## Seminar information archive

Seminar information archive ～02/01｜Today's seminar 02/02 | Future seminars 02/03～

#### Algebraic Geometry Seminar

16:40-18:10 Room #126 (Graduate School of Math. Sci. Bldg.)

On weak Fano varieties with log canonical singularities

**權業 善範**(東大数理)On weak Fano varieties with log canonical singularities

[ Abstract ]

We prove that the anti-canonical divisors of weak Fano

3-folds with log canonical singularities are semiample. Moreover, we consider

semiampleness of the anti-log canonical divisor of any weak log Fano pair

with log canonical singularities. We show semiampleness dose not hold in

general by constructing several examples. Based on those examples, we propose

sufficient conditions which seem to be the best possible and we prove

semiampleness under such conditions. In particular we derive semiampleness of the

anti-canonical divisors of log canonical weak Fano 4-folds whose lc centers

are at most 1-dimensional. We also investigate the Kleiman-Mori cones of

weak log Fano pairs with log canonical singularities.

We prove that the anti-canonical divisors of weak Fano

3-folds with log canonical singularities are semiample. Moreover, we consider

semiampleness of the anti-log canonical divisor of any weak log Fano pair

with log canonical singularities. We show semiampleness dose not hold in

general by constructing several examples. Based on those examples, we propose

sufficient conditions which seem to be the best possible and we prove

semiampleness under such conditions. In particular we derive semiampleness of the

anti-canonical divisors of log canonical weak Fano 4-folds whose lc centers

are at most 1-dimensional. We also investigate the Kleiman-Mori cones of

weak log Fano pairs with log canonical singularities.

### 2010/01/22

#### Lecture Series on Mathematical Sciences in Soceity

16:20-17:50 Room #117 (Graduate School of Math. Sci. Bldg.)

数学者と企業研究者との連携

**中川淳一**(新日本製鐵(株)技術開発本部)数学者と企業研究者との連携

#### Lectures

16:30-18:00 Room #118 (Graduate School of Math. Sci. Bldg.)

Fractional Evolution Equations and Applications 3

**伊東一文**(大学院数理科学研究科)Fractional Evolution Equations and Applications 3

[ Abstract ]

In recent years increasing interests and considerable

researches have been given to the fractional differential equations both

in time and space variables.

These are due to the applications of the fractional calculus

to problems in a wide areas of physics and engineering science and a rapid

development of the corresponding theory. A motivating example includes

the so-called continuous time random walk process

and the Levy process model for the mathematical finance.

In this lecture we develop solution techniques based on the linear and

nonlinear semigroup theory and apply it to solve the associated inverse

and optimal control problems. The property and stability of the solutions

as well as numerical integration methods

are discussed. The lecture also covers the basis and application of the

so-called Crandall-Ligget theory and the locally quasi-dissipative

operator method developed by Kobayashi-Kobayashi-Oharu.

Nonlinear evolution equations, Crandall-Ligget theory,

Locally quasi-dissipative operators approach

In recent years increasing interests and considerable

researches have been given to the fractional differential equations both

in time and space variables.

These are due to the applications of the fractional calculus

to problems in a wide areas of physics and engineering science and a rapid

development of the corresponding theory. A motivating example includes

the so-called continuous time random walk process

and the Levy process model for the mathematical finance.

In this lecture we develop solution techniques based on the linear and

nonlinear semigroup theory and apply it to solve the associated inverse

and optimal control problems. The property and stability of the solutions

as well as numerical integration methods

are discussed. The lecture also covers the basis and application of the

so-called Crandall-Ligget theory and the locally quasi-dissipative

operator method developed by Kobayashi-Kobayashi-Oharu.

Nonlinear evolution equations, Crandall-Ligget theory,

Locally quasi-dissipative operators approach

### 2010/01/21

#### Operator Algebra Seminars

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

On Subfactors Arising from Asymptotic Representations of Symmetric Groups

**山下真**(東大数理)On Subfactors Arising from Asymptotic Representations of Symmetric Groups

#### Applied Analysis

16:00-17:30 Room #122 (Graduate School of Math. Sci. Bldg.)

A finite volume method on general meshes for a degenerate parabolic convection-reaction-diffusion equation

**Danielle Hilhorst**(パリ南大学 / CNRS)A finite volume method on general meshes for a degenerate parabolic convection-reaction-diffusion equation

[ Abstract ]

We propose a finite volume method on general meshes for degenerate parabolic convection-reaction-diffusion equations. Such equations arise for instance in the modeling of contaminant transport in groundwater. After giving a convergence proof, we present the results of numerical tests.

We propose a finite volume method on general meshes for degenerate parabolic convection-reaction-diffusion equations. Such equations arise for instance in the modeling of contaminant transport in groundwater. After giving a convergence proof, we present the results of numerical tests.

### 2010/01/20

#### Geometry Seminar

17:00-18:30 Room #122 (Graduate School of Math. Sci. Bldg.)

Asymptotically conical manifolds and the Monge-Ampere equation

**Craig Van Coevering**(MIT)Asymptotically conical manifolds and the Monge-Ampere equation

[ Abstract ]

Some analysis is considered on manifolds with a conical end. Then we show that in the Kahler case the complex Monge-Ampere equation can be solved with the same regularity as is known in the ALE case. By considering resolutions of toric singularities and hypersurface singularities this can easily be used to produce many Calabi-Yau manifolds with a conical end.

Some analysis is considered on manifolds with a conical end. Then we show that in the Kahler case the complex Monge-Ampere equation can be solved with the same regularity as is known in the ALE case. By considering resolutions of toric singularities and hypersurface singularities this can easily be used to produce many Calabi-Yau manifolds with a conical end.

#### Lectures

16:30-18:00 Room #118 (Graduate School of Math. Sci. Bldg.)

Fractional Evolution Equations and Applications 2

**伊東一文**(大学院数理科学研究科)Fractional Evolution Equations and Applications 2

[ Abstract ]

In recent years increasing interests and considerable

researches have been given to the fractional differential equations both

in time and space variables.

These are due to the applications of the fractional calculus

to problems in a wide areas of physics and engineering science and a rapid

development of the corresponding theory. A motivating example includes

the so-called continuous time random walk process

and the Levy process model for the mathematical finance.

In this lecture we develop solution techniques based on the linear and

nonlinear semigroup theory and apply it to solve the associated inverse

and optimal control problems. The property and stability of the solutions

as well as numerical integration methods

are discussed. The lecture also covers the basis and application of the

so-called Crandall-Ligget theory and the locally quasi-dissipative

operator method developed by Kobayashi-Kobayashi-Oharu.

Existence and Uniqueness by C_0 semigroup theory, dissipative linear

operator

and Hille-Yoshida, Trotter-Kato theory.

In recent years increasing interests and considerable

researches have been given to the fractional differential equations both

in time and space variables.

These are due to the applications of the fractional calculus

to problems in a wide areas of physics and engineering science and a rapid

development of the corresponding theory. A motivating example includes

the so-called continuous time random walk process

and the Levy process model for the mathematical finance.

In this lecture we develop solution techniques based on the linear and

nonlinear semigroup theory and apply it to solve the associated inverse

and optimal control problems. The property and stability of the solutions

as well as numerical integration methods

are discussed. The lecture also covers the basis and application of the

so-called Crandall-Ligget theory and the locally quasi-dissipative

operator method developed by Kobayashi-Kobayashi-Oharu.

Existence and Uniqueness by C_0 semigroup theory, dissipative linear

operator

and Hille-Yoshida, Trotter-Kato theory.

#### Mathematical Biology Seminar

14:40-16:10 Room #052 (Graduate School of Math. Sci. Bldg.)

東京都市圏パーソントリップ調査に基づく新型インフルエンザ感染拡大シミュレーション

**江島啓介**(東京大学情報理工学研究科数理情報専攻修士課程)東京都市圏パーソントリップ調査に基づく新型インフルエンザ感染拡大シミュレーション

[ Abstract ]

新型インフルエンザの感染拡大に対する対応策として,学校施設等の閉鎖など外

出時の感染機会を減らすための措置が考えられるが,その効果は十分に明らかで

はない.そこで本研究では,individual based modelに東京都市圏パーソント

リップ調査を組み合わせることにより感染拡大モデルを構築し,数値シミュレー

ションによって外出規制および施設閉鎖の効果を検討した.外出規制に関して

は,規制日数が12日以上と長い場合には効果が大きいことがわかった.また,施

設閉鎖に関しては,閉鎖期間・閉鎖基準を厳しくすると,ピークまでの日数は変

わらないものの,累積罹患率は低下することがわかった.

新型インフルエンザの感染拡大に対する対応策として,学校施設等の閉鎖など外

出時の感染機会を減らすための措置が考えられるが,その効果は十分に明らかで

はない.そこで本研究では,individual based modelに東京都市圏パーソント

リップ調査を組み合わせることにより感染拡大モデルを構築し,数値シミュレー

ションによって外出規制および施設閉鎖の効果を検討した.外出規制に関して

は,規制日数が12日以上と長い場合には効果が大きいことがわかった.また,施

設閉鎖に関しては,閉鎖期間・閉鎖基準を厳しくすると,ピークまでの日数は変

わらないものの,累積罹患率は低下することがわかった.

### 2010/01/19

#### Tuesday Seminar of Analysis

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

超函数の有界性と Massera 型定理について

**岡田 靖則**(千葉大・理)超函数の有界性と Massera 型定理について

#### Operator Algebra Seminars

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

Entire Cyclic Cohomology of Noncommutative Spheres

**高井博司**(首都大学東京)Entire Cyclic Cohomology of Noncommutative Spheres

#### Tuesday Seminar on Topology

17:00-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Localization via group action and its application to

the period condition of algebraic minimal surfaces

**小林 亮一**(名古屋大学)Localization via group action and its application to

the period condition of algebraic minimal surfaces

[ Abstract ]

The optimal estimate for the number of exceptional

values of the Gauss map of algebraic minimal surfaces is a long

standing problem. In this lecture, I will introduce new ideas

toward the solution of this problem. The ``collective Cohn-Vossen

inequality" is the key idea. From this we have effective

Nevanlinna's lemma on logarithmic derivative for a certain class

of meromorphic functions on the disk. On the other hand, we can

construct a family holomorphic functions on the disk from the

Weierstrass data of the algebraic minimal surface under

consideration, which encodes the period condition.

Applying effective Lemma on logarithmic derivative to these

functions, we can extract an intriguing inequality.

The optimal estimate for the number of exceptional

values of the Gauss map of algebraic minimal surfaces is a long

standing problem. In this lecture, I will introduce new ideas

toward the solution of this problem. The ``collective Cohn-Vossen

inequality" is the key idea. From this we have effective

Nevanlinna's lemma on logarithmic derivative for a certain class

of meromorphic functions on the disk. On the other hand, we can

construct a family holomorphic functions on the disk from the

Weierstrass data of the algebraic minimal surface under

consideration, which encodes the period condition.

Applying effective Lemma on logarithmic derivative to these

functions, we can extract an intriguing inequality.

#### Lectures

16:30-18:00 Room #118 (Graduate School of Math. Sci. Bldg.)

Fractional Evolution Equations and Applications 1

**伊東一文**(大学院数理科学研究科)Fractional Evolution Equations and Applications 1

[ Abstract ]

In recent years increasing interests and considerable

researches have been given to the fractional differential equations both

in time and space variables.

These are due to the applications of the fractional calculus

to problems in a wide areas of physics and engineering science and a rapid

development of the corresponding theory. A motivating example includes

the so-called continuous time random walk process

and the Levy process model for the mathematical finance.

In this lecture we develop solution techniques based on the linear and

nonlinear semigroup theory and apply it to solve the associated inverse

and optimal control problems. The property and stability of the solutions

as well as numerical integration methods

are discussed. The lecture also covers the basis and application of the

so-called Crandall-Ligget theory and the locally quasi-dissipative

operator method developed by Kobayashi-Kobayashi-Oharu.

Motivation: Continuous time random walk (CTRW) process

Fractional differential equations in time and Mittag-Leffler functions

In recent years increasing interests and considerable

researches have been given to the fractional differential equations both

in time and space variables.

These are due to the applications of the fractional calculus

to problems in a wide areas of physics and engineering science and a rapid

development of the corresponding theory. A motivating example includes

the so-called continuous time random walk process

and the Levy process model for the mathematical finance.

In this lecture we develop solution techniques based on the linear and

nonlinear semigroup theory and apply it to solve the associated inverse

and optimal control problems. The property and stability of the solutions

as well as numerical integration methods

are discussed. The lecture also covers the basis and application of the

so-called Crandall-Ligget theory and the locally quasi-dissipative

operator method developed by Kobayashi-Kobayashi-Oharu.

Motivation: Continuous time random walk (CTRW) process

Fractional differential equations in time and Mittag-Leffler functions

### 2010/01/18

#### Seminar on Geometric Complex Analysis

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

スプライス商特異点について

**奥間智弘**(山形大学地域教育文化学部)スプライス商特異点について

#### Algebraic Geometry Seminar

16:40-18:10 Room #126 (Graduate School of Math. Sci. Bldg.)

The divisor class group of terminal Gorenstein Fano 3-folds and rationality questions

**Anne-Sophie Kaloghiros**(RIMS)The divisor class group of terminal Gorenstein Fano 3-folds and rationality questions

[ Abstract ]

Let Y be a quartic hypersurface in CP^4 with mild singularities, e.g. no worse than ordinary double points.

If Y contains a surface that is not a hyperplane section, Y is not Q-factorial and the divisor class group of Y, Cl Y, contains divisors that are not Cartier. However, the rank of Cl Y is bounded.

In this talk, I will show that in most cases, it is possible to describe explicitly the divisor class group Cl Y by running a Minimal Model Program (MMP) on X, a small Q-factorialisation of Y. In this case, the generators of Cl Y/ Pic Y are ``topological traces " of K-negative extremal contractions on X.

This has surprising consequences: it is possible to conclude that a number of families of non-factorial quartic 3-folds are rational.

In particular, I give some examples of rational quartic hypersurfaces Y_4\\subset CP^4 with rk Cl Y=2 and show that when the divisor class group of Y has sufficiently high rank, Y is always rational.

Let Y be a quartic hypersurface in CP^4 with mild singularities, e.g. no worse than ordinary double points.

If Y contains a surface that is not a hyperplane section, Y is not Q-factorial and the divisor class group of Y, Cl Y, contains divisors that are not Cartier. However, the rank of Cl Y is bounded.

In this talk, I will show that in most cases, it is possible to describe explicitly the divisor class group Cl Y by running a Minimal Model Program (MMP) on X, a small Q-factorialisation of Y. In this case, the generators of Cl Y/ Pic Y are ``topological traces " of K-negative extremal contractions on X.

This has surprising consequences: it is possible to conclude that a number of families of non-factorial quartic 3-folds are rational.

In particular, I give some examples of rational quartic hypersurfaces Y_4\\subset CP^4 with rk Cl Y=2 and show that when the divisor class group of Y has sufficiently high rank, Y is always rational.

### 2010/01/15

#### Lecture Series on Mathematical Sciences in Soceity

16:20-17:50 Room #117 (Graduate School of Math. Sci. Bldg.)

製鐵プロセスにおける数学

**中川淳一**(新日本製鐵(株)技術開発本部)製鐵プロセスにおける数学

### 2010/01/14

#### Operator Algebra Seminars

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

Applications of operator algebras in Quantum information theory

**Marius Junge**(Univ. Illinois, Urbana-Champaign)Applications of operator algebras in Quantum information theory

### 2010/01/13

#### Lectures

16:45-17:45 Room #128 (Graduate School of Math. Sci. Bldg.)

Scaled limit for the largest eigenvalue from the generalized Cauchy ensemble

**Felix Rubin**(Zurich 大学)Scaled limit for the largest eigenvalue from the generalized Cauchy ensemble

#### Lectures

15:30-16:30 Room #128 (Graduate School of Math. Sci. Bldg.)

Breaking the chain: slow versus fast pulling

**Michael Allman**(Warwick 大学)Breaking the chain: slow versus fast pulling

### 2010/01/12

#### Lie Groups and Representation Theory

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

代数的差分方程式の可解性と既約性

https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar2010.html#20100112nishioka

**西岡斉治**(東京大学大学院数理科学研究科博士課程)代数的差分方程式の可解性と既約性

[ Abstract ]

差分代数の理論を使って,代数的差分方程式の代数函数解や超幾

何函数解の非存在や,存在する場合の特殊解の分類をする。

[ Reference URL ]差分代数の理論を使って,代数的差分方程式の代数函数解や超幾

何函数解の非存在や,存在する場合の特殊解の分類をする。

https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar2010.html#20100112nishioka

#### Tuesday Seminar on Topology

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

Index problem for generically-wild homoclinic classes in dimension three

On a generalized suspension theorem for directed Fukaya categories

**篠原 克寿**(東京大学大学院数理科学研究科) 16:30-17:30Index problem for generically-wild homoclinic classes in dimension three

[ Abstract ]

In the sphere of non-hyperbolic differentiable dynamical systems, one can construct an example of a homolinic class which does not admit any kind of dominated splittings (a weak form of hyperbolicity) in a robust way. In this talk, we discuss the index (dimension of the unstable manifold) of the periodic points inside such homoclinic classes from a $C^1$-generic viewpoint.

In the sphere of non-hyperbolic differentiable dynamical systems, one can construct an example of a homolinic class which does not admit any kind of dominated splittings (a weak form of hyperbolicity) in a robust way. In this talk, we discuss the index (dimension of the unstable manifold) of the periodic points inside such homoclinic classes from a $C^1$-generic viewpoint.

**二木 昌宏**(東京大学大学院数理科学研究科) 17:30-18:30On a generalized suspension theorem for directed Fukaya categories

[ Abstract ]

The directed Fukaya category $\\mathrm{Fuk} W$ of exact Lefschetz

fibration $W : X \\to \\mathbb{C}$ proposed by Kontsevich is a

categorification of the Milnor lattice of $W$. This is defined as the

directed $A_\\infty$-category $\\mathrm{Fuk} W = \\mathrm{Fuk}^\\to

\\mathbb{V}$ generated by a distinguished basis $\\mathbb{V}$ of

vanishing cycles.

Recently Seidel has proved that this is stable under the suspension $W

+ u^2$ as a consequence of his foundational work on the directed

Fukaya category. We generalize his suspension theorem to the $W + u^d$

case by considering partial tensor product $\\mathrm{Fuk} W \\otimes'

\\mathcal{A}_{d-1}$, where $\\mathcal{A}_{d-1}$ is the category

corresponding to the $A_n$-type quiver. This also generalizes a recent

work by the author with Kazushi Ueda.

The directed Fukaya category $\\mathrm{Fuk} W$ of exact Lefschetz

fibration $W : X \\to \\mathbb{C}$ proposed by Kontsevich is a

categorification of the Milnor lattice of $W$. This is defined as the

directed $A_\\infty$-category $\\mathrm{Fuk} W = \\mathrm{Fuk}^\\to

\\mathbb{V}$ generated by a distinguished basis $\\mathbb{V}$ of

vanishing cycles.

Recently Seidel has proved that this is stable under the suspension $W

+ u^2$ as a consequence of his foundational work on the directed

Fukaya category. We generalize his suspension theorem to the $W + u^d$

case by considering partial tensor product $\\mathrm{Fuk} W \\otimes'

\\mathcal{A}_{d-1}$, where $\\mathcal{A}_{d-1}$ is the category

corresponding to the $A_n$-type quiver. This also generalizes a recent

work by the author with Kazushi Ueda.

### 2010/01/08

#### Colloquium

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

特殊関数とFuchs型常微分方程式

**大島利雄**(東京大学大学院数理科学研究科)特殊関数とFuchs型常微分方程式

[ Abstract ]

岩波全書の数学公式集III「特殊関数」の大部分はGaussの超幾何関数とその特殊化のBessel関数やLegendre多項式などで占められている。この超幾何関数についての最も重要な基本結果は1での値を与えるGaussの和公式とRiemann schemeによる特徴付けとであろう。この関数は一般超幾何関数やJordan-Pochhammer方程式へ、またHeun方程式からPainleve方程式へという解析、さらにAppell,Gelfand-青本,Heckman-Opdamによる多変数化という3つの方向の発展がある。講演ではこれらを含む統一的な理解、Riemann schemeの一般化とuniversal modelの存在定理(Deligne-Katz-Simpson問題)、接続公式(Gaussの和公式の一般化)、無限次元Kac-Moody Weyl群の作用について解説し、特異点の合流、積分表示、ベキ級数表示などについても述べたい。結果は構成的 でコンピュータ・プログラムで実現できる。

岩波全書の数学公式集III「特殊関数」の大部分はGaussの超幾何関数とその特殊化のBessel関数やLegendre多項式などで占められている。この超幾何関数についての最も重要な基本結果は1での値を与えるGaussの和公式とRiemann schemeによる特徴付けとであろう。この関数は一般超幾何関数やJordan-Pochhammer方程式へ、またHeun方程式からPainleve方程式へという解析、さらにAppell,Gelfand-青本,Heckman-Opdamによる多変数化という3つの方向の発展がある。講演ではこれらを含む統一的な理解、Riemann schemeの一般化とuniversal modelの存在定理(Deligne-Katz-Simpson問題)、接続公式(Gaussの和公式の一般化)、無限次元Kac-Moody Weyl群の作用について解説し、特異点の合流、積分表示、ベキ級数表示などについても述べたい。結果は構成的 でコンピュータ・プログラムで実現できる。

### 2010/01/07

#### Operator Algebra Seminars

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

Large deviations, Billiards, and Non-equilibrium Statistical Mechanics

**Luc Rey-Bellet**(Univ. Massachusetts)Large deviations, Billiards, and Non-equilibrium Statistical Mechanics

#### GCOE Seminars

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

Large deviations, Billiards, and Non-equilibrium Statistical Mechanics

**LucRey-Bellet**(Univ. Massachusetts)Large deviations, Billiards, and Non-equilibrium Statistical Mechanics

[ Abstract ]

Large deviations have applications in many aspects of statistical mechanics. New applications for the steady states of non-equilibrium statistical mechanics have emerged during the past ten years and these applications deal with the fluctuations of the entropy production. After discussing some general aspects of entropy production we turn to concrete examples, in particular billiards with and without external forces and discuss large deviations theorems for such systems.

Large deviations have applications in many aspects of statistical mechanics. New applications for the steady states of non-equilibrium statistical mechanics have emerged during the past ten years and these applications deal with the fluctuations of the entropy production. After discussing some general aspects of entropy production we turn to concrete examples, in particular billiards with and without external forces and discuss large deviations theorems for such systems.

### 2010/01/05

#### Tuesday Seminar on Topology

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

The volume growth of hyperkaehler manifolds of type $A_{\\infty}$

**服部 広大**(東京大学大学院数理科学研究科) 16:30-17:30The volume growth of hyperkaehler manifolds of type $A_{\\infty}$

[ Abstract ]

Hyperkaehler manifolds of type $A_{\\infty}$ were constructed due to Anderson-Kronheimer-LeBrun and Goto. These manifolds are 4-demensional, noncompact and their homology groups are infinitely generated. We focus on the volume growth of these hyperkaehler metrics. Here, the volume growth is asymptotic behavior of the volume of a ball of radius $r0$ with the center fixed. There are known examples of hyperkaehler manifolds whose volume growth is $r^4$ (ALE space) or $r^3$ (Taub-NUT space). In this talk we show that there exists a hyperkaehler manifold of type $A_{\\infty}$ whose volume growth is $r^c$ for a given $3

On the Runge theorem for instantons

Hyperkaehler manifolds of type $A_{\\infty}$ were constructed due to Anderson-Kronheimer-LeBrun and Goto. These manifolds are 4-demensional, noncompact and their homology groups are infinitely generated. We focus on the volume growth of these hyperkaehler metrics. Here, the volume growth is asymptotic behavior of the volume of a ball of radius $r0$ with the center fixed. There are known examples of hyperkaehler manifolds whose volume growth is $r^4$ (ALE space) or $r^3$ (Taub-NUT space). In this talk we show that there exists a hyperkaehler manifold of type $A_{\\infty}$ whose volume growth is $r^c$ for a given $3

**松尾 信一郎**(東京大学大学院数理科学研究科) 17:30-18:30

On the Runge theorem for instantons

[ Abstract ]

A classical theorem of Runge in complex analysis asserts that a

meromorphic function on a domain in the Riemann sphere can be

approximated, over compact subsets, by rational functions, that is,

meromorphic functions on the Riemann sphere.

This theorem can be paraphrased by saying that any solution of the

Cauchy-Riemann equations on a domain in the Riemann sphere can be

approximated, over compact subsets, by global solutions.

In this talk we will present an analogous result in which the

Cauchy-Riemann equations on Riemann surfaces are replaced by the

Yang-Mills instanton equations on oriented 4-manifolds.

We will also mention that the Runge theorem for instantons can be

applied to develop Yang-Mills gauge theory on open 4-manifolds.

A classical theorem of Runge in complex analysis asserts that a

meromorphic function on a domain in the Riemann sphere can be

approximated, over compact subsets, by rational functions, that is,

meromorphic functions on the Riemann sphere.

This theorem can be paraphrased by saying that any solution of the

Cauchy-Riemann equations on a domain in the Riemann sphere can be

approximated, over compact subsets, by global solutions.

In this talk we will present an analogous result in which the

Cauchy-Riemann equations on Riemann surfaces are replaced by the

Yang-Mills instanton equations on oriented 4-manifolds.

We will also mention that the Runge theorem for instantons can be

applied to develop Yang-Mills gauge theory on open 4-manifolds.

### 2009/12/25

#### Lectures

17:00-18:00 Room #370 (Graduate School of Math. Sci. Bldg.)

A criterion for the strong solvability of the mixed Cauchy problem for the Laplace equation

**Academician T. Sh. Kalmenov**(Research Centre of Physics and Mathematics Almaty, Kazakhstan)A criterion for the strong solvability of the mixed Cauchy problem for the Laplace equation

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