「可積分系、ランダム行列、代数幾何と幾何学的不変量」研究集会
English
昨年度に引き続き、
日本学術振興会、ロシア基礎科学財団による二国間交流事業共同研究プロジェクト
「可積分系、ランダム行列、代数幾何と幾何学的不変量」の一環として、
プロジェクトメンバーによる以下の講演会を開催します。
- 日程:2011 年 10 月 20 日(木)〜 21 日(金)
- 場所:京都大学吉田南キャンパス、人間環境学研究科棟2階226号室
- アクセスマップ
http://www.h.kyoto-u.ac.jp/access
- 世話人:
-
高崎金久 (京都大学大学院人間・環境学研究科)
塩田隆比呂 (京都大学大学院理学研究科)
武部尚志 (ロシア国立大学経済高等学校数学学部)
プログラム
- 20日(木)
- 10:00-10:30 柳田伸太郎 (神戸大学理学研究科)
- 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 沼田泰英 (東京大学情報理工学系研究科)
- On a system of partial differential equations for hypergeometric
functions of matrix argument
- 14:45-15:15 松本詔 (名古屋大学大学院多元数理科学研究科)
- On Moments of entries of a COE matrix
- 15:30-16:00 近藤智 (東京大学数物連携宇宙研究機構)
- On quasi-generic representations of the general linear group
of a non-archimedean local field
- 16:30-17:30 Leonid Rybnikov (ロシア国立大学経済高等学校数学学部)
- Yangians and cohomology rings of Laumon spaces
- 21日(金)
- 9:30-10:00 池田暁志 (東京大学大学院数理科学研究科)
- The space of stability conditions on local P^2 and related topics
- 10:15-10:45 岩尾慎介 (立教大学理学部)
- 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
講演概要
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.
池田暁志,
"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.
岩尾慎介, "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.
近藤智, "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.
松本詔, "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.
沼田泰英, "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.
柳田伸太郎, "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.