Gorin, Petrov, and Sasamoto.
■ Jan de Gier (Melbourne)
1. Applications of the Bethe ansatz for the ASEP with open boundaries
Abstract: I will discuss the Bethe ansatz for the open ASEP and two applications: (i) the spectral gap or subleading eigenvalue in the various phases of the ASPE and (ii) the asymptotics of the current large deviation function.
2. Multi-species exclusion process and Macdonald polynomials
Abstract: The normalisation of the stationary state of the multi-species ASEP is a specialisation of a Macdonald polynomial. I will discuss how this arises in the framework of integrability and the matrix product method.
■ Shin Isojima (Hosei)
New Airy-type solutions of the ultradiscrete Painlev\'{e} II equation with parity variables
Abstract: The q-difference Painlev\'{e} II equation admits special solutions written in terms of determinant whose entries are the general solution of the q-Airy equation. An ultradiscrete limit of the special solutions is studied by the procedure of ultradiscretization with parity varialbes. Then we obtain new Airy-type solutions of the ultradiscrete Painlev\'{e} II equation with parity variables, and the solutions have richer structure than the known solutions.
■ Ken Ito (Aichi Inst of Tech)
Convex canonical bases of PBW type for the quantum untwisted affine algebras
Abstract: We classified all ``convex orders'' on the positive root system $\Delta_+$ of an arbitrary untwisted affine Lie algebra ${\mathfrak g}$, and gave a concrete method of constructing all convex orders on $\Delta_+$. The aim of this talk is to introduce a description of ``convex bases'' of PBW type of the positive subalgebra $U^+$ of the quantum affine algebra $U=U_q({\mathfrak g})$ by using the concrete method of constructing all convex orders on $\Delta_+$ [Hiroshima Math. J., Vol.40(2) (2010)]. Moreover, we will talk about a realization of G.~Lusztig's canonical bases of $U^+$ as the convex bases for the quantum untwisted affine algebras.
■ Michio Jimbo (Rikkyo)
Finite type modules and Bethe Ansatz for quantum toroidal gl1
Abstract: We introduce and study `finite type' modules of the Borel subalgebra of the quantum toroidal gl1 algebra. These modules are infinite dimensional in general but the Cartan like generator \psi^+(z) has a finite number of eigenvalues. We use them to diagonalize the transfer matrices T(u;p) analogous to those of the six vertex model, wherein the role of C^2 is played by infinite dimensional Fock spaces. We construct auxiliary operators Q(u;p) and \cT(u;p) satisfying a two-term TQ relation, from which the Bethe Ansatz equation is derived. We give an expression for the eigenvalues of T(u;p) in terms of Q(u;p).
This is a joint work with B.Feigin, T.Miwa and E.Mukhin.
■ Saburo Kakei (Rikkyo)
Linearization of the box-ball system: an elementary approach
Abstract: Kuniba, Okado, Takagi, and Yamada found that the time-evolution of the Takahashi-Satsuma box-ball system (BBS) can be linearized by considering the rigged configurations associated with states of the BBS. We introduce a simple way to understand the rigged configuration of $A_1^{(1)}$-type, and give an elementary proof of the linearization property.
[Joint work with Jon J.C. Nimmo (Glasgow), Satoshi Tsujimoto (Kyoto) and Ralph Willox (Tokyo)]
■ Thomas Lam (Michigan)
Geometric R-matrices
Abstract: Geometric R-matrices are remarkable birational transformations that play a role in combinatorics (e.g. jeu-de-taquin), integrable systems (e.g. box-ball systems), and representation theory (e.g. crystal graphs). I will give an introduction to the theory and explain some new results towards the construction of cluster and quantum geometric R-matrices. This is joint work with Rei Inoue and Pavlo Pylyavskyy.
■ Kirone Mallick (Saclay)
1. The Asymmetric Simple Exclusion Process, an integrable model for non-equilibrium physics
Abstract: The asymmetric simple exclusion process plays the role of a paradigm to study various aspects of non-equilibrium statistical physics. It is a minimal model of interacting particles with transport, that displays interesting collective behaviour and breaks time-reversal symmetry. This system appears as a building block in more realistic descriptions of low-dimensional transport with constraints.
During the last twenty years, a large number of exact results have been derived for the exclusion process thanks to its integrability properties. The aim of this talk is to review the main techniques used to study this system and some of the important results that have been obtained.
2. Tensor Matrix Ansatz for multispecies exclusion processes
Abstract: In this talk, we shall explain how the stationary measure of multi-species exclusion processes on a ring can be expressed by a Matrix Product Representation. The algebras involved are constructed recursively by performing tensor products of the fundamental quadratic algebra that appeared in the solution of Derrida, Evans, Hakim and Pasquier, 1993.
In the second part of the talk, we show that similar structures play a role in the calculation of the current fluctuations in the single-species exclusion process with open boundaries.
■ Shouya Maruyama
Multi-species 1D stochastic model and Integrability
Abstract: In the context of non-equilibrium statistical mechanics, many types of 1D stochastic models are studied. Some of them show integrability and play an important role in the study of non-equilibrium systems. In this talk, I want to discuss the construction of a matrix product form of the steady state for two types of such models, multi-species totally asymmetric simple exclusion process (TASEP) and multi-species totally asymmetric zero range process (TAZRP). This talk is based on joint works with Atsuo Kuniba and Masato Okado.
■ Satoshi Naito (Tokyo Tech)
Symmetric Macdonald polynomials and pseudo-quantum Lakshmibai-Seshadri paths
Abstract: In this talk, I would like to explain the relationship between symmetric Macdonald polynomials associated to an arbitrary untwisted affine root system and pseudo-quantum Lakshmibai-Seshadri (pqLS) paths; pqLS paths are introduced as a generalization of quantum LS paths, which provide a realization of crystal bases for tensor products of level-zero fundamental representations of a quantum affine algebra. We remark that the set of pqLS paths can be endowed with (a generalization of) an affine crystal structure, and that the resulting crystal graph turns out to be connected.
■ Yosuke Saito (OCAMI)
Special eigenfunctions for the Ruijsenaars operator
Abstract: The Ruijsenaars operator was introduced by Ruijsenaars in 1985 as a q-analog of the Hamiltonian for the Calogero-Moser system. However, eigenfunctions for the Ruijsenaars operator are not understood well. In this talk, using a free field approach, we show that an elliptic analog of the elementary symmetric polynomials gives eigenfunctions for the Ruijsenaars operator in a special case.
■ Kazumitsu Sakai (Tokyo)
Multiple SLEs
Abstract: We construct multiple Schramm-Loewner evolutions (SLEs) for several conformal field theories. As applications, probabilities of occurrence for certain configurations of SLE traces related to some statistical mechanics models are exactly calculated. This talk is based on a joint work with Yoshiki Fukusumi.
■ Tomohiro Sasamoto (Tokyo Tech)
Determinantal structures for 1D KPZ systems
Abstract: The Kardar-Parisi-Zhang(KPZ) equation is a simple stochastic nonlinear PDE describing a growing interface motion. In the last few years, its one-dimensional version has attracted much attention and many results have been obtained. For instance our understanding of its definition has deepened. Its universality has turned out to be much stronger than previously thought. These developments are to some extent related to the fact that the 1D KPZ equation is exactly solvable; some explicit formulas for important quantities such as the height distribution and two-point correlation function have been obtained.
In this presentation first we briefly review recent developments on the 1D KPZ equations and related models. Then we discuss how these models can be solved, in particular from the point of view of connection to random matrix theory and its generalizations.
This is based on collaborations with T. Imamura.
■ Takashi Takebe (National Research U Higher School of Economics)
Q-operators for higher spin eight vertex models
Abstract: We construct the Q-operator for generalised eight vertex models associated to higher spin representations of the Sklyanin algebra, following Baxter's 1973 paper. As an application, we prove the sum rule for the Bethe roots.
■ Yoshihiro Takeyama (Tsukuba)
Algebraic construction of multi-species q-Boson system
Abstract: The q-Boson system due to Sasamoto and Wadati is an integrable stochastic particle system. In this talk I give an algebraic construction of the q-Boson system and its multi-species version using a representation of a deformation of the affine Hecke algebra of type GL. I also discuss eigenfunctions of the transition rate matrix which we can construct by means of the Bethe ansatz method.