## 過去の記録

過去の記録 ～09/23｜本日 09/24 | 今後の予定 09/25～

### 2006年12月13日(水)

#### 諸分野のための数学研究会

10:30-11:30 数理科学研究科棟(駒場) 056号室

Computational Methods for Geometric PDEs

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

**C. M. Elliott 氏**(University of Sussex)Computational Methods for Geometric PDEs

[ 講演概要 ]

Computational approaches to evolutionary geometric partial differential equations such as anisotropic motion by mean curvature and surface diffusion are reviewed. We consider methods based on graph, parametric , level set and phase field descriptions of the surface. We also discuss the approximation of partial differential equations which hold on the evolving surfaces. Numerical results will be presented along with some approximation results.

[ 参考URL ]Computational approaches to evolutionary geometric partial differential equations such as anisotropic motion by mean curvature and surface diffusion are reviewed. We consider methods based on graph, parametric , level set and phase field descriptions of the surface. We also discuss the approximation of partial differential equations which hold on the evolving surfaces. Numerical results will be presented along with some approximation results.

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

#### 数理ファイナンスセミナー

17:30-19:00 数理科学研究科棟(駒場) 118号室

動的なファンドプロテクションと最適化について

**関根 順 氏**(京都大)動的なファンドプロテクションと最適化について

### 2006年12月12日(火)

#### トポロジー火曜セミナー

16:30-18:00 数理科学研究科棟(駒場) 056号室

Tea: 16:00 - 16:30 コモンルーム

Thom polynomials for maps of curves with isolated singularities

(joint with S. Lando)

Tea: 16:00 - 16:30 コモンルーム

**Maxim Kazarian 氏**(Steklov Math. Institute)Thom polynomials for maps of curves with isolated singularities

(joint with S. Lando)

[ 講演概要 ]

Thom (residual) polynomials in characteristic classes are used in

the analysis of geometry of functional spaces. They serve as a

tool in description of classes Poincar\\'e dual to subvarieties of

functions of prescribed types. We give explicit universal

expressions for residual polynomials in spaces of functions on

complex curves having isolated singularities and

multisingularities, in terms of few characteristic classes. These

expressions lead to a partial explicit description of a

stratification of Hurwitz spaces.

Thom (residual) polynomials in characteristic classes are used in

the analysis of geometry of functional spaces. They serve as a

tool in description of classes Poincar\\'e dual to subvarieties of

functions of prescribed types. We give explicit universal

expressions for residual polynomials in spaces of functions on

complex curves having isolated singularities and

multisingularities, in terms of few characteristic classes. These

expressions lead to a partial explicit description of a

stratification of Hurwitz spaces.

### 2006年12月11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

Modified deficiencies of holomorphic curves and defect relation

**相原義弘 氏**(沼津高専)Modified deficiencies of holomorphic curves and defect relation

### 2006年12月08日(金)

#### 講演会

10:30-12:00 数理科学研究科棟(駒場) 056号室

Computational Methods for Surface Partial Differential Equations

**Charles M. Elliott 氏**(University of Sussex)Computational Methods for Surface Partial Differential Equations

[ 講演概要 ]

In these lectures we discuss the formulation, approximation and applications of partial differential equations on stationary and evolving surfaces. Partial differential equations on surfaces occur in many applications. For example, traditionally they arise naturally in fluid dynamics, materials science, pattern formation on biological organisms and more recently in the mathematics of images. We will derive the conservation law on evolving surfaces and formulate a number of equations.

We propose a surface finite element method (SFEM) for the numerical solution of parabolic partial differential equations on hypersurfaces $\\Gamma$ in $\\mathbb R^{n+1}$. The key idea is based on the approximation of $\\Gamma$ by a polyhedral surface $\\Gamma_h$ consisting of a union of simplices (triangles for $n=2$, intervals for $n=1$) with vertices on $\\Gamma$. A finite element space of functions is then defined by taking the continuous functions on $\\Gamma_h$ which are linear affine on each simplex of the polygonal surface. We use surface gradients to define weak forms of elliptic operators and naturally generate weak formulations of elliptic and parabolic equations on $\\Gamma$. Our finite element method is applied to weak forms of the equations. The computation of the mass and element stiffness matrices are simple and straightforward. We give an example of error bounds in the case of semi-discretization in space for a fourth order linear problem. We extend this approach to pdes on evolving surfaces. We define an Eulerian level set method for partial differential equations on surfaces. The key idea is based on formulating the partial differential equation on all level set surfaces of a prescribed function $\\Phi$ whose zero level set is $\\Gamma$. We use Eulerian surface gradients to define weak forms

of elliptic operators which naturally generate weak formulations

of Eulerian elliptic and parabolic equations. This results in a degenerate equation formulated in anisotropic Sobolev spaces based on the level set function $\\Phi$. The resulting equation is then solved in one space dimension higher but can be solved on a fixed finite element grid.

Numerical experiments are described for several linear and Nonlinear partial differential equations. In particular the power of the method is demonstrated by employing it to solve highly nonlinear second and fourth order problems such as surface Allen-Cahn and Cahn-Hilliard equations and surface level set equations for geodesic mean curvature flow. In particular we show how surface level set and phase field models can be used to compute the motion of curves on surfaces. This is joint work with G. Dziuk(Freiburg).

In these lectures we discuss the formulation, approximation and applications of partial differential equations on stationary and evolving surfaces. Partial differential equations on surfaces occur in many applications. For example, traditionally they arise naturally in fluid dynamics, materials science, pattern formation on biological organisms and more recently in the mathematics of images. We will derive the conservation law on evolving surfaces and formulate a number of equations.

We propose a surface finite element method (SFEM) for the numerical solution of parabolic partial differential equations on hypersurfaces $\\Gamma$ in $\\mathbb R^{n+1}$. The key idea is based on the approximation of $\\Gamma$ by a polyhedral surface $\\Gamma_h$ consisting of a union of simplices (triangles for $n=2$, intervals for $n=1$) with vertices on $\\Gamma$. A finite element space of functions is then defined by taking the continuous functions on $\\Gamma_h$ which are linear affine on each simplex of the polygonal surface. We use surface gradients to define weak forms of elliptic operators and naturally generate weak formulations of elliptic and parabolic equations on $\\Gamma$. Our finite element method is applied to weak forms of the equations. The computation of the mass and element stiffness matrices are simple and straightforward. We give an example of error bounds in the case of semi-discretization in space for a fourth order linear problem. We extend this approach to pdes on evolving surfaces. We define an Eulerian level set method for partial differential equations on surfaces. The key idea is based on formulating the partial differential equation on all level set surfaces of a prescribed function $\\Phi$ whose zero level set is $\\Gamma$. We use Eulerian surface gradients to define weak forms

of elliptic operators which naturally generate weak formulations

of Eulerian elliptic and parabolic equations. This results in a degenerate equation formulated in anisotropic Sobolev spaces based on the level set function $\\Phi$. The resulting equation is then solved in one space dimension higher but can be solved on a fixed finite element grid.

Numerical experiments are described for several linear and Nonlinear partial differential equations. In particular the power of the method is demonstrated by employing it to solve highly nonlinear second and fourth order problems such as surface Allen-Cahn and Cahn-Hilliard equations and surface level set equations for geodesic mean curvature flow. In particular we show how surface level set and phase field models can be used to compute the motion of curves on surfaces. This is joint work with G. Dziuk(Freiburg).

#### 代数幾何学セミナー

15:00-16:25 数理科学研究科棟(駒場) 126号室

Universität zu Köln

)

Rationally connected

foliations

**Stefan Kebekus 氏 氏**(Mathematisches InstitutUniversität zu Köln

)

Rationally connected

foliations

### 2006年12月07日(木)

#### 講演会

13:00-14:30 数理科学研究科棟(駒場) 056号室

大学院イニシアティブ特別講演会(平成18年度-冬)

組織委員:儀我美一(東京大学大学院数理科学研究科)連絡先:丸菱美佳 (labgiga@ms.u-tokyo.ac.jp)

Computational Methods for Surface Partial Differential Equations

https://www.u-tokyo.ac.jp/campusmap/map02_02_j.html

大学院イニシアティブ特別講演会(平成18年度-冬)

組織委員:儀我美一(東京大学大学院数理科学研究科)連絡先:丸菱美佳 (labgiga@ms.u-tokyo.ac.jp)

**Charles M. Elliott 氏**(University of Sussex)Computational Methods for Surface Partial Differential Equations

[ 講演概要 ]

In these lectures we discuss the formulation, approximation and applications of partial differential equations on stationary and evolving surfaces. Partial differential equations on surfaces occur in many applications. For example, traditionally they arise naturally in fluid dynamics, materials science, pattern formation on biological organisms and more recently in the mathematics of images. We will derive the conservation law on evolving surfaces and formulate a number of equations.

We propose a surface finite element method (SFEM) for the numerical solution of parabolic partial differential equations on hypersurfaces $\\Gamma$ in $\\mathbb R^{n+1}$. The key idea is based on the approximation of $\\Gamma$ by a polyhedral surface $\\Gamma_h$ consisting of a union of simplices (triangles for $n=2$, intervals for $n=1$) with vertices on $\\Gamma$. A finite element space of functions is then defined by taking the continuous functions on $\\Gamma_h$ which are linear affine on each simplex of the polygonal surface. We use surface gradients to define weak forms of elliptic operators and naturally generate weak formulations of elliptic and parabolic equations on $\\Gamma$. Our finite element method is applied to weak forms of the equations. The computation of the mass and element stiffness matrices are simple and straightforward. We give an example of error bounds in the case of semi-discretization in space for a fourth order linear problem. We extend this approach to pdes on evolving surfaces. We define an Eulerian level set method for partial differential equations on surfaces. The key idea is based on formulating the partial differential equation on all level set surfaces of a prescribed function $\\Phi$ whose zero level set is $\\Gamma$. We use Eulerian surface gradients to define weak forms

of elliptic operators which naturally generate weak formulations

of Eulerian elliptic and parabolic equations. This results in a degenerate equation formulated in anisotropic Sobolev spaces based on the level set function $\\Phi$. The resulting equation is then solved in one space dimension higher but can be solved on a fixed finite element grid.

Numerical experiments are described for several linear and Nonlinear partial differential equations. In particular the power of the method is demonstrated by employing it to solve highly nonlinear second and fourth order problems such as surface Allen-Cahn and Cahn-Hilliard equations and surface level set equations for geodesic mean curvature flow. In particular we show how surface level set and phase field models can be used to compute the motion of curves on surfaces. This is joint work with G. Dziuk(Freiburg).

[ 参考URL ]In these lectures we discuss the formulation, approximation and applications of partial differential equations on stationary and evolving surfaces. Partial differential equations on surfaces occur in many applications. For example, traditionally they arise naturally in fluid dynamics, materials science, pattern formation on biological organisms and more recently in the mathematics of images. We will derive the conservation law on evolving surfaces and formulate a number of equations.

We propose a surface finite element method (SFEM) for the numerical solution of parabolic partial differential equations on hypersurfaces $\\Gamma$ in $\\mathbb R^{n+1}$. The key idea is based on the approximation of $\\Gamma$ by a polyhedral surface $\\Gamma_h$ consisting of a union of simplices (triangles for $n=2$, intervals for $n=1$) with vertices on $\\Gamma$. A finite element space of functions is then defined by taking the continuous functions on $\\Gamma_h$ which are linear affine on each simplex of the polygonal surface. We use surface gradients to define weak forms of elliptic operators and naturally generate weak formulations of elliptic and parabolic equations on $\\Gamma$. Our finite element method is applied to weak forms of the equations. The computation of the mass and element stiffness matrices are simple and straightforward. We give an example of error bounds in the case of semi-discretization in space for a fourth order linear problem. We extend this approach to pdes on evolving surfaces. We define an Eulerian level set method for partial differential equations on surfaces. The key idea is based on formulating the partial differential equation on all level set surfaces of a prescribed function $\\Phi$ whose zero level set is $\\Gamma$. We use Eulerian surface gradients to define weak forms

of elliptic operators which naturally generate weak formulations

of Eulerian elliptic and parabolic equations. This results in a degenerate equation formulated in anisotropic Sobolev spaces based on the level set function $\\Phi$. The resulting equation is then solved in one space dimension higher but can be solved on a fixed finite element grid.

Numerical experiments are described for several linear and Nonlinear partial differential equations. In particular the power of the method is demonstrated by employing it to solve highly nonlinear second and fourth order problems such as surface Allen-Cahn and Cahn-Hilliard equations and surface level set equations for geodesic mean curvature flow. In particular we show how surface level set and phase field models can be used to compute the motion of curves on surfaces. This is joint work with G. Dziuk(Freiburg).

https://www.u-tokyo.ac.jp/campusmap/map02_02_j.html

#### 作用素環セミナー

16:30-18:00 数理科学研究科棟(駒場) 126号室

An introduction to analytic endomotives (after Connes-Consani-Marcolli)

**山下真 氏**(東大数理)An introduction to analytic endomotives (after Connes-Consani-Marcolli)

### 2006年12月06日(水)

#### 諸分野のための数学研究会

10:30-11:30 数理科学研究科棟(駒場) 056号室

Formation of rims surrounding a chondrule during solidification in 3- dimensions using the phase field model

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

**横山悦郎 氏**(学習院大学)Formation of rims surrounding a chondrule during solidification in 3- dimensions using the phase field model

[ 講演概要 ]

Chondrules are small particles of silicate material of the order of a few millimeters in radius, and are the main component of chondritic meteorite.

In this paper, we present a model of the growth starting from a seed crystal at the location of an outer part of pure melt droplet into spherical single crystal corresponding to a chondrule. The formation of rims surrounding a chondrule during solidification is simulated by using the phase field model in three dimensions. Our results display a well developed rim structure when we choose the initial temperature of a melt droplet more than the melting point under the condition of larger supercooling. Furthermore, we show that the size of a droplet plays an important role in the formation of rims during solidification.

[ 参考URL ]Chondrules are small particles of silicate material of the order of a few millimeters in radius, and are the main component of chondritic meteorite.

In this paper, we present a model of the growth starting from a seed crystal at the location of an outer part of pure melt droplet into spherical single crystal corresponding to a chondrule. The formation of rims surrounding a chondrule during solidification is simulated by using the phase field model in three dimensions. Our results display a well developed rim structure when we choose the initial temperature of a melt droplet more than the melting point under the condition of larger supercooling. Furthermore, we show that the size of a droplet plays an important role in the formation of rims during solidification.

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

#### 代数学コロキウム

16:30-18:45 数理科学研究科棟(駒場) 117号室

2講演です

New applications of the arithmetic Riemann-Roch theorem

Zariski Closures of Automorphic Galois Representations

2講演です

**Vincent Maillot 氏**(Jussieu/京大数理研) 16:30-17:30New applications of the arithmetic Riemann-Roch theorem

**Don Blasius 氏**(UCLA) 17:45-18:45Zariski Closures of Automorphic Galois Representations

#### 統計数学セミナー

15:00-16:10 数理科学研究科棟(駒場) 128号室

Inference problems for the standard and geometric telegraph process

https://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2006/16.html

**Stefano IACUS 氏**(Department of Economics Business and Statistics, University of Milan, Italy)Inference problems for the standard and geometric telegraph process

[ 講演概要 ]

The telegraph process {X(t), t>0}, has been introduced (see Goldstein, 1951) as an alternative model to the Brownian motion B(t). This process describes a motion of a particle on the real line which alternates its velocity, at Poissonian times, from +v to -v. The density of the distribution of the position of the particle at time t solves the hyperbolic differential equation called telegraph equation and hence the name of the process. Contrary to B(t) the process X(t) has finite variation and continuous and differentiable paths. At the same time it is mathematically challenging to handle.

In this talk we will discuss inference problems for the estimation of the intensity of the Poisson process, either homogeneous and non homogeneous, from continuous and discrete time observations of X(t). We further discuss estimation problems for the geometric telegraph process S(t) = S(0) * exp{m - 0.5 * s^2) * t + s X(t)} where m is a known constant and s>0 and the intensity of the underlying Poisson process are two parameter of interest to be estimated. The geometric telegraph process has been recently introduced in Mathematical Finance to describe the dynamics of assets as an alternative to the usual geometric Brownian motion.

For discrete time observations we consider the "high frequency" approach, which means that data are collected at n+1 equidistant time points Ti=i * Dn, i=0,1,..., n, n*Dn = T, T fixed and such that Dn shrinks to 0 as n increases.

The process X(t) in non Markovian, non stationary and not ergodic thus we use approximation arguments to derive estimators. Given the complexity of the equations involved only estimators on the standard telegraph process can be studied analytically. We will also present a Monte Carlo study on the performance of the estimators for small sample size, i.e. Dn not shrinking to 0.

[ 参考URL ]The telegraph process {X(t), t>0}, has been introduced (see Goldstein, 1951) as an alternative model to the Brownian motion B(t). This process describes a motion of a particle on the real line which alternates its velocity, at Poissonian times, from +v to -v. The density of the distribution of the position of the particle at time t solves the hyperbolic differential equation called telegraph equation and hence the name of the process. Contrary to B(t) the process X(t) has finite variation and continuous and differentiable paths. At the same time it is mathematically challenging to handle.

In this talk we will discuss inference problems for the estimation of the intensity of the Poisson process, either homogeneous and non homogeneous, from continuous and discrete time observations of X(t). We further discuss estimation problems for the geometric telegraph process S(t) = S(0) * exp{m - 0.5 * s^2) * t + s X(t)} where m is a known constant and s>0 and the intensity of the underlying Poisson process are two parameter of interest to be estimated. The geometric telegraph process has been recently introduced in Mathematical Finance to describe the dynamics of assets as an alternative to the usual geometric Brownian motion.

For discrete time observations we consider the "high frequency" approach, which means that data are collected at n+1 equidistant time points Ti=i * Dn, i=0,1,..., n, n*Dn = T, T fixed and such that Dn shrinks to 0 as n increases.

The process X(t) in non Markovian, non stationary and not ergodic thus we use approximation arguments to derive estimators. Given the complexity of the equations involved only estimators on the standard telegraph process can be studied analytically. We will also present a Monte Carlo study on the performance of the estimators for small sample size, i.e. Dn not shrinking to 0.

https://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2006/16.html

### 2006年12月04日(月)

#### 代数幾何学セミナー

16:30-18:00 数理科学研究科棟(駒場) 126号室

When does a curve move on a surface, especially over a finite field?

**Professor Burt Totaro**

氏(University of Cambridge)氏

When does a curve move on a surface, especially over a finite field?

#### 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

Invariant CR-Laplacian type operator on the Silov boundary of a Siegel domain of rank one

**伊師英之 氏**(横浜市立大学)Invariant CR-Laplacian type operator on the Silov boundary of a Siegel domain of rank one

### 2006年12月02日(土)

#### 東京無限可積分系セミナー

13:30-14:30 数理科学研究科棟(駒場) 117号室

Spin Hall effect in metals and in insulators

**村上 修一 氏**(東大物工)Spin Hall effect in metals and in insulators

[ 講演概要 ]

We theoretically predicted that by applying an electric field

to a nonmagnetic system, a spin current is induced in a transverse

direction [1,2]. This is called a spin Hall effect. After its

theoretical predictions on semiconductors [1,2], it has been

extensively studied theoretically and experimentally, partly due

to a potential application to spintronics devices.

In particular, one of the topics of interest is quantum spin

Hall systems, which are spin analogues of the quantum Hall systems.

These systems are insulators in bulk, and have gapless edge states

which carry a spin current. These edge states are characterized

by a Z_2 topological number [3] of a bulk Hamiltonian.

If the topological number is odd, there appear gapless edge states

which carry spin current. In my talk I will briefly review the

spin Hall effect including its experimental results and present

understanding. Then I will focus on the quantum spin Hall systems,

and explain various properties of the Z_2 topological number and

its relation to edge states.

[1] S. Murakami, N. Nagaosa, and S.-C. Zhang, Science 301, 1348 (2003).

[2] J. Sinova et al., Phys. Rev. Lett. 92, 126603 (2004)

[3] C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95, 146802, 226801 (2005)

We theoretically predicted that by applying an electric field

to a nonmagnetic system, a spin current is induced in a transverse

direction [1,2]. This is called a spin Hall effect. After its

theoretical predictions on semiconductors [1,2], it has been

extensively studied theoretically and experimentally, partly due

to a potential application to spintronics devices.

In particular, one of the topics of interest is quantum spin

Hall systems, which are spin analogues of the quantum Hall systems.

These systems are insulators in bulk, and have gapless edge states

which carry a spin current. These edge states are characterized

by a Z_2 topological number [3] of a bulk Hamiltonian.

If the topological number is odd, there appear gapless edge states

which carry spin current. In my talk I will briefly review the

spin Hall effect including its experimental results and present

understanding. Then I will focus on the quantum spin Hall systems,

and explain various properties of the Z_2 topological number and

its relation to edge states.

[1] S. Murakami, N. Nagaosa, and S.-C. Zhang, Science 301, 1348 (2003).

[2] J. Sinova et al., Phys. Rev. Lett. 92, 126603 (2004)

[3] C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95, 146802, 226801 (2005)

#### 東京無限可積分系セミナー

15:00-16:00 数理科学研究科棟(駒場) 117号室

Disorder in Quantum Spin Hall Systems

**Yshai Avishai 氏**(Ben-Gurion Univ. , 東大物工)Disorder in Quantum Spin Hall Systems

[ 講演概要 ]

The quantum spin Hall phase is a novel state of matter with

topological properties. It might be realized in graphene and

probably also in type III semiconductors quantum wells.

Most recent theoretical treatments of this phase discuss its

occurrence in clean systems with perfect crystal symmetry.

In this seminar I will report on a recent work (in collaboration

with N. Nagaosa and M. Onoda) on disordered quantum spin Hall

systems. Following a brief introduction and background I will

discuss the persistence of topological terms also in disordered

systems (following a recent work of Sheng and Haldane) and

then present our results on the localization problem in two

dimensional systems. Due to spin-orbit interaction, there

is a metallic phase as is well known

for the symplectic ensemble. Together with the existence of

a topological term it leads to some surprising results regarding

the scaling theory of localization.

The quantum spin Hall phase is a novel state of matter with

topological properties. It might be realized in graphene and

probably also in type III semiconductors quantum wells.

Most recent theoretical treatments of this phase discuss its

occurrence in clean systems with perfect crystal symmetry.

In this seminar I will report on a recent work (in collaboration

with N. Nagaosa and M. Onoda) on disordered quantum spin Hall

systems. Following a brief introduction and background I will

discuss the persistence of topological terms also in disordered

systems (following a recent work of Sheng and Haldane) and

then present our results on the localization problem in two

dimensional systems. Due to spin-orbit interaction, there

is a metallic phase as is well known

for the symplectic ensemble. Together with the existence of

a topological term it leads to some surprising results regarding

the scaling theory of localization.

### 2006年12月01日(金)

#### 講演会

16:00-18:00 数理科学研究科棟(駒場) 126号室

von Neumann 環上の群作用

[ 参考URL ]

https://www.ms.u-tokyo.ac.jp/~yasuyuki/mt.htm

**竹崎正道 氏**(UCLA)von Neumann 環上の群作用

[ 参考URL ]

https://www.ms.u-tokyo.ac.jp/~yasuyuki/mt.htm

#### 談話会・数理科学講演会

16:30-17:30 数理科学研究科棟(駒場) 123号室

お茶&Coffee&お菓子: 16:00～16:30(コモンルーム)

Finite generation of the canonical ring

お茶&Coffee&お菓子: 16:00～16:30(コモンルーム)

**James McKernan 氏**(UC Santa Barbara)Finite generation of the canonical ring

[ 講演概要 ]

One of the most fundamental invariants of any smooth projective variety is the canonical ring, the graded ring of all global pluricanonical holomorphic n-forms. We explain some of the recent ideas behind the proof of finite generation of the canonical ring and its connection with the programme of Iitaka and Mori in the classification of algebraic varieties.

One of the most fundamental invariants of any smooth projective variety is the canonical ring, the graded ring of all global pluricanonical holomorphic n-forms. We explain some of the recent ideas behind the proof of finite generation of the canonical ring and its connection with the programme of Iitaka and Mori in the classification of algebraic varieties.

### 2006年11月30日(木)

#### 講演会

16:00-18:00 数理科学研究科棟(駒場) 126号室

von Neumann 環上の群作用

[ 参考URL ]

https://www.ms.u-tokyo.ac.jp/~yasuyuki/mt.htm

**竹崎正道 氏**(UCLA)von Neumann 環上の群作用

[ 参考URL ]

https://www.ms.u-tokyo.ac.jp/~yasuyuki/mt.htm

### 2006年11月29日(水)

#### 諸分野のための数学研究会

10:30-11:30 数理科学研究科棟(駒場) 056号室

Atomistic view of InAs quantum dot self-assembly from inside the growth chamber

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

**塚本 史郎 氏**(東京大学生産技術研究所)Atomistic view of InAs quantum dot self-assembly from inside the growth chamber

[ 講演概要 ]

A 'quantum dot' is a tiny region of a solid, typically just nanometres in each direction, in which electrons can be confined. Semiconductor quantum dots are the focus of intense research geared towards exploiting this property for electronic devices. The most economical method of producing quantum dots is by self-assembly, where billions of dots can be grown simultaneously. The precise mechanism of self-assembly is not understood and is hampering efforts to control the characteristics of the dots. We have used a unique microscope to directly image semiconductor quantum dots as they are growing, which is a unique scanning tunnelling microscope placed within the molecular beam epitaxy growth chamber. The images elucidate the mechanism of InAs quantum dot nucleation on GaAs(001) substrate, demonstrating directly that not all deposited In is initially incorporated into the lattice, hence providing a large supply of material to rapidly form quantum dots via islands containing tens of atoms. kinetic Monte Carlo simulations based on first-principles calculations show that alloy fluctuations in the InGaAs wetting layer prior to are crucial in determining nucleation sites.

[ 参考URL ]A 'quantum dot' is a tiny region of a solid, typically just nanometres in each direction, in which electrons can be confined. Semiconductor quantum dots are the focus of intense research geared towards exploiting this property for electronic devices. The most economical method of producing quantum dots is by self-assembly, where billions of dots can be grown simultaneously. The precise mechanism of self-assembly is not understood and is hampering efforts to control the characteristics of the dots. We have used a unique microscope to directly image semiconductor quantum dots as they are growing, which is a unique scanning tunnelling microscope placed within the molecular beam epitaxy growth chamber. The images elucidate the mechanism of InAs quantum dot nucleation on GaAs(001) substrate, demonstrating directly that not all deposited In is initially incorporated into the lattice, hence providing a large supply of material to rapidly form quantum dots via islands containing tens of atoms. kinetic Monte Carlo simulations based on first-principles calculations show that alloy fluctuations in the InGaAs wetting layer prior to are crucial in determining nucleation sites.

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

#### 講演会

16:00-18:00 数理科学研究科棟(駒場) 122号室

von Neumann 環上の群作用

[ 参考URL ]

https://www.ms.u-tokyo.ac.jp/~yasuyuki/mt.htm

**竹崎正道 氏**(UCLA)von Neumann 環上の群作用

[ 参考URL ]

https://www.ms.u-tokyo.ac.jp/~yasuyuki/mt.htm

#### 数理ファイナンスセミナー

17:30-19:00 数理科学研究科棟(駒場) 118号室

Gaussian K-Scheme について

**楠岡 成雄 氏**(東京大)Gaussian K-Scheme について

### 2006年11月28日(火)

#### 講演会

16:00-18:00 数理科学研究科棟(駒場) 122号室

von Neumann 環上の群作用

[ 参考URL ]

https://www.ms.u-tokyo.ac.jp/~yasuyuki/mt.htm

**竹崎正道 氏**(UCLA)von Neumann 環上の群作用

[ 参考URL ]

https://www.ms.u-tokyo.ac.jp/~yasuyuki/mt.htm

#### トポロジー火曜セミナー

17:00-18:00 数理科学研究科棟(駒場) 056号室

Tea: 16:40 - 17:00 コモンルーム

The Yamabe constants of infinite coverings and a positive mass theorem

Tea: 16:40 - 17:00 コモンルーム

**芥川 和雄 氏**(東京理科大学理工学部)The Yamabe constants of infinite coverings and a positive mass theorem

[ 講演概要 ]

The {\\it Yamabe constant} $Y(M, C)$ of a given closed conformal manifold

$(M, C)$ is defined by the infimum of

the normalized total-scalar-curavarure functional $E$

among all metrics in $C$.

The study of the second variation of this functional $E$ led O.Kobayashi and Schoen

to independently introduce a natural differential-topological invariant $Y(M)$,

which is obtained by taking the supremum of $Y(M, C)$ over the space of all conformal classes.

This invariant $Y(M)$ is called the {\\it Yamabe invariant} of $M$.

For the study of the Yamabe invariant,

the relationship between $Y(M, C)$ and those of its conformal coverings

is important, the case when $Y(M, C)> 0$ particularly.

When $Y(M, C) \\leq 0$, by the uniqueness of unit-volume constant scalar curvature metrics in $C$,

the desired relation is clear.

When $Y(M, C) > 0$, such a uniqueness does not hold.

However, Aubin proved that $Y(M, C)$ is strictly less than

the Yamabe constant of any of its non-trivial {\\it finite} conformal coverings,

called {\\it Aubin's Lemma}.

In this talk, we generalize this lemma to the one for the Yamabe constant of

any $(M_{\\infty}, C_{\\infty})$ of its {\\it infinite} conformal coverings,

under a certain topological condition on the relation between $\\pi_1(M)$ and $\\pi_1(M_{\\infty})$.

For the proof of this, we aslo establish a version of positive mass theorem

for a specific class of asymptotically flat manifolds with singularities.

The {\\it Yamabe constant} $Y(M, C)$ of a given closed conformal manifold

$(M, C)$ is defined by the infimum of

the normalized total-scalar-curavarure functional $E$

among all metrics in $C$.

The study of the second variation of this functional $E$ led O.Kobayashi and Schoen

to independently introduce a natural differential-topological invariant $Y(M)$,

which is obtained by taking the supremum of $Y(M, C)$ over the space of all conformal classes.

This invariant $Y(M)$ is called the {\\it Yamabe invariant} of $M$.

For the study of the Yamabe invariant,

the relationship between $Y(M, C)$ and those of its conformal coverings

is important, the case when $Y(M, C)> 0$ particularly.

When $Y(M, C) \\leq 0$, by the uniqueness of unit-volume constant scalar curvature metrics in $C$,

the desired relation is clear.

When $Y(M, C) > 0$, such a uniqueness does not hold.

However, Aubin proved that $Y(M, C)$ is strictly less than

the Yamabe constant of any of its non-trivial {\\it finite} conformal coverings,

called {\\it Aubin's Lemma}.

In this talk, we generalize this lemma to the one for the Yamabe constant of

any $(M_{\\infty}, C_{\\infty})$ of its {\\it infinite} conformal coverings,

under a certain topological condition on the relation between $\\pi_1(M)$ and $\\pi_1(M_{\\infty})$.

For the proof of this, we aslo establish a version of positive mass theorem

for a specific class of asymptotically flat manifolds with singularities.

#### 代数解析火曜セミナー

16:30-18:00 数理科学研究科棟(駒場) 052号室

非圧縮性完全流体の特異初期値問題

http://agusta.ms.u-tokyo.ac.jp/alganalysis.html

**打越 敬祐 氏**(防衛大学校)非圧縮性完全流体の特異初期値問題

[ 講演概要 ]

題材は流体力学ですが,内容的には超局所解析の考え方を駆使する問題

[ 参考URL ]題材は流体力学ですが,内容的には超局所解析の考え方を駆使する問題

http://agusta.ms.u-tokyo.ac.jp/alganalysis.html

### 2006年11月27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

Logarithmic connections along Saito free divisors

**Aleksandr G. Aleksandrov 氏**(Institute for Control Sciences, Moscow)Logarithmic connections along Saito free divisors

[ 講演概要 ]

We develop an approach to the study of meromorphic connections with logarithmic poles along a Saito free divisor. In particular, basic properties of Christoffel symbols of such connections are established. We also compute the set of all integrable meromorphic connections with logarithmic poles and describe the corresponding spaces of horizontal sections for some examples of Saito free divisors including the discriminants of the minimal versal deformations of $A_2$- and of $A_3$-singularities, and a divisor in $\mathbf{C}^3$ which appeared in a work of M. Sato in the context of the theory of prehomogeneous spaces.

We develop an approach to the study of meromorphic connections with logarithmic poles along a Saito free divisor. In particular, basic properties of Christoffel symbols of such connections are established. We also compute the set of all integrable meromorphic connections with logarithmic poles and describe the corresponding spaces of horizontal sections for some examples of Saito free divisors including the discriminants of the minimal versal deformations of $A_2$- and of $A_3$-singularities, and a divisor in $\mathbf{C}^3$ which appeared in a work of M. Sato in the context of the theory of prehomogeneous spaces.

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