## セミナー -- Infinite Analysis Seminar

Title

**
Quantization of conic Lagrangian submanifolds of cotangent bundles.**

Date

October 18 (Thu.)2012 14:00-15:30

Room

Room 110, RIMS, Kyoto University

Speaker

Professor Stephane Guillermou (Grenoble University)

Abstract

Several recent works make use of the microlocal theory of sheaves
of M. Kashiwara and P. Schapira to obtain results in symplectic geometry.
The link between sheaves on a manifold *M* and the symplectic geometry
of the cotangent bundle of *M* is given by the microsupport of a sheaf,
which is a conic co-isotropic subset of the cotangent bundle.
In the above mentioned works, properties of a given Lagrangian
submanifold $\Lambda$ are deduced from the existence of
a sheaf with microsupport $\Lambda$,
which we call a quantization of $\Lambda$.

In this talk I will give sufficient conditions on a Lagrangian
submanifold which imply
the existence of a quantization.

Title

**
On Tamarkin's proof of the non displaceability theorem and open questions
**

Date

February 24 (Thursday) 2011, 14:00--17:00

Room

RIMS Room 110

Speaker

Professor Pierre Schapira

Title

**
Bernstein-Sato polynomials and double affine Hecke algebras**

Date

August 20-th (Thurs.) 14 o'clock

Room

Room 110, the first floor of Building 3-rd, Fac. of Science, Kyoto Univ.

(where the workshop in honor of Miwa took place)

Speaker

Professor Ivan Cherednik

Title

**
Difference Whittaker functions in rank one**

Date

May 2, (Sat), 2009 11:00-12:30

Room

Room 206, RIMS, Kyoto University

Speaker

Professor Ivan Cherednik (Department of Mathematics, UNC, Chapel Hill, USA)

Abstract

The q-Whittaker functions, which are eigenfunctions of the q-Toda difference operators, can be obtained as a limit of the q,t-spherical (hypergeometric) functions, generalizing the Macdonald polynomials. The limiting procedure is t->0 upon the Inozemtsev-Etingof transformation, which was shown recently to be compatible with the so-called global difference spherical functions, which are meromorpic for all values of the coordinates and spectral parameters. In contrast to the spherical functions, the q-Whittaker functions are not symmetric. However their coefficients have important integrality-positivity properties (which are, generally, missing in the Macdonald theory) and are closely connected with the level one Demazure characters of Kac-Moody algebras and Q-Hermite polynomials. These functions are known/expected to serve the quantum K-theory of affine flag varieties and their IC-theory. Hopefully, they are related to the quantum Langlands program. We will discuss them in the 1D case almost from scratch.

Title

**
On some uses of (super) $\bar\partial$-connections.**

Date

October 11, (Sat), 2008 11:00-12:30

Room

Room 206, RIMS, Kyoto University

Speaker

Alexei Rosly (IPMU, Kashiva / ITEP, Moscow)

Abstract

Starting with a trivial observation that the properties of Dolbeault's $\bar\partial$-operator are analogous to de Rham's $d$,we shall see how far one can proceed with this analogy. This will lead us to a holomorphic analogue of topological homology and linking number. The latter notion finds certain motivations and applications in physics. We shall see also that the symplectic structure on the moduli of holomorphic bundles over complex surfaces can be understood in the context of the above analogy. To extend our discussion from holomorphic vector bundles to arbitrary coherent sheaves we shall consider super $\bar\partial$-connections. The notion of a super connection is understood in the same way as in a work of Quillen.

Title

**
Representations of the elliptic quantum group
$U_{q,p}({\hat {sl}}_2)$**

Date

December 22 (Sat), 2007 11:00-12:30

Room

Room 402, RIMS, Kyoto University

Speaker

Hitoshi Konno (Department of Mathematics, Hiroshima University)

Abstract

We introduce an h-Hopf algebroid structure into the elliptic algebra $U_{q,p}({\hat {sl}}_2)$ and formulate it as an elliptic quantum group. We then discuss dynamical representations of $U_{q,p}({\hat {sl}}_2).$ In particular, we extend the Chari-Pressley classification theorem of the finite-dimensional irreducible representations of the quantum affine algebra $U_q({\hat {sl}}_2)$ to the elliptic case. As an application, we investigate a structure of the tensor product of two evaluation representations and derive an elliptic analogue of the Clebsch-Gordan coefficients. We show that they are expressed by using the terminating very-well-poised balanced elliptic hypergeometric series $12V11.$

----------- Lunch 12:30 --- 14:00 -----------

Title

**
The connection problem associated with a Selberg type
integral and the q-Racah polynomials
**

Date

December 22 (Sat), 2007, 14:00-15:30

Room

Room 402, RIMS, Kyoto University

Speaker

Katsuhisa Mimachi (Department of Mathematics, Tokyo Institute of Technology)

Abstract

The purpose of our talk is a report on our recent progress in the connection problem associated with a Selberg type integral, which satisfies an ordinary differential equation of order $m+1$ with three regular singular points $0, 1$ and $\infty.$ The connection problem we mean here is to give linear relations between the fundamental sets of solutions around the singularities.

Title

**
The symplectic groupoid of triangular bilinear forms
and Poisson-Lie groups.**

Date

November 17 (Sat), 2007, 11:00-12:30

Room

Room 402, RIMS, Kyoto University

Speaker

Alexey Bondal (Steklov Mathematical Institute, Moscow, Russia)

Abstract

We will construct a natural symplectic groupoid built on triangular bilinear forms. The corresponding Poisson structure on the space of triangular bilinear forms will be described and investigated. Relation to the theory of Poisson-Lie groups will be established. The dual symplectic groupoid will be outlined.

----------- Lunch 12:30 --- 14:00 -----------

Title

**
Actions of quantum toroidal algebras on higher q-Fock spaces
**

Date

November 17 (Sat), 2007, 14:00-15:30

Room

Room 402, RIMS, Kyoto University

Speaker

Kentaro Nagao (Kyoto University, Kyoto,Japan)

Abstract

The quantum toroidal algebras are quantum affinizations of the affine Kac-Moody algebras. Takemura-Uglov have constructed actions of quantum toroidal algebras of type-A on higher q-Fock spaces. In this talk we will construct simultaneous eigenvectors (for the actions of the affinizations of Cartan subalgebras) and give combinatorial descriptions of these representations. In particular, we can see q-characters of higher q-Fock spaces equal to products of q-characters of level-1 q-Fock spaces.

Title

**
Combinatorial Bethe ansatz and box-ball system**

Date

September 29 (Sat), 2007, 11:00-12:30

Room

Room 402, RIMS, Kyoto University

Speaker

Atsuo Kuniba (Institute of Physics, Graduate School of Arts and Science, University of Tokyo)

Abstract

I explain combinatorial versions of Bethe ansatz and corner transfer matrix, how they produce action-angle variables and tau functions of box-ball systems on infinite and periodic lattices, and a solution of the initial value problem in terms of an ultradiscrete analogue of the Riemann theta function.

----------- Lunch 12:30 --- 14:00 -----------

Title

**
Tropical spectral curves and integrable automata
**

Date

September 29 (Sat), 2007, 14:00-15:30

Room

Room 402, RIMS, Kyoto University

Speaker

Rei Inoue Yamazaki (Department of Physics, Graduate School of Science, The University of Tokyo)

Abstract

We study an integrable automata, the ultradiscrete periodic Toda lattice, via the tropical algebraic geometry. We construct the tropical version of the eigenvector map from the isolevel set to the Jacobi variety of the tropical spectral curve. We also mention the algebra-geometrical meaning of the initial value problem for the periodic box and ball system related to the $U_q(A_1^{(1)})$ Bethe equation at $q=0$. (This talk is based on the joint work with T. Takenawa.)

Title

**
Elliptic beta integrals on root systems **

Date

March 24 (Sat), 2007, 11:00-12:30

Room

Room 206, RIMS, Kyoto University

Speaker

Vyacheslav Spiridonov (BLTP, Dubna, Russia and RIMS, Kyoto, Japan)

Abstract

The univariate elliptic beta integral, discovered by the author in 2000,
provides the most general known exact integration formula generalizing
Euler's beta integral. It represents the simplest example of the
very-well-poised elliptic hypergeometric integrals --- functions of
hypergeometric type obeying properties of classical special functions.

In this talk we give a brief review of the theory and applications of
elliptic hypergeometric integrals on root systems $A_n$ and $C_n$,
introduced by van Diejen, Rains, Warnaar and the author.

Title

**
Filtrations on Schur algebras for $GL_2$ **

Date

February 15 (Thu), 2007, 16:00-17:00

Room

Room 402, RIMS, Kyoto University

Speaker

Vanessa Miemietz (Koln)

Abstract

We give a filtration by ideals of the Schur algebras $S(n,r)$ for $GL_2$ which is conjectured to induce a grading. We describe an iterative procedure how to construct a graded quasihereditary algebra $\mathcal(C)_p(A)$ with the same filtration and give a stable equivalence between two infinite-dimensional algebras which have heredity subquotients $\mathcal(C)_p(A)$ resp. $S(n,r)$.

Title

**
Higher Representation Theory **

Date

Nov. 22 (Wed.), 2006, 15:00--16:00

Nov. 23 (Thur.), 2006 10:00--12:00, 14:00--16:00

Nov. 24 (Fri.), 2006 10:00--12:00, 14:00--16:00

Room

Nov. 22, 24 : Room 402, Bldg 6, Graduate School of Science, Kyoto University

Nov. 23 : Room 102, RIMS, Kyoto University

Speaker

Raphael Rouquier (University of Leeds)

Title

**
q-Demazure characters (joint work with F. Descouens) **

Date

March 25, 2006, 11:00-12:30

Room

RIMS, Room 206

Speaker

Alain Lascoux ( C.N.R.S., Institut Gaspard Monge Université de Marne-la-Vallee, FRANCE )

Abstract

Schur functions (i.e. irreducible characters of the linear group) have been extended in different directions : non symmetric characters (Demazure), or symmetric functions with a parameter $q$ (Hall-Littlewood polynomials). I shall give a linear basis of the ring of polynomials which specializes to Demazure characters for $q=0$ and contains as a subfamily the Hall-Littlewood polynomials. The Yang-Baxter operators play an esential role in this construction.