## Seminar information archive

Seminar information archive ～12/08｜Today's seminar 12/09 | Future seminars 12/10～

#### Algebraic Geometry Seminar

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

Homogical methods in Non-commutative Geometry

**Dmitry KALEDIN**(Steklov研究所, 東大数理)Homogical methods in Non-commutative Geometry

[ Abstract ]

Of all the approaches to non-commutative geometry, probably the most promising is the homological one, developed by Keller, Kontsevich, Toen and others, where non-commutative eometry is understood as "geometry of triangulated categories". Examples of "geometric" triangulated categories come from representation theory, symplectic geometry (Fukaya category) and algebraic geometry (the derived category of coherent sheaves on an algebraic variety and

various generalizations). Non-commutative point of view is expected to be helpful even in traditional questions of algebraic geometry such as the termination of flips.

We plan to give an introduction to the subject, with emphasis on homological methods (such as e.g. Hodge theory which, as it turns out, can be mostly formulated in the non-commutative setting).

No knowledge of non-commutative geometry whatsoever is assumed. However, familiarity with basic homological algebra and algebraic geometry will be helpful.

Of all the approaches to non-commutative geometry, probably the most promising is the homological one, developed by Keller, Kontsevich, Toen and others, where non-commutative eometry is understood as "geometry of triangulated categories". Examples of "geometric" triangulated categories come from representation theory, symplectic geometry (Fukaya category) and algebraic geometry (the derived category of coherent sheaves on an algebraic variety and

various generalizations). Non-commutative point of view is expected to be helpful even in traditional questions of algebraic geometry such as the termination of flips.

We plan to give an introduction to the subject, with emphasis on homological methods (such as e.g. Hodge theory which, as it turns out, can be mostly formulated in the non-commutative setting).

No knowledge of non-commutative geometry whatsoever is assumed. However, familiarity with basic homological algebra and algebraic geometry will be helpful.

### 2007/10/15

#### Kavli IPMU Komaba Seminar

17:00-18:30 Room #002 (Graduate School of Math. Sci. Bldg.)

Topics on string theory, mirror symmetry, and Gromov-Witten invariants

**Shinobu Hosono**(The University of Tokyo)Topics on string theory, mirror symmetry, and Gromov-Witten invariants

[ Abstract ]

Recently, some technical developments in solving BCOV

(Bershadsky-Cecotti-Ooguri-Vafa) holomorphic anomaly equation has been

made and it has become possible to predict higher genus Gromov-Witten

invariants for some class of Calabi-Yau 3 folds.

With a brief introduction to BCOV equation, I will present some

predictions for Gromov-Witten invariants of certain Calabi-Yau 3 folds,

which are not birational but derived equivalent. (This is based on

a work with Y. Konishi which appeared in mathAG/0704.2928.)

Before coming to this specific topic, I will review some recent

topics of the homological mirror symmetry focusing on

its connection to the `classical' mirror symmetry, where the

variation theory of Hodge structures (VHS) plays a central role.

The BCOV equation and its open string generalization have their grounds

on the VHS.

Recently, some technical developments in solving BCOV

(Bershadsky-Cecotti-Ooguri-Vafa) holomorphic anomaly equation has been

made and it has become possible to predict higher genus Gromov-Witten

invariants for some class of Calabi-Yau 3 folds.

With a brief introduction to BCOV equation, I will present some

predictions for Gromov-Witten invariants of certain Calabi-Yau 3 folds,

which are not birational but derived equivalent. (This is based on

a work with Y. Konishi which appeared in mathAG/0704.2928.)

Before coming to this specific topic, I will review some recent

topics of the homological mirror symmetry focusing on

its connection to the `classical' mirror symmetry, where the

variation theory of Hodge structures (VHS) plays a central role.

The BCOV equation and its open string generalization have their grounds

on the VHS.

#### Seminar on Geometric Complex Analysis

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

On the curvature of holomorphic foliations

**大沢健夫**(名古屋大学)On the curvature of holomorphic foliations

### 2007/10/13

#### Monthly Seminar on Arithmetic of Automorphic Forms

13:30-16:00 Room #123 (Graduate School of Math. Sci. Bldg.)

2次のジーゲルカスプ形式の空間上のヘッケ作用素の明示的跡公式について

2次のジーゲルカスプ形式の空間上のヘッケ作用素の明示的跡公式について

A propagation formula for principal series Whittaker functions on $GL(3,C)$

**若槻聡**(金沢大学理学部) 13:30-14:302次のジーゲルカスプ形式の空間上のヘッケ作用素の明示的跡公式について

2次のジーゲルカスプ形式の空間上のヘッケ作用素の明示的跡公式について

[ Abstract ]

2次のジーゲルカスプ形式の空間上のヘッケ作用素の跡に、ある明示的公式を与

える。まだ公式から跡の具体的な数値を得ることはできないが、この公式は数値を得る

ための一つのステップとなっている。一変数の場合や一般論と比較しながら、得られた公式と今後の目標について解説する。

2次のジーゲルカスプ形式の空間上のヘッケ作用素の跡に、ある明示的公式を与

える。まだ公式から跡の具体的な数値を得ることはできないが、この公式は数値を得る

ための一つのステップとなっている。一変数の場合や一般論と比較しながら、得られた公式と今後の目標について解説する。

**平野幹**(成蹊大学理工学部) 15:00-16:00A propagation formula for principal series Whittaker functions on $GL(3,C)$

[ Abstract ]

$GL(n,\\mathbf{R})$上のクラス1Whittaker関数を$GL(n-1,\\mathbf{R})$上の同関数で表す公式が石井-Stadeにより得られてる(J. Funct. Anal. 244 (2007))。また、$GL(n,\\mathbf{R})$および$GL(n,\\mathbf{C})$上のクラス1Whittaker関数のelementaryな関係(Stade (1995)) により、この公式は$GL(n,\\mathbf{C})$上のクラス1Whittaker関数に対しても成立する。ここでは$GL(3,\\mathbf{C})$上のクラス1でない主系列Whittaker関数の明示公式(織田孝幸氏との共同研究)に基づき、これを$GL(2,\\mathbf{C})$上のクラス1でない主系列Whittaker関数で表す類似の公式を紹介する。

$GL(n,\\mathbf{R})$上のクラス1Whittaker関数を$GL(n-1,\\mathbf{R})$上の同関数で表す公式が石井-Stadeにより得られてる(J. Funct. Anal. 244 (2007))。また、$GL(n,\\mathbf{R})$および$GL(n,\\mathbf{C})$上のクラス1Whittaker関数のelementaryな関係(Stade (1995)) により、この公式は$GL(n,\\mathbf{C})$上のクラス1Whittaker関数に対しても成立する。ここでは$GL(3,\\mathbf{C})$上のクラス1でない主系列Whittaker関数の明示公式(織田孝幸氏との共同研究)に基づき、これを$GL(2,\\mathbf{C})$上のクラス1でない主系列Whittaker関数で表す類似の公式を紹介する。

### 2007/10/11

#### Operator Algebra Seminars

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

Construction of local nets from a wedge algebra

**Gandalf Lechner**(Erwin Schroedinger Institute)Construction of local nets from a wedge algebra

### 2007/10/10

#### Algebraic Geometry Seminar

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

Abel-Jacobi Maps Associated to Algebraic Cycles, I.

**James Lewis**(University of Alberta)Abel-Jacobi Maps Associated to Algebraic Cycles, I.

[ Abstract ]

This talk concerns the Bloch cycle class map from the higher Chow groups to Deligne cohomology of a projective algebraic manifold. We provide an explicit formula for this map in terms of polylogarithmic type currents.

This talk concerns the Bloch cycle class map from the higher Chow groups to Deligne cohomology of a projective algebraic manifold. We provide an explicit formula for this map in terms of polylogarithmic type currents.

#### Number Theory Seminar

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

Abel-Jacobi Maps Associated to Algebraic Cycles I

**James Lewis**(University of Alberta)Abel-Jacobi Maps Associated to Algebraic Cycles I

[ Abstract ]

This talk concerns the Bloch cycle class map from the higher Chow groups to Deligne cohomology of a projective algebraic manifold. We provide an explicit formula for this map in terms of polylogarithmic type currents.

This talk concerns the Bloch cycle class map from the higher Chow groups to Deligne cohomology of a projective algebraic manifold. We provide an explicit formula for this map in terms of polylogarithmic type currents.

#### Geometry Seminar

14:40-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)

A multiplicative analogue of quiver variety

AdS/CFT 対応における変分問題について

**山川 大亮**(京都大学大学院 理学研究科) 14:40-16:10A multiplicative analogue of quiver variety

[ Abstract ]

本講演では,箙(quiver)に付随して現れる新しい複素シンプレクティック多様体を紹介する.これは中島によって導入された箙多様体と非常に良く似た構成をする事で得られるが,違いは運動量写像ではなく群値運動量写像と呼ばれるものを使って商を取るところにある.この多様体は箙多様体と良く似た幾何学的性質を有し,一方,星型箙の場合に点付きRiemann球面上の放物接続のモジュライ空間とRiemann-Hilbert対応によって関係している.また箙多様体との直接的な関係も存在している.これらについて説明したい.

本講演では,箙(quiver)に付随して現れる新しい複素シンプレクティック多様体を紹介する.これは中島によって導入された箙多様体と非常に良く似た構成をする事で得られるが,違いは運動量写像ではなく群値運動量写像と呼ばれるものを使って商を取るところにある.この多様体は箙多様体と良く似た幾何学的性質を有し,一方,星型箙の場合に点付きRiemann球面上の放物接続のモジュライ空間とRiemann-Hilbert対応によって関係している.また箙多様体との直接的な関係も存在している.これらについて説明したい.

**加藤 晃史**(東京大学大学院数理科学研究科) 16:30-18:00AdS/CFT 対応における変分問題について

[ Abstract ]

弦双対性の一つである AdS/CFT 対応は,重力場(時空の幾何学)とゲージ理論(共形場理論)との間に対応があるという予想である.講演ではこの予想について概観するとともに,その一例として,佐々木・アインシュタイン多様体の体積に関する変分問題と quiver ゲージ理論の a-maximization の関係を説明したい.

弦双対性の一つである AdS/CFT 対応は,重力場(時空の幾何学)とゲージ理論(共形場理論)との間に対応があるという予想である.講演ではこの予想について概観するとともに,その一例として,佐々木・アインシュタイン多様体の体積に関する変分問題と quiver ゲージ理論の a-maximization の関係を説明したい.

#### Seminar on Probability and Statistics

16:20-17:30 Room #122 (Graduate School of Math. Sci. Bldg.)

勾配モデルの摂動解析と許容領域の評価

https://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2007/06.html

**清 智也**(東大情報理工)勾配モデルの摂動解析と許容領域の評価

[ Abstract ]

多変量標準正規分布を凸関数の勾配写像によって 引き戻すと, 一つの確率分布が得られる. さらにパラメトリックな勾配写像を考えれば, 統計モデルが得られる. この統計モデルを勾配モデルと呼ぶことにする. 本講演は二つの内容からなる. 第一に, 恒等写像に摂動を加えた勾配写像を考え, 対応する密度関数, キュムラント母関数, ダイバージェンスなどの摂動展開を求める. 第二に, より具体的な勾配モデルに対して, パラメータが定義域に属すための十分条件を示す. このような考察の必要性は, 定義域が無限個の 制約式で与えられることによる.

[ Reference URL ]多変量標準正規分布を凸関数の勾配写像によって 引き戻すと, 一つの確率分布が得られる. さらにパラメトリックな勾配写像を考えれば, 統計モデルが得られる. この統計モデルを勾配モデルと呼ぶことにする. 本講演は二つの内容からなる. 第一に, 恒等写像に摂動を加えた勾配写像を考え, 対応する密度関数, キュムラント母関数, ダイバージェンスなどの摂動展開を求める. 第二に, より具体的な勾配モデルに対して, パラメータが定義域に属すための十分条件を示す. このような考察の必要性は, 定義域が無限個の 制約式で与えられることによる.

https://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2007/06.html

#### Algebraic Geometry Seminar

15:00-16:00 Room #122 (Graduate School of Math. Sci. Bldg.)

p-adic Hodge theory in the non-commutative setting

**Dmitry Kaledin**(Steklov Institute)p-adic Hodge theory in the non-commutative setting

[ Abstract ]

We will explain what is the natural replacement of the notion of Hodge structure in the p-adic setting, and how to construct such a structure for non-commutative manifolds (something which at present cannot be done for the usual Hodge structures, but works perfectly well for the p-adic analog).

We will explain what is the natural replacement of the notion of Hodge structure in the p-adic setting, and how to construct such a structure for non-commutative manifolds (something which at present cannot be done for the usual Hodge structures, but works perfectly well for the p-adic analog).

### 2007/10/09

#### Tuesday Seminar on Topology

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

Classification of codimension-one locally free actions of the affine group of the real line.

**浅岡 正幸**(京都大学大学院理学研究科)Classification of codimension-one locally free actions of the affine group of the real line.

[ Abstract ]

By GA, we denote the group of affine and orientation-preserving transformations

of the real line. In this talk, I will report on classification of locally free action of

GA on closed three manifolds, which I obtained recently. In 1979, E.Ghys proved

that if such an action preserves a volume, then it is smoothly conjugate to a homogeneous action. However, it was unknown whether non-homogeneous action exists. As a consequence of the classification, we will see that the unit tangent bundle of a closed surface of higher genus admits a finite-parameter family of

non-homogeneous actions.

By GA, we denote the group of affine and orientation-preserving transformations

of the real line. In this talk, I will report on classification of locally free action of

GA on closed three manifolds, which I obtained recently. In 1979, E.Ghys proved

that if such an action preserves a volume, then it is smoothly conjugate to a homogeneous action. However, it was unknown whether non-homogeneous action exists. As a consequence of the classification, we will see that the unit tangent bundle of a closed surface of higher genus admits a finite-parameter family of

non-homogeneous actions.

#### Lie Groups and Representation Theory

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

Rankin-Cohen brackets and covariant quantization

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

**Michael Pevzner**(Reims University and University of Tokyo)Rankin-Cohen brackets and covariant quantization

[ Abstract ]

The particular geometric structure of causal symmetric spaces permits the definition of a covariant quantization of these homogeneous manifolds.

Composition formulae (#-products) of quantizad operators give rise to a new interpretation of Rankin-Cohen brackets and allow to connect them with the branching laws of tensor products of holomorphic discrete series representations.

[ Reference URL ]The particular geometric structure of causal symmetric spaces permits the definition of a covariant quantization of these homogeneous manifolds.

Composition formulae (#-products) of quantizad operators give rise to a new interpretation of Rankin-Cohen brackets and allow to connect them with the branching laws of tensor products of holomorphic discrete series representations.

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

### 2007/10/04

#### Operator Algebra Seminars

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

Local nets of von Neumann algebras and modular theory

**Gandalf Lechner**(Erwin Schroedinger Institute)Local nets of von Neumann algebras and modular theory

### 2007/10/02

#### Lie Groups and Representation Theory

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

Invariant integral operators on affine G-varieties and their kernels

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

**Pablo Ramacher**(Gottingen University)Invariant integral operators on affine G-varieties and their kernels

[ Abstract ]

We consider certain invariant integral operators on a smooth affine variety M carrying the action of a reductive algebraic group G, and assume that G acts on M with an open orbit. Then M is isomorphic to a homogeneous vector bundle, and can locally be described via the theory of prehomogenous vector spaces. We then study the Schwartz kernels of the considered operators, and give a description of their singularities using the calculus of b-pseudodifferential operators developed by Melrose. In particular, the restrictions of the kernels to the diagonal can be described in terms of local zeta functions.

[ Reference URL ]We consider certain invariant integral operators on a smooth affine variety M carrying the action of a reductive algebraic group G, and assume that G acts on M with an open orbit. Then M is isomorphic to a homogeneous vector bundle, and can locally be described via the theory of prehomogenous vector spaces. We then study the Schwartz kernels of the considered operators, and give a description of their singularities using the calculus of b-pseudodifferential operators developed by Melrose. In particular, the restrictions of the kernels to the diagonal can be described in terms of local zeta functions.

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

### 2007/09/28

#### Colloquium

16:30-17:30 Room #123 (Graduate School of Math. Sci. Bldg.)

Irreducible unitary representations and automorphic forms

**Marko Tadic'**(University of Zagreb)Irreducible unitary representations and automorphic forms

[ Abstract ]

Unitary representations of adelic groups in the spaces of automorphic forms are big source of important irreducible unitary representations of classical groups over local fields.

We shall present classifications of some classes of irreducible unitary representations (older, as well as quite new), describe

isolated unitary representations among them, and discuss which of them can be obtained from spaces of automorphic forms.

Unitary representations of adelic groups in the spaces of automorphic forms are big source of important irreducible unitary representations of classical groups over local fields.

We shall present classifications of some classes of irreducible unitary representations (older, as well as quite new), describe

isolated unitary representations among them, and discuss which of them can be obtained from spaces of automorphic forms.

### 2007/09/26

#### Algebraic Geometry Seminar

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

Floor diagrams and enumeration of tropical curves

**Grigory Mikhalkin**(Toronto大学)Floor diagrams and enumeration of tropical curves

[ Abstract ]

The enumerative problems considered in this talk are finding the number of curves in projective spaces (over complex, real and tropical numbers) of given genus and degree constrained by certain incidence conditions (e.g. passing via points or lines). Floor diagrams are a combinatorial tool that reduces an enumerative problem in dimension n to the corresponding problem n dimension n-1. Floor diagrams give a constructive (and rather efficient) way to find all tropical curves for a given enumerative problem. And once we have a tropical solution of the problem we can use it to solve the corresponding problems over the complex and real numbers.

The enumerative problems considered in this talk are finding the number of curves in projective spaces (over complex, real and tropical numbers) of given genus and degree constrained by certain incidence conditions (e.g. passing via points or lines). Floor diagrams are a combinatorial tool that reduces an enumerative problem in dimension n to the corresponding problem n dimension n-1. Floor diagrams give a constructive (and rather efficient) way to find all tropical curves for a given enumerative problem. And once we have a tropical solution of the problem we can use it to solve the corresponding problems over the complex and real numbers.

### 2007/09/19

#### Number Theory Seminar

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

Etale cobordism

**Gereon Quick**(Universitaet Muenster)Etale cobordism

[ Abstract ]

We define and study a new candidate of etale topological cohomology theories for schemes over a field of abritrary characteristic: etale cobordism. As etale K-theory is related to algebraic K-theory, etale cobordism is related to algebraic cobordism of Voevodsky and Levine/Morel. It shares some nice properties of topological theories, e.g. it is equipped with an Atiyah-Hirzebruch spectral sequence from etale cohomology. We discuss in particular a comparison theorem between etale and algebraic cobordism after inverting a Bott element and, finally, we give an outlook to further possible applications of this theory.

We define and study a new candidate of etale topological cohomology theories for schemes over a field of abritrary characteristic: etale cobordism. As etale K-theory is related to algebraic K-theory, etale cobordism is related to algebraic cobordism of Voevodsky and Levine/Morel. It shares some nice properties of topological theories, e.g. it is equipped with an Atiyah-Hirzebruch spectral sequence from etale cohomology. We discuss in particular a comparison theorem between etale and algebraic cobordism after inverting a Bott element and, finally, we give an outlook to further possible applications of this theory.

### 2007/09/15

#### Monthly Seminar on Arithmetic of Automorphic Forms

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

Siegl principal series Whittaker functions on $Sp(2,\\mathbf{R})$

(部屋は056室)

A higher rank version of Abel-Jacobi's theorem (Room 056)

**長谷川泰子**(東京大学数理科学) 13:30-14:30Siegl principal series Whittaker functions on $Sp(2,\\mathbf{R})$

(部屋は056室)

[ Abstract ]

2次シンプレクティック群のSiegel極大放物型部分群から誘導された一般型主系列表現に対するWhittaker関数の級数表示と積分表示を与えることを目的とし,Whittaker関数の満たす微分評定式を与え,その解の構成に向けて現在進めている研究の方針を述べる.

(部屋は,冷房効く056室に変更です)

2次シンプレクティック群のSiegel極大放物型部分群から誘導された一般型主系列表現に対するWhittaker関数の級数表示と積分表示を与えることを目的とし,Whittaker関数の満たす微分評定式を与え,その解の構成に向けて現在進めている研究の方針を述べる.

(部屋は,冷房効く056室に変更です)

**市川尚志**(佐賀大学理工学部) 15:00-16:00A higher rank version of Abel-Jacobi's theorem (Room 056)

[ Abstract ]

極大退化曲線に近いリーマン面上のベクトル束とそのモジュライについて話す.次数0の安定ベクトル束が,リーマン面を一意化するショットキー群の線形表現から得られることを述べ,ショットキー群の線形表現の空間とベクトル束のモジュライ空間の関係を,アーベル・ヤコビの定理,フェアリンデ公式を用いて考察する.

(部屋は117号室です)

極大退化曲線に近いリーマン面上のベクトル束とそのモジュライについて話す.次数0の安定ベクトル束が,リーマン面を一意化するショットキー群の線形表現から得られることを述べ,ショットキー群の線形表現の空間とベクトル束のモジュライ空間の関係を,アーベル・ヤコビの定理,フェアリンデ公式を用いて考察する.

(部屋は117号室です)

### 2007/09/12

#### Algebraic Geometry Seminar

15:00-18:00 Room #117 (Graduate School of Math. Sci. Bldg.)

Classification of p-divisible groups by displays and duality

Applications of the theory of displays

Presentation of mapping class groups from algebraic geometry

**E. Lau**(Univ. of Bielefeld) 15:00-15:45Classification of p-divisible groups by displays and duality

**T. Zink**(Univ. of Bielefeld) 16:00-16:45Applications of the theory of displays

**E. Looijenga**(Univ. of Utrecht) 17:00-18:00Presentation of mapping class groups from algebraic geometry

[ Abstract ]

A presentation of the mapping class group of a genus g surface with one hole is due to Wajnryb with later improvements due to M. Matsumoto. The generators are Dehn twists defined by 2g+1 closed curves on the surface. The relations involving only two Dehn twists are the familiar Artin relations, we show that those involving more than two can be derived from algebro-geometry considerations.

A presentation of the mapping class group of a genus g surface with one hole is due to Wajnryb with later improvements due to M. Matsumoto. The generators are Dehn twists defined by 2g+1 closed curves on the surface. The relations involving only two Dehn twists are the familiar Artin relations, we show that those involving more than two can be derived from algebro-geometry considerations.

#### Number Theory Seminar

15:00-18:00 Room #117 (Graduate School of Math. Sci. Bldg.)

Classification of p-divisible groups by displays and duality

Applications of the theory of displays

Presentation of mapping class groups from algebraic geometry

**E. Lau**(Univ. of Bielefeld) 15:00-15:45Classification of p-divisible groups by displays and duality

**T. Zink**(Univ. of Bielefeld) 16:00-16:45Applications of the theory of displays

**E. Looijenga**(Univ. of Utrecht) 17:00-18:00Presentation of mapping class groups from algebraic geometry

[ Abstract ]

A presentation of the mapping class group of a genus g surface with one hole is due to Wajnryb with later improvements due to M. Matsumoto. The generators are Dehn twists defined by 2g+1 closed curves on the surface. The relations involving only two Dehn twists are the familiar Artin relations, we show that those involving more than two can be derived from algebro-geometry considerations.

A presentation of the mapping class group of a genus g surface with one hole is due to Wajnryb with later improvements due to M. Matsumoto. The generators are Dehn twists defined by 2g+1 closed curves on the surface. The relations involving only two Dehn twists are the familiar Artin relations, we show that those involving more than two can be derived from algebro-geometry considerations.

### 2007/09/05

#### PDE Real Analysis Seminar

10:30-11:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Reguarity of Weak Solutions to the Navier-Stokes System beyond Serrin's Criterion

http://coe.math.sci.hokudai.ac.jp/sympo/pde_ra/index.html

**Reinhard Farwig**(Darmstadt University of Technology)Reguarity of Weak Solutions to the Navier-Stokes System beyond Serrin's Criterion

[ Abstract ]

Consider a weak instationary solution $u(x,t)$ of the Navier-Stokes equations in a domain $\\Omega \\subset \\mathbb{R}^3$ in the sense of Leray-Hopf. As is well-known, $u$ is is unique and regular if $u\\in L^s(0,T;L^q(\\Omega))$ satisfies the {\\it strong energy inequality} and $s,q$ satisfy Serrin's condition $\\frac{2}{s} + \\frac{3}{q}=1$, $s>2,\\, q>3$. Now consider $u$ such that $$u\\in L^r(0,T;L^q(\\Omega))\\quad \\mbox{ where }\\quad \\frac{2}{r} + \\frac{3}{q}>1$$ and has a sufficiently small norm in $L^r(0,T;L^q(\\Omega))$. Then we will prove that $u$ is regular. Similar results of local rather than global type in space will be proved provided that $u$ satisfies the {\\it localized energy inequality}. Finally H\\"older continuity of the kinetic energy in time will imply regularity.

The proofs use local in time regularity results which are based on the {\\it theory of very weak solutions} and on uniqueness arguments for weak solutions.

[ Reference URL ]Consider a weak instationary solution $u(x,t)$ of the Navier-Stokes equations in a domain $\\Omega \\subset \\mathbb{R}^3$ in the sense of Leray-Hopf. As is well-known, $u$ is is unique and regular if $u\\in L^s(0,T;L^q(\\Omega))$ satisfies the {\\it strong energy inequality} and $s,q$ satisfy Serrin's condition $\\frac{2}{s} + \\frac{3}{q}=1$, $s>2,\\, q>3$. Now consider $u$ such that $$u\\in L^r(0,T;L^q(\\Omega))\\quad \\mbox{ where }\\quad \\frac{2}{r} + \\frac{3}{q}>1$$ and has a sufficiently small norm in $L^r(0,T;L^q(\\Omega))$. Then we will prove that $u$ is regular. Similar results of local rather than global type in space will be proved provided that $u$ satisfies the {\\it localized energy inequality}. Finally H\\"older continuity of the kinetic energy in time will imply regularity.

The proofs use local in time regularity results which are based on the {\\it theory of very weak solutions} and on uniqueness arguments for weak solutions.

http://coe.math.sci.hokudai.ac.jp/sympo/pde_ra/index.html

### 2007/08/29

#### Algebraic Geometry Seminar

17:00-18:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Computations on the moduli spaces of weighted log pairs

**Valery Alexeev**(Georgia大学)Computations on the moduli spaces of weighted log pairs

### 2007/08/27

#### Number Theory Seminar

16:30-17:30 Room #002 (Graduate School of Math. Sci. Bldg.)

The reductive Borel-Serre motive

**Steven Zucker**(Johns Hopkins大学)The reductive Borel-Serre motive

### 2007/08/02

#### Algebraic Geometry Seminar

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

Dynamics of automorphisms on algebraic varieties

**De-Qi Zhang**(Singapore大学)Dynamics of automorphisms on algebraic varieties

[ Abstract ]

The building blocks of automorphisms / endomorphisms of compact varieties are determined --- an algebro geometric approach towards dynamics.

The building blocks of automorphisms / endomorphisms of compact varieties are determined --- an algebro geometric approach towards dynamics.

### 2007/07/25

#### Seminar on Probability and Statistics

16:20-17:30 Room #122 (Graduate School of Math. Sci. Bldg.)

大規模ランダム行列のスペクトル理論とデータ解析への応用(Review)

https://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2007/05.html

**小林 景**(統計数理研究所, 学振特別研究員)大規模ランダム行列のスペクトル理論とデータ解析への応用(Review)

[ Abstract ]

The empirical spectral distribution of random matrices have been studied since Wigner's pioneering work on the semicircular law in the 1950's. The result says that the empirical spectral distribution of a symmetric matrix with i.i.d. random elements converges to the semicircular law as the size of the matrix becomes large. Though this result is beautiful in theory, its application has been limited to a few problems in nuclear physics and coding theory. The next breakthrough was the Marcenko-Pastur (M-P) law for the asymptotic spectral distribution of sample covariance matrices. The M-P law has found more applications, in particular high dimensional statistical data analysis, than the semicircular law.

In this talk I will first review these two significant results. Each of them has three completely different proofs. Then I will explain several other theoretical results that have mostly been studied this decade. Finally, I will present some of the applications of these results. This review is partly based on lectures on random matrices given by P. Bickel, N. El-Karoui and A. Guionnet, and also some seminars at UC Berkeley.

(# This talk is almost the same as the talk I gave at ISM on June 1.)

[ Reference URL ]The empirical spectral distribution of random matrices have been studied since Wigner's pioneering work on the semicircular law in the 1950's. The result says that the empirical spectral distribution of a symmetric matrix with i.i.d. random elements converges to the semicircular law as the size of the matrix becomes large. Though this result is beautiful in theory, its application has been limited to a few problems in nuclear physics and coding theory. The next breakthrough was the Marcenko-Pastur (M-P) law for the asymptotic spectral distribution of sample covariance matrices. The M-P law has found more applications, in particular high dimensional statistical data analysis, than the semicircular law.

In this talk I will first review these two significant results. Each of them has three completely different proofs. Then I will explain several other theoretical results that have mostly been studied this decade. Finally, I will present some of the applications of these results. This review is partly based on lectures on random matrices given by P. Bickel, N. El-Karoui and A. Guionnet, and also some seminars at UC Berkeley.

(# This talk is almost the same as the talk I gave at ISM on June 1.)

https://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2007/05.html

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