Integrable Systems, Random Matrices, Algebraic Geometry and Geometric Invariants

Workshop ``Integrable Systems, Random Matrices, Algebraic Geometry and Geometric Invariants''

This workshop is supported by the Japan Society of Promotion of Sciences and the Russian Foundation for Basic Research under the Japan-Russia Research Cooperative Project. [Last year's workshop]

Date: 20 - 21 October 2011
Place: Kyoto University, Yoshida South Campus, Graduate School of Human and Environmental Studies Bldg, room 226 (2nd Floor): Access map: http://www.kyoto-u.ac.jp/en/access/campus/y_south.htm
Organizers:: Kanehisa Takasaki (Graduate School of Human and Environmental Studies, Kyoto University)
Takahiro Shiota (Department of Mathematics, Kyoto University)
Takashi Takebe (Faculty of Mathematics, Higher School of Economics, Moscow)

Program

Thursday, 20 October
10:00-10:30 Shintaro Yanagida (Department of Mathematics, Kobe University): Trace of intertwiner for Ding-Iohara algebra
11:00-12:00 Anton Zabrodin (Institute of Biochemical Physics): Quantum Painleve-Calogero correspondence
14:00-14:30 Yasuhide Numata (Graduate School of Information Science and Technology, University of Tokyo): On a system of partial differential equations for hypergeometric functions of matrix argument
14:45-15:15 Sho Matsumoto (Graduate School of Mathematics, Nagoya University): On Moments of entries of a COE matrix
15:30-16:00 Satoshi Kondo (Institute for the Physics and Mathematics of Universe, University of Tokyo): On quasi-generic representations of the general linear group of a non-archimedean local field
16:30-17:30 Leonid Rybnikov (Faculty of Mathematics, Higher School of Economics, Moscow): Yangians and cohomology rings of Laumon spaces
Friday, 21 October
9:30-10:00 Akishi Ikeda (Graduate School of Mathematical Sciences, University of Tokyo): The space of stability conditions on local P^2 and related topics
10:15-10:45 Shinsuke Iwao (Faculty of Science, Rikkyo University): Periodic and Non-periodic Ultradiscrete Integrable Systems
11:15-12:15 Alexander Orlov (P.P. Shirshov Institute of Oceanology): Pfaffian structure and certain solutions of $O(2\infty)$ BKP hierarchy

Abstracts of talks

Alexander Orlov, "Pfaffian structure and certain solutions of $O(2\infty)$ BKP hierarchy"
We introduce a useful and rather simple class of DKP and fermionic BKP tau functions which generalizes hypergeometric functions. It may be presented in form of multiple integrals or multiple sums and series. We also present few applications of such tau functions: as partition functions for certain matrix models (circular $\beta=1,2,4$ ensembles, interpolating ensembles) together with perturbation series for these ensembles and as a partition function of certain random processes. We also consider (neutral) 2-BKP hierarchy and DKP coupled to neutral BKP ones.

Leonid Rybnikov, "Yangians and cohomology rings of Laumon spaces" (joint with A.Braverman, B.Feigin, M.Finkelberg, A.Negut)
Laumon moduli spaces are certain smooth closures of the moduli spaces of maps from the projective line to the flag variety of $GL_n$. We construct the action of the Yangian of $sl_n$ in the cohomology of Laumon spaces by certain natural correspondences. We construct the action of the affine Yangian (two-parametric deformation of the universal enveloping algebra of the universal central extension of $sl_n[s^{\pm1},t]$) in the cohomology of the affine version of Laumon spaces. We compute the matrix coefficients of the generators of the affine Yangian in the fixed point basis of cohomology. This basis is an affine analogue of the Gelfand-Tsetlin basis. The affine analogue of the Gelfand-Tsetlin algebra surjects onto the equivariant cohomology rings of the affine Laumon spaces. I will discuss the version of this picture for partial flag varieties of $GL_n$ and its relation to the AGT conjecture.

Anton Zabrodin, "Quantum Painleve-Calogero correspondence"
The Painleve-Calogero correspondence is extended to auxiliary linear problems associated with Painleve equations. The linear problems are represented in a new form which has a suggestive interpretation as a "quantized" version of the Painleve-Calogero correspondence. Namely, the linear problem responsible for the time evolution is brought into the form of non-stationary Schrodinger equation in imaginary time whose Hamiltonian is a natural quantization of the classical Calogero-like Hamiltonian for the corresponding Painleve equation. The talk is based on our recent joint work with A.Zotov.

Akishi Ikeda, "The space of stability conditions on local P^2 and related topics"
The space of stability conditions on a triangulated category is introduced by T. Bridgeland and this space has a structrure of complex manifold. In this talk, following his work, we consider the description of the space of stability conditions on the derived category of coherent sheaves on the local P^2 and study some properties of this space. In particular, we discuss the conjectual relationship between this space and the Frobenius manifold of the quantum cohomology of P^2.

Shinsuke Iwao, "Periodic and Non-periodic Ultradiscrete Integrable Systems"
The periodic Box-Ball system (PBBS) is a typical example of the ultradiscrete integrable systems that can be analyzed by the inverse scattering method. In this talk, I will introduce the ultradiscrete version of the inverse scattering method and its variant, which is valid also for the non-periodic Box-Ball system.

Satoshi Kondo, "On quasi-generic representations of the general linear group of a non-archimedean local field"
There is a notion of generic representation of the general linear group of a nonarchimedean local field. Jacquet-Piatetskii-Shapiro-Shalika define the conductor of a generic representation and show that there exists a (nontrivial) new vector. We define the notion of quasi-generic representation. One of the characterizations is the existence of (an analogue of) a new vector; in particular, generic representations are quasi-generic. Further properties and the characterizations are discussed.

Sho Matsumoto, "On Moments of entries of a COE matrix"
We consider a symmetric unitary random matrix from Dyson's circular orthogonal ensemble and give explicit formulas for mixed moments of entries of the matrix. Our method is the Weingarten calculus for Haar-distributed unitary matrices.

Yasuhide Numata, "On a system of partial differential equations for hypergeometric functions of matrix argument"
The hypergeometric function ${}_{1}F_{1}(a,c;X)$ of matrix argument is a function of eigenvalues of the square matrix $X$. The hypergeometric function can be written as the infinite summation of zonal polynomials. It is known that there exists a system of partial differential equations such that the hypergeometric function is the unique symmetric function satisfying the system. We discuss the holonomicity of the system. This talk is based on a joint work with H. Hashiguchi and A. Takemura.

Shintaro Yanagida, "Trace of intertwiner for Ding-Iohara algebra"
This talk is based on the collaboration with H. Awata, B. Feigin, A. Hoshino, M. Kanai and J. Shiraishi. In this short talk, I will discuss a formula, which is analogous to the Cauchy formula for Macdonald symmetric functions. Our deformed formula is obtained from two ways of computations of trace of intertwiner for Ding-Iohara algebra.