## 過去の記録

過去の記録 ～06/17｜本日 06/18 | 今後の予定 06/19～

#### 複素解析幾何セミナー

10:30-12:00 数理科学研究科棟(駒場) 128号室

The Fubini-distance functions to pseudoconvex domains in $\mathbb{C}\mathbb{P}^2$ (Japanese)

**松本 和子 氏**(東京理科大学)The Fubini-distance functions to pseudoconvex domains in $\mathbb{C}\mathbb{P}^2$ (Japanese)

[ 講演概要 ]

In this talk, we would like to present two explicit formulas for the Levi forms of the Fubini-Study distance functions to complex or real hypersurfaces in $\mathbb{C}\mathbb{P}^2$. This is the first step for us to approach the non-existence conjecture of Levi-flat real hypersurfaces in $\mathbb{C}\mathbb{P}^2$. We would like to also discuss a certain important quantity found in the formulas.

In this talk, we would like to present two explicit formulas for the Levi forms of the Fubini-Study distance functions to complex or real hypersurfaces in $\mathbb{C}\mathbb{P}^2$. This is the first step for us to approach the non-existence conjecture of Levi-flat real hypersurfaces in $\mathbb{C}\mathbb{P}^2$. We would like to also discuss a certain important quantity found in the formulas.

#### 数値解析セミナー

16:30-18:00 数理科学研究科棟(駒場) 056号室

Numerical approximation of spinodal decomposition in the presence of noise (English)

**Fredrik Lindgren 氏**(大阪大学)Numerical approximation of spinodal decomposition in the presence of noise (English)

[ 講演概要 ]

Numerical approximations of stochastic partial differential equations (SPDE) has evolved to a vivid subfield of computational mathematics in the last decades. It poses new challenges both for numerical analysis and the theory of SPDE.

In this talk we will discuss the strength and weaknesses of the \emph{semigroup approach} to SPDE when it is combined with the idea of viewing a single-step method in time as a \emph{rational approximation of a semigroup}. We shall apply this framework to the stochastic Allen-Cahn equation, a parabolic semi-linear SPDE where the non-linearity is non-globally Lipschitz continuous, but has a \emph{one-sided Lipschitz condition}, and the deterministic equation has a Lyapunov functional.

We focus on semi-discretisation in time, the first step in Rothe's method, and show how the semigroup approach allows for convergence proofs under the assumption that the numerical solution admits moment bounds. However, this assumption turns out to be difficult to verify in the semi-group framework, and the rates achieved are not sharp. This is due to the fact that the one-sided Lipschitz condition, being a variational inequality, can't be utilised. We thus turn to variational methods to solve this issue.

If time admits we shall also comment on the stochastic Cahn-Hilliard equation where the non-linearity has a one-sided Lipschitz condition in a lower norm, only. However, the fact of convergence can still be proved.

This is joint work with Daisuke Furihata (Osaka University), Mih\'aly Kov\'acs (University of Otago, New Zealand), Stig Larsson (Chalmers University of Technology, Sweden) and Shuji Yoshikawa (Ehime University).

Numerical approximations of stochastic partial differential equations (SPDE) has evolved to a vivid subfield of computational mathematics in the last decades. It poses new challenges both for numerical analysis and the theory of SPDE.

In this talk we will discuss the strength and weaknesses of the \emph{semigroup approach} to SPDE when it is combined with the idea of viewing a single-step method in time as a \emph{rational approximation of a semigroup}. We shall apply this framework to the stochastic Allen-Cahn equation, a parabolic semi-linear SPDE where the non-linearity is non-globally Lipschitz continuous, but has a \emph{one-sided Lipschitz condition}, and the deterministic equation has a Lyapunov functional.

We focus on semi-discretisation in time, the first step in Rothe's method, and show how the semigroup approach allows for convergence proofs under the assumption that the numerical solution admits moment bounds. However, this assumption turns out to be difficult to verify in the semi-group framework, and the rates achieved are not sharp. This is due to the fact that the one-sided Lipschitz condition, being a variational inequality, can't be utilised. We thus turn to variational methods to solve this issue.

If time admits we shall also comment on the stochastic Cahn-Hilliard equation where the non-linearity has a one-sided Lipschitz condition in a lower norm, only. However, the fact of convergence can still be proved.

This is joint work with Daisuke Furihata (Osaka University), Mih\'aly Kov\'acs (University of Otago, New Zealand), Stig Larsson (Chalmers University of Technology, Sweden) and Shuji Yoshikawa (Ehime University).

### 2015年10月23日(金)

#### 幾何コロキウム

10:00-11:30 数理科学研究科棟(駒場) 126号室

球面束上のスカラー曲率一定計量 (Japanese)

**大鳥羽 暢彦 氏**(慶應義塾大学)球面束上のスカラー曲率一定計量 (Japanese)

[ 講演概要 ]

この講演は Jimmy Petean 氏との共同研究に基づく.

2つの Riemann 多様体の捻れ積 (より正確には, 全測地的ファイバーを持つ Riemann 沈め込みの全空間) の上で山辺 PDE を解析する我々の試みについて話す. 例として, それぞれ内積つきのベクトル束 $E$ と直線束 $L$ の Whitney 和の単位球面束 $U(E \oplus L)$ の上にスカラー曲率一定計量を構成し, この場合にはどのようにして山辺 PDE の解の個数を評価できるか説明する.

この講演は Jimmy Petean 氏との共同研究に基づく.

2つの Riemann 多様体の捻れ積 (より正確には, 全測地的ファイバーを持つ Riemann 沈め込みの全空間) の上で山辺 PDE を解析する我々の試みについて話す. 例として, それぞれ内積つきのベクトル束 $E$ と直線束 $L$ の Whitney 和の単位球面束 $U(E \oplus L)$ の上にスカラー曲率一定計量を構成し, この場合にはどのようにして山辺 PDE の解の個数を評価できるか説明する.

### 2015年10月22日(木)

#### FMSPレクチャーズ

16:00-16:50 数理科学研究科棟(駒場) 002号室

応用解析セミナーとの共催

The semiflow of a delay differential equation on its solution manifold (ENGLISH)

http://fmsp.ms.u-tokyo.ac.jp/Walther-abstract-1.pdf

応用解析セミナーとの共催

**Hans-Otto Walther 氏**(University of Giessen)The semiflow of a delay differential equation on its solution manifold (ENGLISH)

[ 講演概要 ]

The lecture surveys work after the turn of the millenium on well-posedness of initial value problems for di_erential equations with variable delay. The focus is on results which provide continuously di_erentiable solution operators, so that in case studies ingredients of dynamical systems theory, such as local invariant manifolds or Poincar_e return maps, become available. We explain why the familar theory of retarded functional di_erential equations [1,2,4] fails for equations with variable delay, discuss what has been achieved for the latter, for autonomous and for nonautonomous equations, with delays bounded or unbounded, and address open problems.

[ 参考URL ]The lecture surveys work after the turn of the millenium on well-posedness of initial value problems for di_erential equations with variable delay. The focus is on results which provide continuously di_erentiable solution operators, so that in case studies ingredients of dynamical systems theory, such as local invariant manifolds or Poincar_e return maps, become available. We explain why the familar theory of retarded functional di_erential equations [1,2,4] fails for equations with variable delay, discuss what has been achieved for the latter, for autonomous and for nonautonomous equations, with delays bounded or unbounded, and address open problems.

http://fmsp.ms.u-tokyo.ac.jp/Walther-abstract-1.pdf

#### FMSPレクチャーズ

17:00-17:50 数理科学研究科棟(駒場) 002号室

応用解析セミナーとの共催

Shilnikov chaos due to state-dependent delay, by means of the fixed point index

(ENGLISH)

http://fmsp.ms.u-tokyo.ac.jp/Walther-abstract-2.pdf

応用解析セミナーとの共催

**Hans-Otto Walther 氏**(University of Giessen)Shilnikov chaos due to state-dependent delay, by means of the fixed point index

(ENGLISH)

[ 講演概要 ]

What can variability of a delay in a delay differential equation do to the dynamics? We find a bounded delay functional d(¥phi), with d(¥phi)=1 on a neighborhood of ¥phi=0, such that the equation x'(t)=-a x(t-d(x_t)) has a solution which is homoclinic to 0, with shift dynamics in its vicinity, whereas the linear equation x'(t)=-a x(t-1) with constant time lag, for small solutions, is hyperbolic with 2-dimensional unstable space.

The proof involves regularity properties of the semiflow close to the homoclinic loop in the solution manifold and a generalization of a method due to Piotr Zgliczynsky which uses the fixed point index and a closing argument in order to establish shift dynamics when certain covering relations hold. (Joint work with Bernhard Lani-Wayda)

[ 参考URL ]What can variability of a delay in a delay differential equation do to the dynamics? We find a bounded delay functional d(¥phi), with d(¥phi)=1 on a neighborhood of ¥phi=0, such that the equation x'(t)=-a x(t-d(x_t)) has a solution which is homoclinic to 0, with shift dynamics in its vicinity, whereas the linear equation x'(t)=-a x(t-1) with constant time lag, for small solutions, is hyperbolic with 2-dimensional unstable space.

The proof involves regularity properties of the semiflow close to the homoclinic loop in the solution manifold and a generalization of a method due to Piotr Zgliczynsky which uses the fixed point index and a closing argument in order to establish shift dynamics when certain covering relations hold. (Joint work with Bernhard Lani-Wayda)

http://fmsp.ms.u-tokyo.ac.jp/Walther-abstract-2.pdf

#### 応用解析セミナー

16:00-17:50 数理科学研究科棟(駒場) 002号室

2つ講演があります．

(Part I) The semiflow of a delay differential equation on its solution manifold

(Part II) Shilnikov chaos due to state-dependent delay, by means of the fixed point index

(ENGLISH)

http://fmsp.ms.u-tokyo.ac.jp/Walther-abstract-1.pdf

2つ講演があります．

**Hans-Otto Walther 氏**(ギーセン大学)(Part I) The semiflow of a delay differential equation on its solution manifold

(Part II) Shilnikov chaos due to state-dependent delay, by means of the fixed point index

(ENGLISH)

[ 講演概要 ]

(Part I) 16:00 - 16:50

The semiflow of a delay differential equation on its solution manifold

(Part II) 17:00 - 17:50

Shilnikov chaos due to state-dependent delay, by means of the fixed point index

(Part I)

The lecture surveys recent work on initial value problems for differential equations with variable delay. The focus is on differentiable solution operators.

The lecture explains why the theory for retarded functional differential equations which is familiar from monographs before the turn of the millenium fails in case of variable delay, discusses what has been achieved in this case, for autonomous and non-autonomous equations, with delays bounded and unbounded, and addresses open problems.

[detailed abstract]

http://fmsp.ms.u-tokyo.ac.jp/Walther-abstract-1.pdf

(Part II)

What can variability of a delay in a delay differential equation do to the dynamics? We find a bounded delay functional $d(\phi)$, with $d(\phi)=1$ on a neighborhood of $\phi=0$, such that the equation $x'(t)=-a x(t-d(x_t))$ has a solution which is homoclinic to $0$, with shift dynamics in its vicinity, whereas the linear equation $x'(t)=-a x(t-1)$ with constant time lag, for small solutions, is hyperbolic with 2-dimensional unstable space.

The proof involves regularity properties of the semiflow close to the homoclinic loop in the solution manifold and a generalization of a method due to Piotr Zgliczynsky which uses the fixed point index and a closing argument in order to establish shift dynamics when certain covering relations hold. (Joint work with Bernhard Lani-Wayda)

[detailed abstract]

http://fmsp.ms.u-tokyo.ac.jp/Walther-abstract-2.pdf

[ 参考URL ](Part I) 16:00 - 16:50

The semiflow of a delay differential equation on its solution manifold

(Part II) 17:00 - 17:50

Shilnikov chaos due to state-dependent delay, by means of the fixed point index

(Part I)

The lecture surveys recent work on initial value problems for differential equations with variable delay. The focus is on differentiable solution operators.

The lecture explains why the theory for retarded functional differential equations which is familiar from monographs before the turn of the millenium fails in case of variable delay, discusses what has been achieved in this case, for autonomous and non-autonomous equations, with delays bounded and unbounded, and addresses open problems.

[detailed abstract]

http://fmsp.ms.u-tokyo.ac.jp/Walther-abstract-1.pdf

(Part II)

What can variability of a delay in a delay differential equation do to the dynamics? We find a bounded delay functional $d(\phi)$, with $d(\phi)=1$ on a neighborhood of $\phi=0$, such that the equation $x'(t)=-a x(t-d(x_t))$ has a solution which is homoclinic to $0$, with shift dynamics in its vicinity, whereas the linear equation $x'(t)=-a x(t-1)$ with constant time lag, for small solutions, is hyperbolic with 2-dimensional unstable space.

The proof involves regularity properties of the semiflow close to the homoclinic loop in the solution manifold and a generalization of a method due to Piotr Zgliczynsky which uses the fixed point index and a closing argument in order to establish shift dynamics when certain covering relations hold. (Joint work with Bernhard Lani-Wayda)

[detailed abstract]

http://fmsp.ms.u-tokyo.ac.jp/Walther-abstract-2.pdf

http://fmsp.ms.u-tokyo.ac.jp/Walther-abstract-1.pdf

### 2015年10月21日(水)

#### 数理人口学・数理生物学セミナー

14:55-16:40 数理科学研究科棟(駒場) 128演習室号室

The distribution of the duration of immunity determines the periodicity of Mycoplasma pneumoniae incidence. (JAPANESE)

http://researchers.general.hokudai.ac.jp/profile/ja.e3OkdvtshzEabOVZ2w5OYw==.html

**大森亮介 氏**(北海道大学人獣共通感染症リサーチセンター バイオインフォマティクス部門)The distribution of the duration of immunity determines the periodicity of Mycoplasma pneumoniae incidence. (JAPANESE)

[ 講演概要 ]

Estimating the periodicity of outbreaks is sometimes equivalent to the

prediction of future outbreaks. However, the periodicity may change

over time so understanding the mechanism of outbreak periodicity is

important. So far, mathematical modeling studies suggest several

drivers for outbreak periodicity including, 1) environmental factors

(e.g. temperature) and 2) host behavior (contact patterns between host

individuals). Among many diseases, multiple determinants can be

considered to cause the outbreak periodicity and it is difficult to

understand the periodicity quantitatively. Here we introduce our case

study of Mycoplasma pneumoniae (MP) which shows three to five year

periodic outbreaks, with multiple candidates for determinants for the

outbreak periodicity being narrowed down to the last one, the variance

of the length of the immunity duration. To our knowledge this is the

first study showing that the variance in the length of the immunity

duration is essential for the periodicity of the outbreaks.

[ 参考URL ]Estimating the periodicity of outbreaks is sometimes equivalent to the

prediction of future outbreaks. However, the periodicity may change

over time so understanding the mechanism of outbreak periodicity is

important. So far, mathematical modeling studies suggest several

drivers for outbreak periodicity including, 1) environmental factors

(e.g. temperature) and 2) host behavior (contact patterns between host

individuals). Among many diseases, multiple determinants can be

considered to cause the outbreak periodicity and it is difficult to

understand the periodicity quantitatively. Here we introduce our case

study of Mycoplasma pneumoniae (MP) which shows three to five year

periodic outbreaks, with multiple candidates for determinants for the

outbreak periodicity being narrowed down to the last one, the variance

of the length of the immunity duration. To our knowledge this is the

first study showing that the variance in the length of the immunity

duration is essential for the periodicity of the outbreaks.

http://researchers.general.hokudai.ac.jp/profile/ja.e3OkdvtshzEabOVZ2w5OYw==.html

### 2015年10月20日(火)

#### FMSPレクチャーズ

16:50-18:20 数理科学研究科棟(駒場) 128号室

解析学火曜セミナーと共催

Existence of an entropy solution in the sense of Young measures for a first order conservation law with a stochastic source term (ENGLISH)

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Hilhorst151020.pdf

解析学火曜セミナーと共催

**Danielle Hilhorst 氏**(CNRS / University of Paris-Sud)Existence of an entropy solution in the sense of Young measures for a first order conservation law with a stochastic source term (ENGLISH)

[ 講演概要 ]

We consider a finite volume scheme for a first order conservation law with a monotone flux function and a multiplicative source term involving a Q-Wiener process. We define a stochastic entropy solution in the sense of Young measures. We present some a priori estimates for the discrete solution including a weak BV estimate. After performing a time interpolation, we prove two entropy inequalities and show that the discrete solution converges along a subsequence to an entropy solution in the sense of Young measures.

This is joint work with T. Funaki, Y. Gao and H. Weber.

[ 参考URL ]We consider a finite volume scheme for a first order conservation law with a monotone flux function and a multiplicative source term involving a Q-Wiener process. We define a stochastic entropy solution in the sense of Young measures. We present some a priori estimates for the discrete solution including a weak BV estimate. After performing a time interpolation, we prove two entropy inequalities and show that the discrete solution converges along a subsequence to an entropy solution in the sense of Young measures.

This is joint work with T. Funaki, Y. Gao and H. Weber.

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Hilhorst151020.pdf

#### 解析学火曜セミナー

16:50-18:20 数理科学研究科棟(駒場) 128号室

FMSPレクチャーズと共催／部屋が普段と異なりますのでご注意ください

Existence of an entropy solution in the sense of Young measures for a first order conservation law with a stochastic source term (ENGLISH)

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Hilhorst151020.pdf

FMSPレクチャーズと共催／部屋が普段と異なりますのでご注意ください

**Danielle Hilhorst 氏**(CNRS / University of Paris-Sud)Existence of an entropy solution in the sense of Young measures for a first order conservation law with a stochastic source term (ENGLISH)

[ 講演概要 ]

We consider a finite volume scheme for a first order conservation law with a monotone flux function and a multiplicative source term involving a Q-Wiener process. We define a stochastic entropy solution in the sense of Young measures. We present some a priori estimates for the discrete solution including a weak BV estimate. After performing a time interpolation, we prove two entropy inequalities and show that the discrete solution converges along a subsequence to an entropy solution in the sense of Young measures.

This is joint work with T. Funaki, Y. Gao and H. Weber.

[ 参考URL ]We consider a finite volume scheme for a first order conservation law with a monotone flux function and a multiplicative source term involving a Q-Wiener process. We define a stochastic entropy solution in the sense of Young measures. We present some a priori estimates for the discrete solution including a weak BV estimate. After performing a time interpolation, we prove two entropy inequalities and show that the discrete solution converges along a subsequence to an entropy solution in the sense of Young measures.

This is joint work with T. Funaki, Y. Gao and H. Weber.

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Hilhorst151020.pdf

#### トポロジー火曜セミナー

17:30-18:30 数理科学研究科棟(駒場) 056号室

Tea : Common Room 17:00 -- 17:30

On the existence of stable compact leaves for

transversely holomorphic foliations (ENGLISH)

Tea : Common Room 17:00 -- 17:30

**Bruno Scardua 氏**(Universidade Federal do Rio de Janeiro)On the existence of stable compact leaves for

transversely holomorphic foliations (ENGLISH)

[ 講演概要 ]

One of the most important results in the theory of foliations is

the celebrated Local stability theorem of Reeb :

A compact leaf of a foliation having finite holonomy group is

stable, indeed, it admits a fundamental system of invariant

neighborhoods where each leaf is compact with finite holonomy

group. This result, together with the Global stability theorem of Reeb

(for codimension one real foliations), has many important consequences

and motivates several questions in the theory of foliations. In this talk

we show how to prove:

A transversely holomorphic foliation on a compact complex manifold, exhibits a compact stable

leaf if and only if the set of compact leaves is not a zero measure subset of the manifold.

This is a joint work with Cesar Camacho.

One of the most important results in the theory of foliations is

the celebrated Local stability theorem of Reeb :

A compact leaf of a foliation having finite holonomy group is

stable, indeed, it admits a fundamental system of invariant

neighborhoods where each leaf is compact with finite holonomy

group. This result, together with the Global stability theorem of Reeb

(for codimension one real foliations), has many important consequences

and motivates several questions in the theory of foliations. In this talk

we show how to prove:

A transversely holomorphic foliation on a compact complex manifold, exhibits a compact stable

leaf if and only if the set of compact leaves is not a zero measure subset of the manifold.

This is a joint work with Cesar Camacho.

### 2015年10月19日(月)

#### 複素解析幾何セミナー

10:30-12:00 数理科学研究科棟(駒場) 128号室

Semiample invertible sheaves with semipositive continuous hermitian metrics (Japanese)

**森脇 淳 氏**(京都大学)Semiample invertible sheaves with semipositive continuous hermitian metrics (Japanese)

[ 講演概要 ]

Let $(L,h)$ be a pair of a semi ample invertible sheaf and a semipositive continuous hermitian metric on a proper algebraic variety over $C$. In this talk, we would like to present the result that $(L, h)$ has the extension property, answering a generalization of a question of S. Zhang. Moreover, we consider its non-archimedean analogue.

Let $(L,h)$ be a pair of a semi ample invertible sheaf and a semipositive continuous hermitian metric on a proper algebraic variety over $C$. In this talk, we would like to present the result that $(L, h)$ has the extension property, answering a generalization of a question of S. Zhang. Moreover, we consider its non-archimedean analogue.

#### 東京確率論セミナー

16:50-18:20 数理科学研究科棟(駒場) 128号室

Entropy and hypo-coercive methods in hydrodynamic limits

**Stefano Olla 氏**(University of Paris-Dauphine)Entropy and hypo-coercive methods in hydrodynamic limits

[ 講演概要 ]

Relative Entropy and entropy production have been main tools

in obtaining hydrodynamic limits Entropic hypo-coercivity can be used to

extend this method to dynamics with highly degenerate noise. I will

apply it to a chain of anharmonic oscillators immersed in a temperature

gradient. Stationary states of these dynamics are of ’non equilibrium’,

and their entropy production does not allow the application of previous

techniques. These dynamics model microscopically an isothermal

thermodynamic transformation between non-equilibrium stationary states.

Ref: http://arxiv.org/abs/1505.05002

Relative Entropy and entropy production have been main tools

in obtaining hydrodynamic limits Entropic hypo-coercivity can be used to

extend this method to dynamics with highly degenerate noise. I will

apply it to a chain of anharmonic oscillators immersed in a temperature

gradient. Stationary states of these dynamics are of ’non equilibrium’,

and their entropy production does not allow the application of previous

techniques. These dynamics model microscopically an isothermal

thermodynamic transformation between non-equilibrium stationary states.

Ref: http://arxiv.org/abs/1505.05002

#### 統計数学セミナー

13:00-16:40 数理科学研究科棟(駒場) 052号室

ビッグデータブームと統計学

**水田 正弘 氏**(北海道大学 情報基盤センター)ビッグデータブームと統計学

[ 講演概要 ]

ビッグデータおよび関連する事項について扱います。

以下の内容を予定しております。

１．ビッグデータの定義と特徴

２．ビッグデータブーム・・・？

３．統計学の流れ

４．ビッグデータの解析に使える統計学

５．可視化は有効か？

６．ミニデータ

７．SDAとFDA

８．その他

ビッグデータおよび関連する事項について扱います。

以下の内容を予定しております。

１．ビッグデータの定義と特徴

２．ビッグデータブーム・・・？

３．統計学の流れ

４．ビッグデータの解析に使える統計学

５．可視化は有効か？

６．ミニデータ

７．SDAとFDA

８．その他

### 2015年10月16日(金)

#### FMSPレクチャーズ

15:00-17:30 数理科学研究科棟(駒場) 056号室

Introduction to BV quantization IV (ENGLISH)

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Reshetikhin.pdf

**Nicolai Reshetikhin 氏**(University of California, Berkeley)Introduction to BV quantization IV (ENGLISH)

[ 講演概要 ]

The lectures will focus on Batalin-Vilkovisky (BV) framework for gauge field theories. We will start with examples of gauge theories such Yang-Mills, BF-theory, Chern-Simons and others. The Hamiltonian structure for field theories will be explained on these examples. Then the classical BV-BFV (Batalin-Fradkin-Vilkovisky) setting will be introduced as a Z-graded extension of the Hamiltonian structure of field theories. The AKSZ (Aleksandrov-Kontsevich-Swartz-Zaboronskij) construction of topological field theories will be introduced. We will construct corresponding BV-BFV theory and its extension to strata of all codimensions. We will also see that Chern-Simons theory, BF theory are of the AKSZ type. The geometry of BV theories is also known as derived geometry. The classical part will be followed by an outline of what is a quantum gauge theory and what is a path integral quantization of a classical gauge theory in the BV-BFV setting. Then we will discuss BV-integrals, fibered BV integrals and perturbative quantization.

[ 参考URL ]The lectures will focus on Batalin-Vilkovisky (BV) framework for gauge field theories. We will start with examples of gauge theories such Yang-Mills, BF-theory, Chern-Simons and others. The Hamiltonian structure for field theories will be explained on these examples. Then the classical BV-BFV (Batalin-Fradkin-Vilkovisky) setting will be introduced as a Z-graded extension of the Hamiltonian structure of field theories. The AKSZ (Aleksandrov-Kontsevich-Swartz-Zaboronskij) construction of topological field theories will be introduced. We will construct corresponding BV-BFV theory and its extension to strata of all codimensions. We will also see that Chern-Simons theory, BF theory are of the AKSZ type. The geometry of BV theories is also known as derived geometry. The classical part will be followed by an outline of what is a quantum gauge theory and what is a path integral quantization of a classical gauge theory in the BV-BFV setting. Then we will discuss BV-integrals, fibered BV integrals and perturbative quantization.

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Reshetikhin.pdf

#### 幾何コロキウム

10:00-11:30 数理科学研究科棟(駒場) 126号室

射影直線上の有理型接続と箙多様体 (Japanese)

**山川大亮 氏**(東京工業大学)射影直線上の有理型接続と箙多様体 (Japanese)

[ 講演概要 ]

Crawley-Boevey によって，射影直線上のある種の対数型接続のモジュライ空間が複素シンプレクティック多様体として中島箙多様体と同型である事が示された．この講演では，Boalch によって予想され廣惠一希との共同研究によって正当化された彼の結果の一般化について紹介し，またそれと関連して有理型接続のモノドロミー保存変形のワイル群対称性についても述べる．

Crawley-Boevey によって，射影直線上のある種の対数型接続のモジュライ空間が複素シンプレクティック多様体として中島箙多様体と同型である事が示された．この講演では，Boalch によって予想され廣惠一希との共同研究によって正当化された彼の結果の一般化について紹介し，またそれと関連して有理型接続のモノドロミー保存変形のワイル群対称性についても述べる．

### 2015年10月15日(木)

#### FMSPレクチャーズ

15:00-18:00 数理科学研究科棟(駒場) 126号室

Introduction to 1-summability and resurgence (ENGLISH)

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Sauzin.pdf

**David Sauzin 氏**(CNRS - Centro di Ricerca Matematica Ennio De Giorgi Scuola Normale Superiore di Pisa)Introduction to 1-summability and resurgence (ENGLISH)

[ 講演概要 ]

The theories of summability and resurgence deal with the mathematical use of certain divergent power series. The first part of the lecure is an introduction to 1-summability. The definitions rely on the formal Borel transform and the Laplace transform along an arbitrary direction of the complex plane. Given an arc of directions, if a power series is 1-summable in that arc, then one can attach to it a Borel-Laplace sum, i.e. a holomorphic function defined in a large enough sector and asymptotic to that power series in Gevrey sense. The second part is an introduction to Ecalle's resurgence theory. A power series is said to be resurgent when its Borel transform is convergent and has good analytic continuation properties: there may be singularities but they must be isolated. The analysis of these singularities, through the so-called alien calculus, allows one to compare the various Borel-Laplace sums attached to the same resurgent 1-summable series. In the context of analytic difference-or-differential equations, this sheds light on the Stokes phenomenon. A few elementary or classical examples will be considered (the Euler series, the Stirling series, a less known example by Poincaré). Special attention must be devoted to non-linear operations: 1-summable series as well as resurgent series form algebras which are stable by composition. An example of a class of non-linear differential equations giving rise to resurgent solutions will be analyzed. The exposition requires only some familiarity with holomorphic functions of one complex variable.

[ 参考URL ]The theories of summability and resurgence deal with the mathematical use of certain divergent power series. The first part of the lecure is an introduction to 1-summability. The definitions rely on the formal Borel transform and the Laplace transform along an arbitrary direction of the complex plane. Given an arc of directions, if a power series is 1-summable in that arc, then one can attach to it a Borel-Laplace sum, i.e. a holomorphic function defined in a large enough sector and asymptotic to that power series in Gevrey sense. The second part is an introduction to Ecalle's resurgence theory. A power series is said to be resurgent when its Borel transform is convergent and has good analytic continuation properties: there may be singularities but they must be isolated. The analysis of these singularities, through the so-called alien calculus, allows one to compare the various Borel-Laplace sums attached to the same resurgent 1-summable series. In the context of analytic difference-or-differential equations, this sheds light on the Stokes phenomenon. A few elementary or classical examples will be considered (the Euler series, the Stirling series, a less known example by Poincaré). Special attention must be devoted to non-linear operations: 1-summable series as well as resurgent series form algebras which are stable by composition. An example of a class of non-linear differential equations giving rise to resurgent solutions will be analyzed. The exposition requires only some familiarity with holomorphic functions of one complex variable.

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Sauzin.pdf

#### FMSPレクチャーズ

15:00-17:30 数理科学研究科棟(駒場) 056号室

Introduction to BV quantization III (ENGLISH)

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Reshetikhin.pdf

**Nicolai Reshetikhin 氏**(University of California, Berkeley)Introduction to BV quantization III (ENGLISH)

[ 講演概要 ]

The lectures will focus on Batalin-Vilkovisky (BV) framework for gauge field theories. We will start with examples of gauge theories such Yang-Mills, BF-theory, Chern-Simons and others. The Hamiltonian structure for field theories will be explained on these examples. Then the classical BV-BFV (Batalin-Fradkin-Vilkovisky) setting will be introduced as a Z-graded extension of the Hamiltonian structure of field theories. The AKSZ (Aleksandrov-Kontsevich-Swartz-Zaboronskij) construction of topological field theories will be introduced. We will construct corresponding BV-BFV theory and its extension to strata of all codimensions. We will also see that Chern-Simons theory, BF theory are of the AKSZ type. The geometry of BV theories is also known as derived geometry. The classical part will be followed by an outline of what is a quantum gauge theory and what is a path integral quantization of a classical gauge theory in the BV-BFV setting. Then we will discuss BV-integrals, fibered BV integrals and perturbative quantization.

[ 参考URL ]The lectures will focus on Batalin-Vilkovisky (BV) framework for gauge field theories. We will start with examples of gauge theories such Yang-Mills, BF-theory, Chern-Simons and others. The Hamiltonian structure for field theories will be explained on these examples. Then the classical BV-BFV (Batalin-Fradkin-Vilkovisky) setting will be introduced as a Z-graded extension of the Hamiltonian structure of field theories. The AKSZ (Aleksandrov-Kontsevich-Swartz-Zaboronskij) construction of topological field theories will be introduced. We will construct corresponding BV-BFV theory and its extension to strata of all codimensions. We will also see that Chern-Simons theory, BF theory are of the AKSZ type. The geometry of BV theories is also known as derived geometry. The classical part will be followed by an outline of what is a quantum gauge theory and what is a path integral quantization of a classical gauge theory in the BV-BFV setting. Then we will discuss BV-integrals, fibered BV integrals and perturbative quantization.

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Reshetikhin.pdf

### 2015年10月14日(水)

#### FMSPレクチャーズ

15:00-17:30 数理科学研究科棟(駒場) 056号室

Introduction to BV quantization II (ENGLISH)

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Reshetikhin.pdf

**Nicolai Reshetikhin 氏**(University of California, Berkeley)Introduction to BV quantization II (ENGLISH)

[ 講演概要 ]

The lectures will focus on Batalin-Vilkovisky (BV) framework for gauge field theories. We will start with examples of gauge theories such Yang-Mills, BF-theory, Chern-Simons and others. The Hamiltonian structure for field theories will be explained on these examples. Then the classical BV-BFV (Batalin-Fradkin-Vilkovisky) setting will be introduced as a Z-graded extension of the Hamiltonian structure of field theories. The AKSZ (Aleksandrov-Kontsevich-Swartz-Zaboronskij) construction of topological field theories will be introduced. We will construct corresponding BV-BFV theory and its extension to strata of all codimensions. We will also see that Chern-Simons theory, BF theory are of the AKSZ type. The geometry of BV theories is also known as derived geometry. The classical part will be followed by an outline of what is a quantum gauge theory and what is a path integral quantization of a classical gauge theory in the BV-BFV setting. Then we will discuss BV-integrals, fibered BV integrals and perturbative quantization.

[ 参考URL ]The lectures will focus on Batalin-Vilkovisky (BV) framework for gauge field theories. We will start with examples of gauge theories such Yang-Mills, BF-theory, Chern-Simons and others. The Hamiltonian structure for field theories will be explained on these examples. Then the classical BV-BFV (Batalin-Fradkin-Vilkovisky) setting will be introduced as a Z-graded extension of the Hamiltonian structure of field theories. The AKSZ (Aleksandrov-Kontsevich-Swartz-Zaboronskij) construction of topological field theories will be introduced. We will construct corresponding BV-BFV theory and its extension to strata of all codimensions. We will also see that Chern-Simons theory, BF theory are of the AKSZ type. The geometry of BV theories is also known as derived geometry. The classical part will be followed by an outline of what is a quantum gauge theory and what is a path integral quantization of a classical gauge theory in the BV-BFV setting. Then we will discuss BV-integrals, fibered BV integrals and perturbative quantization.

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Reshetikhin.pdf

#### 作用素環セミナー

16:45-18:15 数理科学研究科棟(駒場) 118号室

Fraisse Theory for Metric Structures (English)

**増本周平 氏**(東大数理)Fraisse Theory for Metric Structures (English)

### 2015年10月13日(火)

#### FMSPレクチャーズ

15:00-17:30 数理科学研究科棟(駒場) 056号室

Introduction to BV quantization I (ENGLISH)

[ 講演概要 ]

The lectures will focus on Batalin-Vilkovisky (BV) framework for gauge field theories. We will start with examples of gauge theories such Yang-Mills, BF-theory, Chern-Simons and others. The Hamiltonian structure for field theories will be explained on these examples. Then the classical BV-BFV (Batalin-Fradkin-Vilkovisky) setting will be introduced as a Z-graded extension of the Hamiltonian structure of field theories. The AKSZ (Aleksandrov-Kontsevich-Swartz-Zaboronskij) construction of topological field theories will be introduced. We will construct corresponding BV-BFV theory and its extension to strata of all codimensions. We will also see that Chern-Simons theory, BF theory are of the AKSZ type. The geometry of BV theories is also known as derived geometry. The classical part will be followed by an outline of what is a quantum gauge theory and what is a path integral quantization of a classical gauge theory in the BV-BFV setting. Then we will discuss BV-integrals, fibered BV integrals and perturbative quantization.
[ 参考URL ]

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Reshetikhin.pdf

**Nicolai Reshetikhin 氏**(University of California, Berkeley)Introduction to BV quantization I (ENGLISH)

The lectures will focus on Batalin-Vilkovisky (BV) framework for gauge field theories. We will start with examples of gauge theories such Yang-Mills, BF-theory, Chern-Simons and others. The Hamiltonian structure for field theories will be explained on these examples. Then the classical BV-BFV (Batalin-Fradkin-Vilkovisky) setting will be introduced as a Z-graded extension of the Hamiltonian structure of field theories. The AKSZ (Aleksandrov-Kontsevich-Swartz-Zaboronskij) construction of topological field theories will be introduced. We will construct corresponding BV-BFV theory and its extension to strata of all codimensions. We will also see that Chern-Simons theory, BF theory are of the AKSZ type. The geometry of BV theories is also known as derived geometry. The classical part will be followed by an outline of what is a quantum gauge theory and what is a path integral quantization of a classical gauge theory in the BV-BFV setting. Then we will discuss BV-integrals, fibered BV integrals and perturbative quantization.

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Reshetikhin.pdf

#### FMSPレクチャーズ

15:00-16:30 数理科学研究科棟(駒場) 128号室

Implicit multiscale analysis of the macroscopic behaviour in microscopic models (ENGLISH)

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Starke.pdf

**Jens Starke 氏**(Queen Mary University of London)Implicit multiscale analysis of the macroscopic behaviour in microscopic models (ENGLISH)

[ 講演概要 ]

A numerical multiscale approach (equation-free analysis) is further improved in the framework of slow-fast dynamical systems and demonstrated for the example of a particle model for traffic flow. The method allows to perform numerical investigations of the macroscopic behavior of microscopically defined systems including continuation and bifurcation analysis on the coarse or macroscopic level where no explicit equations are available. This approach fills a gap in the analysis of many complex real-world applications including particle models with intermediate number of particles where the microscopic system is too large for a direct numerical analysis of the full system and too small to justify large-particle limits.

An implicit equation-free method is presented which reduces numerical errors of the equation-free analysis considerably. It can be shown that the implicitly defined coarse-level time stepper converges to the true dynamics on the slow manifold. The method is applied to perform a coarse bifurcation analysis of microscopic particle models describing car traffic on single lane highways. The results include an equation-free continuation of traveling wave solutions, identification of bifurcations as well as two-parameter continuations of bifurcation points. This is joint work with Christian Marschler and Jan Sieber.

[ 参考URL ]A numerical multiscale approach (equation-free analysis) is further improved in the framework of slow-fast dynamical systems and demonstrated for the example of a particle model for traffic flow. The method allows to perform numerical investigations of the macroscopic behavior of microscopically defined systems including continuation and bifurcation analysis on the coarse or macroscopic level where no explicit equations are available. This approach fills a gap in the analysis of many complex real-world applications including particle models with intermediate number of particles where the microscopic system is too large for a direct numerical analysis of the full system and too small to justify large-particle limits.

An implicit equation-free method is presented which reduces numerical errors of the equation-free analysis considerably. It can be shown that the implicitly defined coarse-level time stepper converges to the true dynamics on the slow manifold. The method is applied to perform a coarse bifurcation analysis of microscopic particle models describing car traffic on single lane highways. The results include an equation-free continuation of traveling wave solutions, identification of bifurcations as well as two-parameter continuations of bifurcation points. This is joint work with Christian Marschler and Jan Sieber.

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Starke.pdf

#### FMSPレクチャーズ

17:00-17:50 数理科学研究科棟(駒場) 002号室

応用解析セミナーとの共催

Shilnikov chaos due to state-dependent delay, by means of the fixed point index (ENGLISH)

応用解析セミナーとの共催

**Hans-Otto Walther 氏**(University of Giessen)Shilnikov chaos due to state-dependent delay, by means of the fixed point index (ENGLISH)

#### 解析学火曜セミナー

16:50-18:20 数理科学研究科棟(駒場) 126号室

Nonlinear analysis with endlessly continuable functions (joint work with Shingo Kamimoto) (English)

**David Sauzin 氏**(CNRS, France)Nonlinear analysis with endlessly continuable functions (joint work with Shingo Kamimoto) (English)

[ 講演概要 ]

We give estimates for the convolution products of an arbitrary number of endlessly continuable functions. This allows us to deal with nonlinear operations for the corresponding resurgent series, e.g. substitution into a convergent power series.

We give estimates for the convolution products of an arbitrary number of endlessly continuable functions. This allows us to deal with nonlinear operations for the corresponding resurgent series, e.g. substitution into a convergent power series.

### 2015年10月06日(火)

#### トポロジー火曜セミナー

17:00-18:30 数理科学研究科棟(駒場) 056号室

Tea : Common Room 16:30 -- 17:00

Delooping theorem in K-theory (JAPANESE)

Tea : Common Room 16:30 -- 17:00

**齋藤 翔 氏**(カブリ数物連携宇宙研究機構)Delooping theorem in K-theory (JAPANESE)

[ 講演概要 ]

There is an important special class of infinite dimensional vector spaces, formed by those called Tate vector spaces. Since their first appearance in Tate’s work on residues of differentials on curves, they have been playing important roles in several different contexts including the study of formal loop spaces and semi-infinite Hodge theory. They have more sophisticated linear algebraic invariant than finite dimensional vector spaces, for instance the dimension of a Tate vector spaces is not a single integer, but a torsor acted upon by the all integers, and the determinant of an automorphism is not a single invertible scalar, but a torsor acted upon by the all invertible scalars. In this talk I will show how a delooping theorem in K-theory provides a clarified perspective on this phenomenon, using the recently developed higher categorical framework of infinity-topoi.

There is an important special class of infinite dimensional vector spaces, formed by those called Tate vector spaces. Since their first appearance in Tate’s work on residues of differentials on curves, they have been playing important roles in several different contexts including the study of formal loop spaces and semi-infinite Hodge theory. They have more sophisticated linear algebraic invariant than finite dimensional vector spaces, for instance the dimension of a Tate vector spaces is not a single integer, but a torsor acted upon by the all integers, and the determinant of an automorphism is not a single invertible scalar, but a torsor acted upon by the all invertible scalars. In this talk I will show how a delooping theorem in K-theory provides a clarified perspective on this phenomenon, using the recently developed higher categorical framework of infinity-topoi.

#### 諸分野のための数学研究会

13:30-14:30 数理科学研究科棟(駒場) 056号室

※ 通常の時間と異なります。

Multi-material phase field approach to structural topology optimization and its relation to sharp interface approach (English)

※ 通常の時間と異なります。

**Mohammad Hassan Farshbaf Shaker 氏**(Weierstrass Institute, Berlin)Multi-material phase field approach to structural topology optimization and its relation to sharp interface approach (English)

[ 講演概要 ]

A phase field approach for structural topology optimization which allows for topology changes and multiple materials is analyzed. First order optimality conditions are rigorously derived and it is shown via formally matched asymptotic expansions that these conditions converge to classical first order conditions obtained in the context of shape calculus. Finally, we present several numerical results for mean compliance problems and a cost involving the least square error to a target displacement. This is joint work with Luise Blank, Harald Garcke and Vanessa Styles.

A phase field approach for structural topology optimization which allows for topology changes and multiple materials is analyzed. First order optimality conditions are rigorously derived and it is shown via formally matched asymptotic expansions that these conditions converge to classical first order conditions obtained in the context of shape calculus. Finally, we present several numerical results for mean compliance problems and a cost involving the least square error to a target displacement. This is joint work with Luise Blank, Harald Garcke and Vanessa Styles.

< 前へ 123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960616263646566676869707172737475767778798081828384858687888990919293949596979899100101102103104105106107108109110111112113114115116117118119120121122123124125126127128129130131132133134135136137138139140141142143144145146147148149150151152153154155156157158159160161162163164165166167168169170171172173174175176177178179180181182183184185186187 次へ >