## Seminar information archive

Seminar information archive ～02/17｜Today's seminar 02/18 | Future seminars 02/19～

### 2007/05/15

#### Algebraic Geometry Seminar

14:30-16:00 Room #122 (Graduate School of Math. Sci. Bldg.)

Riemann-Roch for determinantal gerbes and smooth manifolds

**Mikhail Kapranov**(Yale 大学)Riemann-Roch for determinantal gerbes and smooth manifolds

[ Abstract ]

A version of the Riemann-Roch theorem for curves due to Deligne, describes the determinant of the cohomology of a vector bundle E on a curve.

If one realizes E via the Krichever construction, the determinant of the cohomology becomes a Hom-space in the determinantal gerbe for the vector space over the field of power series. So one has a ``local" Riemann-Roch problem of description of this gerbe itself. The talk will present the results of a joint work with E. Vasserot describing the class of such a gerbe in a family which geometrically can be seen as a circle fibration. This can be further generalized to the case of a fibration with fibers being smooth compact manifolds of any dimension d (joint work with P. Bressler, B. Tsygan and E. Vasserot).

A version of the Riemann-Roch theorem for curves due to Deligne, describes the determinant of the cohomology of a vector bundle E on a curve.

If one realizes E via the Krichever construction, the determinant of the cohomology becomes a Hom-space in the determinantal gerbe for the vector space over the field of power series. So one has a ``local" Riemann-Roch problem of description of this gerbe itself. The talk will present the results of a joint work with E. Vasserot describing the class of such a gerbe in a family which geometrically can be seen as a circle fibration. This can be further generalized to the case of a fibration with fibers being smooth compact manifolds of any dimension d (joint work with P. Bressler, B. Tsygan and E. Vasserot).

#### Tuesday Seminar of Analysis

16:30-18:00 Room #128 (Graduate School of Math. Sci. Bldg.)

量子電磁場中を運動する原子の安定性について

**宮尾 忠宏**(岡山大自然科学研究科 学振特別研究員 )量子電磁場中を運動する原子の安定性について

#### Tuesday Seminar on Topology

16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Kontsevich's characteristic classes for higher dimensional homology sphere bundles

**渡邉 忠之**(京都大学数理解析研究所)Kontsevich's characteristic classes for higher dimensional homology sphere bundles

[ Abstract ]

As an analogue of the perturbative Chern-Simons theory, Maxim Kontsevich

constructed universal characteristic classes of smooth fiber bundles with fiber

diffeomorphic to a singularly framed odd dimensional homology sphere.

In this talk, I will give a sketch proof of our result on non-triviality of the

Kontsevich classes for 7-dimensional homology sphere bundles.

As an analogue of the perturbative Chern-Simons theory, Maxim Kontsevich

constructed universal characteristic classes of smooth fiber bundles with fiber

diffeomorphic to a singularly framed odd dimensional homology sphere.

In this talk, I will give a sketch proof of our result on non-triviality of the

Kontsevich classes for 7-dimensional homology sphere bundles.

### 2007/05/14

#### Seminar on Geometric Complex Analysis

10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Unicity problems with truncated multiplicities of mermorphic mappings in several complex variables

**Si, Quang Duc**(東大数理)Unicity problems with truncated multiplicities of mermorphic mappings in several complex variables

### 2007/05/10

#### Operator Algebra Seminars

16:30-18:00 Room #128 (Graduate School of Math. Sci. Bldg.)

自己準同型の収束と近似的内部性

**戸松玲治**(東大数理)自己準同型の収束と近似的内部性

### 2007/05/09

#### Number Theory Seminar

16:30-17:30 Room #117 (Graduate School of Math. Sci. Bldg.)

$(g,K)$-module structures of principal series representations

of $Sp(3,R)$

**宮崎 直**(東京大学大学院数理科学研究科)$(g,K)$-module structures of principal series representations

of $Sp(3,R)$

### 2007/05/08

#### Tuesday Seminar on Topology

16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)

On the vanishing of the Rohlin invariant

**森山 哲裕**(東京大学大学院数理科学研究科)On the vanishing of the Rohlin invariant

[ Abstract ]

The vanishing of the Rohlin invariant of an amphichiral integral

homology $3$-sphere $M$ (i.e. $M \\cong -M$) is a natural consequence

of some elementary properties of the Casson invariant. In this talk, we

give a new direct (and more elementary) proof of this vanishing

property. The main idea comes from the definition of the degree 1

part of the Kontsevich-Kuperberg-Thurston invariant, and we progress

by constructing some $7$-dimensional manifolds in which $M$ is embedded.

The vanishing of the Rohlin invariant of an amphichiral integral

homology $3$-sphere $M$ (i.e. $M \\cong -M$) is a natural consequence

of some elementary properties of the Casson invariant. In this talk, we

give a new direct (and more elementary) proof of this vanishing

property. The main idea comes from the definition of the degree 1

part of the Kontsevich-Kuperberg-Thurston invariant, and we progress

by constructing some $7$-dimensional manifolds in which $M$ is embedded.

#### Lie Groups and Representation Theory

17:00-18:00 Room #126 (Graduate School of Math. Sci. Bldg.)

Affine W-algebras and their representations

http://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar.html

**荒川知幸**(奈良女子大学)Affine W-algebras and their representations

[ Abstract ]

The W-algebras are an interesting class of vertex algebras, which can be understood as a generalization of Virasoro algebra. It was originally introduced by Zamolodchikov in his study of conformal field theory. Later Feigin-Frenkel discovered that the W-algebras can be defined via the method of quantum BRST reduction. A few years ago this method was generalized by Kac-Roan-Wakimoto in full generality, producing many interesting vertex algebras. Almost at the same time Premet re-discovered the finite-dimensional version of W-algebras (finite W-algebras), in connection with the modular representation theory.

In the talk we quickly recall the Feigin-Frenkel theory which connects the Whittaker models of the center of $U({\\mathfrak g})$ and affine (principal) W-algebras, and discuss their representation theory. Next we recall the construction of Kac-Roan-Wakimoto and discuss the representation theory of affine W-algebras associated with general nilpotent orbits. In particular, I explain how the representation theory of finite W-algebras (=the endmorphism ring of the generalized Gelfand-Graev representation) applies to the representation of affine W-algebras.

[ Reference URL ]The W-algebras are an interesting class of vertex algebras, which can be understood as a generalization of Virasoro algebra. It was originally introduced by Zamolodchikov in his study of conformal field theory. Later Feigin-Frenkel discovered that the W-algebras can be defined via the method of quantum BRST reduction. A few years ago this method was generalized by Kac-Roan-Wakimoto in full generality, producing many interesting vertex algebras. Almost at the same time Premet re-discovered the finite-dimensional version of W-algebras (finite W-algebras), in connection with the modular representation theory.

In the talk we quickly recall the Feigin-Frenkel theory which connects the Whittaker models of the center of $U({\\mathfrak g})$ and affine (principal) W-algebras, and discuss their representation theory. Next we recall the construction of Kac-Roan-Wakimoto and discuss the representation theory of affine W-algebras associated with general nilpotent orbits. In particular, I explain how the representation theory of finite W-algebras (=the endmorphism ring of the generalized Gelfand-Graev representation) applies to the representation of affine W-algebras.

http://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar.html

### 2007/05/07

#### Seminar on Geometric Complex Analysis

10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Canonical metrics on relative canonical bundles and Extension of pluri log canonical systems

**辻 元**(上智大学)Canonical metrics on relative canonical bundles and Extension of pluri log canonical systems

#### Algebraic Geometry Seminar

16:30-18:00 Room #122 (Graduate School of Math. Sci. Bldg.)

Pathologies on ruled surfaces in positive characteristic

**謝啓鴻(Xie Qihong)**(東大・数理)Pathologies on ruled surfaces in positive characteristic

[ Abstract ]

We discuss some pathologies of log varieties in positive characteristic. Mainly, we show that on ruled surfaces there are counterexamples of several logarithmic type theorems. On the other hand, we also give a characterization of the counterexamples of the Kawamata-Viehweg vanishing theorem on a geometrically ruled surface by means of the Tango invariant of the base curve.

We discuss some pathologies of log varieties in positive characteristic. Mainly, we show that on ruled surfaces there are counterexamples of several logarithmic type theorems. On the other hand, we also give a characterization of the counterexamples of the Kawamata-Viehweg vanishing theorem on a geometrically ruled surface by means of the Tango invariant of the base curve.

### 2007/05/02

#### Number Theory Seminar

16:30-17:30 Room #117 (Graduate School of Math. Sci. Bldg.)

Cohen-Eisenstein series and modular forms associated to imaginary quadratic fields

**長谷川 泰子**(東京大学大学院数理科学研究科)Cohen-Eisenstein series and modular forms associated to imaginary quadratic fields

### 2007/05/01

#### Tuesday Seminar of Analysis

16:30-18:00 Room #128 (Graduate School of Math. Sci. Bldg.)

非線型シュレディンガー方程式の解の長時間挙動について

**下村 明洋**(学習院大理学部)非線型シュレディンガー方程式の解の長時間挙動について

#### Lie Groups and Representation Theory

16:30-18:00 Room #126 (Graduate School of Math. Sci. Bldg.)

Harish-Chandra expansion of the matrix coefficients of $P_J$ Principal series Representation of $Sp(2,R)$

http://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar.html

**飯田正敏**(城西大学)Harish-Chandra expansion of the matrix coefficients of $P_J$ Principal series Representation of $Sp(2,R)$

[ Abstract ]

Asymptotic expansion of the matrix coefficents of class-1 principal series representation was considered by Harish-Chandra. The coefficient of the leading exponent of the expansion is called the c-function which plays an important role in the harmonic analysis on the Lie group.

In this talk, we consider the Harish-Chandra expansion of the matrix coefficients of the standard representation which is the parabolic induction with respect to a non-minimal parabolic subgroup of $Sp(2,R)$.

This is the joint study with Professor T. Oda.

[ Reference URL ]Asymptotic expansion of the matrix coefficents of class-1 principal series representation was considered by Harish-Chandra. The coefficient of the leading exponent of the expansion is called the c-function which plays an important role in the harmonic analysis on the Lie group.

In this talk, we consider the Harish-Chandra expansion of the matrix coefficients of the standard representation which is the parabolic induction with respect to a non-minimal parabolic subgroup of $Sp(2,R)$.

This is the joint study with Professor T. Oda.

http://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar.html

### 2007/04/26

#### Seminar for Mathematical Past of Asia

17:00-18:30 Room #123 (Graduate School of Math. Sci. Bldg.)

明治前期の日本において学ばれたユークリッド幾何学

[ Reference URL ]

http://www.ms.u-tokyo.ac.jp/~kawazumi/asia.html

**公田 藏**(立教大学名誉教授)明治前期の日本において学ばれたユークリッド幾何学

[ Reference URL ]

http://www.ms.u-tokyo.ac.jp/~kawazumi/asia.html

#### Operator Algebra Seminars

16:30-18:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Nonequilibrium steady states in quantum systems

**緒方芳子**(東大数理)Nonequilibrium steady states in quantum systems

### 2007/04/25

#### Number Theory Seminar

16:30-17:30 Room #117 (Graduate School of Math. Sci. Bldg.)

Localized Characteristic Class and Swan Class

**津嶋 貴弘**(東京大学大学院数理科学研究科)Localized Characteristic Class and Swan Class

### 2007/04/24

#### Tuesday Seminar on Topology

16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Realization of twisted $K$-theory and

finite-dimensional approximation of Fredholm operators

**五味 清紀**(東京大学大学院数理科学研究科)Realization of twisted $K$-theory and

finite-dimensional approximation of Fredholm operators

[ Abstract ]

A problem in twisted $K$-theory is to realize twisted $K$-groups generally by means of finite-dimensional geometric objects, like vector bundles. I would like to talk about an approach toward the problem by means of Mikio Furuta's generalized vector bundles. By using a twisted version of the generalized vector bundle and a finite-dimensional approximation of Fredholm operators, I construct a group into which there exists a natural injection from the twisted $K$-group twisted by any third integral cohomology class.

A problem in twisted $K$-theory is to realize twisted $K$-groups generally by means of finite-dimensional geometric objects, like vector bundles. I would like to talk about an approach toward the problem by means of Mikio Furuta's generalized vector bundles. By using a twisted version of the generalized vector bundle and a finite-dimensional approximation of Fredholm operators, I construct a group into which there exists a natural injection from the twisted $K$-group twisted by any third integral cohomology class.

#### Lie Groups and Representation Theory

16:30-18:00 Room #126 (Graduate School of Math. Sci. Bldg.)

Existence problem of compact Clifford-Klein forms of the infinitesimal homogeneous space of indefinite Stiefle manifolds

http://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar2007.html#20070424yoshino

**Taro Yoshino (吉野太郎)**(University of Tokyo)Existence problem of compact Clifford-Klein forms of the infinitesimal homogeneous space of indefinite Stiefle manifolds

[ Abstract ]

The existence problem of compact Clifford-Klein forms is important in the study of discrete groups. There are several open problems on it, even in the reductive cases, which is most studied. For a homogeneous space of reductive type, one can define its `infinitesimal' homogeneous space.

This homogeneous space is easier to consider the existence problem of compact Clifford-Klein forms.

In this talk, we especially consider the infinitesimal homogeneous spaces of indefinite Stiefel manifolds. And, we reduce the existence problem of compact Clifford-Klein forms to certain algebraic problem, which was already studied from other motivation.

[ Reference URL ]The existence problem of compact Clifford-Klein forms is important in the study of discrete groups. There are several open problems on it, even in the reductive cases, which is most studied. For a homogeneous space of reductive type, one can define its `infinitesimal' homogeneous space.

This homogeneous space is easier to consider the existence problem of compact Clifford-Klein forms.

In this talk, we especially consider the infinitesimal homogeneous spaces of indefinite Stiefel manifolds. And, we reduce the existence problem of compact Clifford-Klein forms to certain algebraic problem, which was already studied from other motivation.

http://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar2007.html#20070424yoshino

### 2007/04/23

#### Seminar on Geometric Complex Analysis

10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Generalization of Q-curvature in CR geometry

**平地健吾**(東京大学)Generalization of Q-curvature in CR geometry

### 2007/04/19

#### Operator Algebra Seminars

16:30-18:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Graph algebras, Exel-Laca algebras and ultragraph algebras

**勝良健史**(東大数理)Graph algebras, Exel-Laca algebras and ultragraph algebras

### 2007/04/17

#### Tuesday Seminar of Analysis

16:30-18:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Some $L^r$-decomposition of $3D$-vector fields and its application to the stationary Navier-Stokes equations in multi-connected domains.

**小薗 英雄**(東北大学・大学院理学研究科)Some $L^r$-decomposition of $3D$-vector fields and its application to the stationary Navier-Stokes equations in multi-connected domains.

#### Tuesday Seminar on Topology

16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Existence Problem of Compact Locally Symmetric Spaces

**小林 俊行**(東京大学大学院数理科学研究科)Existence Problem of Compact Locally Symmetric Spaces

[ Abstract ]

The local to global study of geometries was a major trend of 20th century geometry, with remarkable developments achieved particularly in Riemannian geometry. In contrast, in areas such as Lorentz geometry, familiar to us as the space-time of relativity theory, and more generally in pseudo-Riemannian geometry, as well as in various other kinds of geometry (symplectic, complex geometry, ...), surprising little is known about global properties of the geometry even if we impose a locally homogeneous structure.

In this talk, I will give a survey on the recent developments on the question about how the local geometric structure affects the global nature of non-Riemannian manifolds with emphasis on the existence problem of compact models of locally symmetric spaces.

The local to global study of geometries was a major trend of 20th century geometry, with remarkable developments achieved particularly in Riemannian geometry. In contrast, in areas such as Lorentz geometry, familiar to us as the space-time of relativity theory, and more generally in pseudo-Riemannian geometry, as well as in various other kinds of geometry (symplectic, complex geometry, ...), surprising little is known about global properties of the geometry even if we impose a locally homogeneous structure.

In this talk, I will give a survey on the recent developments on the question about how the local geometric structure affects the global nature of non-Riemannian manifolds with emphasis on the existence problem of compact models of locally symmetric spaces.

### 2007/04/16

#### Seminar on Geometric Complex Analysis

10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Joyce計量のツイスター空間の具体的な構成方法

**本多 宣博**(東京工業大学)Joyce計量のツイスター空間の具体的な構成方法

#### Lectures

16:30-17:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Rearrangement inequalities and isoperimetric eigenvalue problems for second-order differential operators

**Francois Hamel**(エクス・マルセーユ第3大学 (Universite Aix-Marseille III))Rearrangement inequalities and isoperimetric eigenvalue problems for second-order differential operators

[ Abstract ]

The talk is concerned with various optimization results for the principal eigenvalues of general second-order elliptic operators in divergence form with Dirichlet boundary condition in bounded domains of R^n. We show that, to each operator in a given domain, we can associate a radially symmetric operator in the ball with the same Lebesgue measure, with a smaller principal eigenvalue. The constraints on the coefficients are of integral, pointwise or geometric types.

The results are new even for symmetric operators or in dimension 1. In particular, we generalize the Rayleigh-Faber-Krahn inequality for the principal eigenvalue of the Dirichlet Laplacian. The proofs use a new rearrangement technique, different from the Schwarz symmetrization. This talk is based on a joint work with N. Nadirashvili and E. Russ.

The talk is concerned with various optimization results for the principal eigenvalues of general second-order elliptic operators in divergence form with Dirichlet boundary condition in bounded domains of R^n. We show that, to each operator in a given domain, we can associate a radially symmetric operator in the ball with the same Lebesgue measure, with a smaller principal eigenvalue. The constraints on the coefficients are of integral, pointwise or geometric types.

The results are new even for symmetric operators or in dimension 1. In particular, we generalize the Rayleigh-Faber-Krahn inequality for the principal eigenvalue of the Dirichlet Laplacian. The proofs use a new rearrangement technique, different from the Schwarz symmetrization. This talk is based on a joint work with N. Nadirashvili and E. Russ.

### 2007/04/14

#### Infinite Analysis Seminar Tokyo

13:00-16:30 Room #117 (Graduate School of Math. Sci. Bldg.)

q-Fock空間と非対称Macdonald多項式

The Quantum Knizhnik-Zamolodchikov Equation

and Non-symmetric Macdonald Polynomials

**長尾健太郎**(京大理) 13:00-14:30q-Fock空間と非対称Macdonald多項式

[ Abstract ]

斎藤-竹村-Uglov,Varagnolo-Vasserotによって,q-Fock空間に

A型量子トロイダル代数のレベル(0,1)表現の構造が入ることが知られています.

この表現をある可換部分代数に制限して得られる作用の同時固有ベクトルを,

非対称Macdonald多項式を用いて構成することができます.

さらにこの同時固有ベクトルをq-Fock空間の基底とすることで,

量子トロイダル代数の作用を組合せ論的に記述することができます.

今回のセミナーでは,斎藤-竹村-Uglov,Varagnolo-Vasserotの構成を

振り返ったあとで,同時固有ベクトルの構成法を紹介します.

最後に箙多様体の同変K群との関連について少しだけ言及します.

斎藤-竹村-Uglov,Varagnolo-Vasserotによって,q-Fock空間に

A型量子トロイダル代数のレベル(0,1)表現の構造が入ることが知られています.

この表現をある可換部分代数に制限して得られる作用の同時固有ベクトルを,

非対称Macdonald多項式を用いて構成することができます.

さらにこの同時固有ベクトルをq-Fock空間の基底とすることで,

量子トロイダル代数の作用を組合せ論的に記述することができます.

今回のセミナーでは,斎藤-竹村-Uglov,Varagnolo-Vasserotの構成を

振り返ったあとで,同時固有ベクトルの構成法を紹介します.

最後に箙多様体の同変K群との関連について少しだけ言及します.

**笠谷昌弘**(京大理) 15:00-16:30The Quantum Knizhnik-Zamolodchikov Equation

and Non-symmetric Macdonald Polynomials

[ Abstract ]

We construct special solutions of the quantum Knizhnik-Zamolodchikov equation

on the tensor product of the vector representation of

the quantum algebra of type $A_{N-1}$.

They are constructed from non-symmetric Macdonald polynomials

through the action of the affine Hecke algebra.

As special cases,

(i) the matrix element of the vertex operators

of level one is reproduced, and

(ii) we give solutions of level $\\frac{N+1}{N}-N$.

(ii) is a generalization of the solution of

level $-\\frac{1}{2}$ by V.Pasquier and me.

This is a jount work with Y.Takeyama.

We construct special solutions of the quantum Knizhnik-Zamolodchikov equation

on the tensor product of the vector representation of

the quantum algebra of type $A_{N-1}$.

They are constructed from non-symmetric Macdonald polynomials

through the action of the affine Hecke algebra.

As special cases,

(i) the matrix element of the vertex operators

of level one is reproduced, and

(ii) we give solutions of level $\\frac{N+1}{N}-N$.

(ii) is a generalization of the solution of

level $-\\frac{1}{2}$ by V.Pasquier and me.

This is a jount work with Y.Takeyama.

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