## Seminar information archive

Seminar information archive ～02/21｜Today's seminar 02/22 | Future seminars 02/23～

### 2009/12/02

#### PDE Real Analysis Seminar

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

A hyperbolic fluid model based on Cattaneo's law

**Juergen Saal**(University of Konstanz)A hyperbolic fluid model based on Cattaneo's law

[ Abstract ]

In various applications a delay of the propagation speed of a fluid (temperature, ...) has been observed. Such phenomena cannot be described by standard parabolic models, whose derivation relies on Fourier's law (Paradoxon of infinite propagation speed).

One way to give account to these observations and which was successfully applied to several models, is to replace Fourier's law by the law of Cattaneo. In the case of a fluid, this leads to a hyperbolicly perturbed quasilinear Navier-Stokes system for which existence and uniqueness results will be presented.

In various applications a delay of the propagation speed of a fluid (temperature, ...) has been observed. Such phenomena cannot be described by standard parabolic models, whose derivation relies on Fourier's law (Paradoxon of infinite propagation speed).

One way to give account to these observations and which was successfully applied to several models, is to replace Fourier's law by the law of Cattaneo. In the case of a fluid, this leads to a hyperbolicly perturbed quasilinear Navier-Stokes system for which existence and uniqueness results will be presented.

### 2009/12/01

#### Tuesday Seminar on Topology

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

Non-Abelian Novikov homology

**Andrei Pajitnov**(Univ. de Nantes)Non-Abelian Novikov homology

[ Abstract ]

Classical construction of S.P. Novikov

associates to each circle-valued Morse map

a chain complex defined over a ring

of Laurent power series in one variable.

In this survey talk we shall explain several

results related to the construction and

properties of non-Abelian generalizations of the

Novikov complex.

Classical construction of S.P. Novikov

associates to each circle-valued Morse map

a chain complex defined over a ring

of Laurent power series in one variable.

In this survey talk we shall explain several

results related to the construction and

properties of non-Abelian generalizations of the

Novikov complex.

### 2009/11/30

#### Seminar on Geometric Complex Analysis

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

Equidistribution and Nevanlinna theory

**奥山裕介**(京都工芸繊維大学)Equidistribution and Nevanlinna theory

#### Kavli IPMU Komaba Seminar

16:30-18:00 Room #002 (Graduate School of Math. Sci. Bldg.)

Chiral Algebras of (0,2) Models: Beyond Perturbation Theory

**Junya Yagi**(Rutgers University)Chiral Algebras of (0,2) Models: Beyond Perturbation Theory

[ Abstract ]

The chiral algebras of two-dimensional sigma models with (0,2)

supersymmetry are infinite-dimensional generalizations of the chiral

rings of (2,2) models. Perturbatively, they enjoy rich mathematical

structures described by sheaves of chiral differential operators.

Nonperturbatively, however, they vanish completely for certain (0,2)

models with no left-moving fermions. In this talk, I will explain how

this vanishing phenomenon takes places. The vanishing of the chiral

algebra of a (0, 2) model implies that supersymmetry is spontaneously

broken in the model, which in turn suggests that no harmonic spinors

exist on the loop space of the target space. In particular, the

elliptic genus of the model vanishes, thereby providing a physics

proof of a special case of the Hoelhn-Stolz conjecture.

The chiral algebras of two-dimensional sigma models with (0,2)

supersymmetry are infinite-dimensional generalizations of the chiral

rings of (2,2) models. Perturbatively, they enjoy rich mathematical

structures described by sheaves of chiral differential operators.

Nonperturbatively, however, they vanish completely for certain (0,2)

models with no left-moving fermions. In this talk, I will explain how

this vanishing phenomenon takes places. The vanishing of the chiral

algebra of a (0, 2) model implies that supersymmetry is spontaneously

broken in the model, which in turn suggests that no harmonic spinors

exist on the loop space of the target space. In particular, the

elliptic genus of the model vanishes, thereby providing a physics

proof of a special case of the Hoelhn-Stolz conjecture.

### 2009/11/27

#### Lecture Series on Mathematical Sciences in Soceity

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

金融リスク管理と数理(実践)

**池森俊文氏(高野 康 氏)**(みずほ第一フィナンシャルテクノロジー(株))金融リスク管理と数理(実践)

#### Seminar on Probability and Statistics

13:40-14:50 Room #128 (Graduate School of Math. Sci. Bldg.)

非線形時系列モデルのイノベーション密度の推定

http://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2009/09.html

**加藤 賢悟**(広島大学大学院理学研究科数学専攻)非線形時系列モデルのイノベーション密度の推定

[ Abstract ]

In this talk, we consider the problem of estimating the innovation density in nonlinear autoregressive models. Specifically, we establish the convergence rate of the supremum distance between the residual-based kernel density estimator and the kernel density estimator using the unobservable actual innovation variables. The proof of the main theorem relies on empirical process theory instead of the conventional Taylor expansion approach. As applications, we obtain the exact rate of weak uniform consistency on the whole line, pointwise asymptotic normality of the residual-based kernel density estimator and the asymptotic distribution of a Bickel-Rosenblatt type global measure statistic related to it. We also examine the conditions of the main theorem for some specic time series model.

[ Reference URL ]In this talk, we consider the problem of estimating the innovation density in nonlinear autoregressive models. Specifically, we establish the convergence rate of the supremum distance between the residual-based kernel density estimator and the kernel density estimator using the unobservable actual innovation variables. The proof of the main theorem relies on empirical process theory instead of the conventional Taylor expansion approach. As applications, we obtain the exact rate of weak uniform consistency on the whole line, pointwise asymptotic normality of the residual-based kernel density estimator and the asymptotic distribution of a Bickel-Rosenblatt type global measure statistic related to it. We also examine the conditions of the main theorem for some specic time series model.

http://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2009/09.html

### 2009/11/26

#### Applied Analysis

16:00-17:30 Room #002 (Graduate School of Math. Sci. Bldg.)

L^p 粘性解の弱ハルナック不等式の最近の進展

**小池 茂昭**(埼玉大学・理学部数学科)L^p 粘性解の弱ハルナック不等式の最近の進展

[ Abstract ]

Caffarelli による粘性解の regularity 研究 (1989 年) を基に, 1996 年に Caffarelli- Crandall-Kocan-Swiech によって L^p 粘性解の概念が導入された. L^p 粘性解とは, 通 常の粘性解理論では扱えなかった, 非有界非斉次項を持つ (非発散型) 偏微分方程 式にも適用可能な弱解である.

しかしながら, 係数に関しては有界係数しか研究されていなかった. その後, Swiech との共同研究により, 係数が非有界だが適当なべき乗可積分性を仮定して Aleksandrov-Bakelman-Pucci 型の最大値原理を導くことが可能になった.

本講演では, 非有界係数・非斉事項を持った, 完全非線形 2 階一様楕円型方程式 の L^p 粘性解の弱ハルナック不等式に関する最近のSwiech との共同研究の結果を紹 介する.

Caffarelli による粘性解の regularity 研究 (1989 年) を基に, 1996 年に Caffarelli- Crandall-Kocan-Swiech によって L^p 粘性解の概念が導入された. L^p 粘性解とは, 通 常の粘性解理論では扱えなかった, 非有界非斉次項を持つ (非発散型) 偏微分方程 式にも適用可能な弱解である.

しかしながら, 係数に関しては有界係数しか研究されていなかった. その後, Swiech との共同研究により, 係数が非有界だが適当なべき乗可積分性を仮定して Aleksandrov-Bakelman-Pucci 型の最大値原理を導くことが可能になった.

本講演では, 非有界係数・非斉事項を持った, 完全非線形 2 階一様楕円型方程式 の L^p 粘性解の弱ハルナック不等式に関する最近のSwiech との共同研究の結果を紹 介する.

### 2009/11/25

#### PDE Real Analysis Seminar

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

Recent results on weak and strong solutions of the Navier-Stokes equations

**Hermann Sohr**(University Paderborn)Recent results on weak and strong solutions of the Navier-Stokes equations

[ Abstract ]

Our purpose is to develop the optimal initial value condition for the existence of a unique local strong solution of the Navier-Stokes equations in a smooth bounded domain.

This condition is not only sufficient

- there are several well-known sufficient conditions in this context

- but also necessary, and yields therefore the largest possible class of such strong solutions.

As an application we obtain several extensions of Serrin's regularity condition. A restricted result also holds for completely general domains. Furthermore we extend the well-known class of Leray-Hopf weak solutions with zero boundary conditions and zero divergence to a larger class with corresponding nonzero conditions.

Our purpose is to develop the optimal initial value condition for the existence of a unique local strong solution of the Navier-Stokes equations in a smooth bounded domain.

This condition is not only sufficient

- there are several well-known sufficient conditions in this context

- but also necessary, and yields therefore the largest possible class of such strong solutions.

As an application we obtain several extensions of Serrin's regularity condition. A restricted result also holds for completely general domains. Furthermore we extend the well-known class of Leray-Hopf weak solutions with zero boundary conditions and zero divergence to a larger class with corresponding nonzero conditions.

### 2009/11/24

#### Tuesday Seminar of Analysis

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

Analytic Properties of Eigen Values of Daubechies Localization Operator

**吉野 邦生**(東京都市大学)Analytic Properties of Eigen Values of Daubechies Localization Operator

[ Abstract ]

1)ドーベシー局在化作用素の固有値の解析的性質、積分表示式、

(2)ドーベシー局在化作用素のシンボルの再現公式、

(3)ドーベシー局在化作用素のバーグマンーフォック空間での表示

等について述べる。

1)ドーベシー局在化作用素の固有値の解析的性質、積分表示式、

(2)ドーベシー局在化作用素のシンボルの再現公式、

(3)ドーベシー局在化作用素のバーグマンーフォック空間での表示

等について述べる。

#### Tuesday Seminar on Topology

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

A topological approach to left orderable groups

**Adam Clay**(University of British Columbia)A topological approach to left orderable groups

[ Abstract ]

A group G is said to be left orderable if there is a strict

total ordering of its elements such that gin G. Left orderable groups have been useful in solving many problems in topology, and now we find that topology is returning the favour: the set of all left orderings of a group is denoted by LO(G), and it admits a natural topology, under which LO(G) becomes a compact topological

space. In general, the structure of the space LO(G) is not well understood, although there are surprising results in a few special cases.

For example, the space of left orderings of the braid group B_n for n>2

contains isolated points (yet it is uncountable), while the space of left

orderings of the fundamental group of the Klein bottle is finite.

Twice in recent years, the space of left orderings has been used very

successfully to solve difficult open problems from the field of left

orderable groups, even though the connection between the topology of LO(G) and the algebraic properties of G was still unclear. I will explain the

newest understanding of this connection, and highlight some potential

applications of further advances.

A group G is said to be left orderable if there is a strict

total ordering of its elements such that g

space. In general, the structure of the space LO(G) is not well understood, although there are surprising results in a few special cases.

For example, the space of left orderings of the braid group B_n for n>2

contains isolated points (yet it is uncountable), while the space of left

orderings of the fundamental group of the Klein bottle is finite.

Twice in recent years, the space of left orderings has been used very

successfully to solve difficult open problems from the field of left

orderable groups, even though the connection between the topology of LO(G) and the algebraic properties of G was still unclear. I will explain the

newest understanding of this connection, and highlight some potential

applications of further advances.

### 2009/11/20

#### Lecture Series on Mathematical Sciences in Soceity

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

金融リスク管理と数理(概論)

**池森俊文**(みずほ第一フィナンシャルテクノロジー(株)取締役社長)金融リスク管理と数理(概論)

#### Colloquium

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

On solving fully nonlinear elliptic Partial Differential Equations

**Louis Nirenberg**

(New York University)On solving fully nonlinear elliptic Partial Differential Equations

[ Abstract ]

The talk will present some results in recent work by R.Harvey and B. Lawson: Dirichlet duality and the nonlinear Dirichlet problem, Comm. Pure Appl. Math. 62 (2009), 396-443. It concerns solving boundary value problems for elliptic equations of the form F(D'2u) = 0. They find generalized solutions which are merely continuous . The talk will be expository. No knowledge of Partial Differential Equations will be necessary.

The talk will present some results in recent work by R.Harvey and B. Lawson: Dirichlet duality and the nonlinear Dirichlet problem, Comm. Pure Appl. Math. 62 (2009), 396-443. It concerns solving boundary value problems for elliptic equations of the form F(D'2u) = 0. They find generalized solutions which are merely continuous . The talk will be expository. No knowledge of Partial Differential Equations will be necessary.

### 2009/11/19

#### Lectures

15:00-16:00 Room #123 (Graduate School of Math. Sci. Bldg.)

数論的D加群の特性サイクルと分岐理論

**阿部知行**(東京大学大学院数理科学研究科)数論的D加群の特性サイクルと分岐理論

### 2009/11/18

#### Lectures

15:30-17:00 Room #128 (Graduate School of Math. Sci. Bldg.)

The stochastic Burgers equation and its discretization

**Herbert Spohn**(ミュンヘン工科大学・九州大学)The stochastic Burgers equation and its discretization

#### Number Theory Seminar

16:30-18:45 Room #056 (Graduate School of Math. Sci. Bldg.)

Elementary computation of ramified component of the Jacobi sum

P-divisible groups and the p-adic Corona problem

**津嶋 貴弘**(東京大学大学院数理科学研究科) 16:30-17:30Elementary computation of ramified component of the Jacobi sum

[ Abstract ]

R. Coleman and W. McCallum calculated the Jacobi sum Hecke characters using their computation of the stable reduction of the Fermat curve in 1988. In my talk, we give an elementary proof of the main result of them without using rigid geometry or the stable model of the Fermat curve.

R. Coleman and W. McCallum calculated the Jacobi sum Hecke characters using their computation of the stable reduction of the Fermat curve in 1988. In my talk, we give an elementary proof of the main result of them without using rigid geometry or the stable model of the Fermat curve.

**Christopher Deninger**(Universität Münster) 17:45-18:45P-divisible groups and the p-adic Corona problem

### 2009/11/17

#### Tuesday Seminar on Topology

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

On the $SO(N)$ and $Sp(N)$ free energy of a closed oriented 3-manifold

**高田 敏恵**(新潟大学)On the $SO(N)$ and $Sp(N)$ free energy of a closed oriented 3-manifold

[ Abstract ]

We give an explicit formula of the $SO(N)$ and $Sp(N)$ free energy

of a lens space and show that the genus $g$ terms of it are analytic

in a neighborhood at zero, where we can choose the neighborhood

independently of $g$.

Moreover, it is proved that for any closed oriented 3-manifold $M$

and any $g$, the genus $g$ terms of $SO(N)$ and $Sp(N)$ free energy

of $M$ coincide up to sign.

We give an explicit formula of the $SO(N)$ and $Sp(N)$ free energy

of a lens space and show that the genus $g$ terms of it are analytic

in a neighborhood at zero, where we can choose the neighborhood

independently of $g$.

Moreover, it is proved that for any closed oriented 3-manifold $M$

and any $g$, the genus $g$ terms of $SO(N)$ and $Sp(N)$ free energy

of $M$ coincide up to sign.

### 2009/11/16

#### Seminar on Geometric Complex Analysis

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

Weighted Green functions of polynomial skew products on C^2

**上野康平**(京都大学大学院理学研究科)Weighted Green functions of polynomial skew products on C^2

#### Algebraic Geometry Seminar

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

Rationality of the Brauer-Severi Varieties of Skylanin algebras

**Colin Ingalls**(University of New Brunswick and RIMS)Rationality of the Brauer-Severi Varieties of Skylanin algebras

[ Abstract ]

Iskovskih's conjecture states that a conic bundle over

a surface is rational if and only if the surface has a pencil of

rational curves which meet the discriminant in 3 or fewer points,

(with one exceptional case). We generalize Iskovskih's proof that

such conic bundles are rational, to the case of projective space

bundles of higher dimension. The proof involves maximal orders

and toric geometry. As a corollary we show that the Brauer-Severi

variety of a Sklyanin algebra is rational.

Iskovskih's conjecture states that a conic bundle over

a surface is rational if and only if the surface has a pencil of

rational curves which meet the discriminant in 3 or fewer points,

(with one exceptional case). We generalize Iskovskih's proof that

such conic bundles are rational, to the case of projective space

bundles of higher dimension. The proof involves maximal orders

and toric geometry. As a corollary we show that the Brauer-Severi

variety of a Sklyanin algebra is rational.

### 2009/11/14

#### Monthly Seminar on Arithmetic of Automorphic Forms

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

On weak endoscopic lift (117号室)

Derivations and Automorphisms on the noncommutative algebra of power series.

**岡崎武生**(京都大学) 13:30-14:30On weak endoscopic lift (117号室)

[ Abstract ]

rank 2のsymplectic 群の保型表現$\\pi$の spinor L-関数(4次)が殆どの素点で楕円保型形式のL-関数の積になっているものをweak ndoscopic liftと呼びます. $\\pi$がtemperedならば, 全てのweak endoscopic liftはrank 4のtheta関数(theta lift)でかける事がBrooks Roberts氏により知られています.

本公演では, このtheta liftの明示的な構成法やその周辺に関する話題(Siegel 三次多様体など)についてお話したいと思います.

rank 2のsymplectic 群の保型表現$\\pi$の spinor L-関数(4次)が殆どの素点で楕円保型形式のL-関数の積になっているものをweak ndoscopic liftと呼びます. $\\pi$がtemperedならば, 全てのweak endoscopic liftはrank 4のtheta関数(theta lift)でかける事がBrooks Roberts氏により知られています.

本公演では, このtheta liftの明示的な構成法やその周辺に関する話題(Siegel 三次多様体など)についてお話したいと思います.

**井原健太郎**(POSTEC) 15:00-16:00Derivations and Automorphisms on the noncommutative algebra of power series.

[ Abstract ]

We discuss a relationship between a class of derivations and a class of automorphisms on the noncommutative algebra of formal power series in two variables. Each class relates bijectively by exponential and logarithm maps. In this talk we define a specific class of derivations, which generates a noncommutaive Lie algebra whose defining relations are related to a classical Witt algebra. The main claim is the explicit description of the

automorphisms which are corresponding to the derivations via exponential map.

We discuss a relationship between a class of derivations and a class of automorphisms on the noncommutative algebra of formal power series in two variables. Each class relates bijectively by exponential and logarithm maps. In this talk we define a specific class of derivations, which generates a noncommutaive Lie algebra whose defining relations are related to a classical Witt algebra. The main claim is the explicit description of the

automorphisms which are corresponding to the derivations via exponential map.

### 2009/11/12

#### Lectures

10:40-12:10 Room #128 (Graduate School of Math. Sci. Bldg.)

冨田竹崎理論とその応用 (6)

**竹崎正道**(UCLA)冨田竹崎理論とその応用 (6)

#### Operator Algebra Seminars

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

Recent results for amalgamated free products of type II$_1$ factors

**酒匂宏樹**(東大数理)Recent results for amalgamated free products of type II$_1$ factors

### 2009/11/11

#### Lectures

14:40-16:10 Room #128 (Graduate School of Math. Sci. Bldg.)

冨田竹崎理論とその応用 (5)

**竹崎正道**(UCLA)冨田竹崎理論とその応用 (5)

### 2009/11/10

#### Lectures

14:40-16:10 Room #128 (Graduate School of Math. Sci. Bldg.)

冨田竹崎理論とその応用 (4)

**竹崎正道**(UCLA)冨田竹崎理論とその応用 (4)

#### Tuesday Seminar on Topology

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

Resurgent analysis of the Witten Laplacian in one dimension

**Alexander Getmanenko**(IPMU)Resurgent analysis of the Witten Laplacian in one dimension

[ Abstract ]

I will recall Witten's approach to the Morse theory through properties of a certain differential operator. Then I will introduce resurgent analysis -- an asymptotic method used, in particular, for studying quantum-mechanical tunneling. In conclusion I will discuss how the methods of resurgent analysis can help us "see" pseudoholomorphic discs in the eigenfunctions of the Witten Laplacian.

I will recall Witten's approach to the Morse theory through properties of a certain differential operator. Then I will introduce resurgent analysis -- an asymptotic method used, in particular, for studying quantum-mechanical tunneling. In conclusion I will discuss how the methods of resurgent analysis can help us "see" pseudoholomorphic discs in the eigenfunctions of the Witten Laplacian.

### 2009/11/09

#### Kavli IPMU Komaba Seminar

16:30-18:00 Room #002 (Graduate School of Math. Sci. Bldg.)

Differential Graded Categories and heterotic string theory

**Makoto Sakurai**(東京大学大学院数理科学研究科)Differential Graded Categories and heterotic string theory

[ Abstract ]

The saying "category theory is an abstract nonsense" is even physically not true.

The schematic language of triangulated category presents a new stage of string theory.

To illuminate this idea, I will draw your attention to the blow-up minimal model

of complex algebraic surfaces. This is done under the hypothetical assumptions

of "generalized complex structure" of cotangent bundle due to Hitchin school.

The coordinate transformation Jacobian matrices of the measure of sigma model

with spin structures cause one part of the gravitational "anomaly cancellation"

of smooth Kahler manifold $X$ and Weyl anomaly of compact Riemann surface $\\Sigma$.

$Anom = c_1 (X) c_1 (\\Sigma) \\oplus ch_2 (X)$,

in terms of 1st and 2nd Chern characters. Note that when $\\Sigma$ is a puctured disk

with flat metric, the chiral algebra is nothing but the ordinary vertex algebra.

Note that I do not explain the complex differential geometry,

but essentially more recent works with the category of DGA (Diffenreial Graded Algebra),

which is behind the super conformal field theory of chiral algebras.

My result of "vanishing tachyon" (nil-radical part of vertex algebras)

and "causality resortation" in compactified non-critical heterotic sigma model

is physically a promising idea of new solution to unitary representation of operator algebras.

This idea is realized in the formalism of BRST cohomology and its generalization

in $\\mathcal{N} = (0,2)$ supersymmetry, that is, non-commutative geometry

with non-linear constraint condition of pure spinors for covariant quantization.

The saying "category theory is an abstract nonsense" is even physically not true.

The schematic language of triangulated category presents a new stage of string theory.

To illuminate this idea, I will draw your attention to the blow-up minimal model

of complex algebraic surfaces. This is done under the hypothetical assumptions

of "generalized complex structure" of cotangent bundle due to Hitchin school.

The coordinate transformation Jacobian matrices of the measure of sigma model

with spin structures cause one part of the gravitational "anomaly cancellation"

of smooth Kahler manifold $X$ and Weyl anomaly of compact Riemann surface $\\Sigma$.

$Anom = c_1 (X) c_1 (\\Sigma) \\oplus ch_2 (X)$,

in terms of 1st and 2nd Chern characters. Note that when $\\Sigma$ is a puctured disk

with flat metric, the chiral algebra is nothing but the ordinary vertex algebra.

Note that I do not explain the complex differential geometry,

but essentially more recent works with the category of DGA (Diffenreial Graded Algebra),

which is behind the super conformal field theory of chiral algebras.

My result of "vanishing tachyon" (nil-radical part of vertex algebras)

and "causality resortation" in compactified non-critical heterotic sigma model

is physically a promising idea of new solution to unitary representation of operator algebras.

This idea is realized in the formalism of BRST cohomology and its generalization

in $\\mathcal{N} = (0,2)$ supersymmetry, that is, non-commutative geometry

with non-linear constraint condition of pure spinors for covariant quantization.

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