## Seminar information archive

Seminar information archive ～02/23｜Today's seminar 02/24 | Future seminars 02/25～

#### Operator Algebra Seminars

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

On uniqueness of the moonshine vertex operator algebra

**Chongying Dong**(UC Santa Cruz)On uniqueness of the moonshine vertex operator algebra

#### Applied Analysis

16:00-17:30 Room #056 (Graduate School of Math. Sci. Bldg.)

特異点を持つ極小部分多様体の変分原理

**山田 澄生**

(東北大学・大学院理学研究科)特異点を持つ極小部分多様体の変分原理

[ Abstract ]

与えられた境界を持つ極小部分集合に特異点が必然的に現れることは今までによく知られている現象である.幾何学的測度論は,それらの特異点を許容する存在定理の枠組みを提供する為に発展してきた.こうして現れる部分集合の幾何学的特徴付けを,写像の持つエネルギー関数の最小化というJ.Douglas の方法論を発展させることによって試みる.また特異点周辺の面積密度の単調性公式についても言及したい.

与えられた境界を持つ極小部分集合に特異点が必然的に現れることは今までによく知られている現象である.幾何学的測度論は,それらの特異点を許容する存在定理の枠組みを提供する為に発展してきた.こうして現れる部分集合の幾何学的特徴付けを,写像の持つエネルギー関数の最小化というJ.Douglas の方法論を発展させることによって試みる.また特異点周辺の面積密度の単調性公式についても言及したい.

### 2006/12/13

#### Seminar on Mathematics for various disciplines

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

Computational Methods for Geometric PDEs

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

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

[ Abstract ]

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.

[ Reference 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

#### Mathematical Finance

17:30-19:00 Room #118 (Graduate School of Math. Sci. Bldg.)

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**(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

### 2006/12/12

#### Tuesday Seminar on Topology

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

Thom polynomials for maps of curves with isolated singularities

(joint with S. Lando)

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

(joint with S. Lando)

[ Abstract ]

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.

### 2006/12/11

#### Seminar on Geometric Complex Analysis

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

Modified deficiencies of holomorphic curves and defect relation

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

### 2006/12/08

#### Lectures

10:30-12:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Computational Methods for Surface Partial Differential Equations

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

[ Abstract ]

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).

#### Algebraic Geometry Seminar

15:00-16:25 Room #126 (Graduate School of Math. Sci. Bldg.)

Universität zu Köln

)

Rationally connected

foliations

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

)

Rationally connected

foliations

### 2006/12/07

#### Lectures

13:00-14:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Computational Methods for Surface Partial Differential Equations

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

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

[ Abstract ]

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).

[ Reference 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).

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

#### Operator Algebra Seminars

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

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

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

### 2006/12/06

#### Seminar on Mathematics for various disciplines

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

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

[ Abstract ]

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.

[ Reference 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

#### Number Theory Seminar

16:30-18:45 Room #117 (Graduate School of Math. Sci. Bldg.)

New applications of the arithmetic Riemann-Roch theorem

Zariski Closures of Automorphic Galois Representations

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

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

#### Seminar on Probability and Statistics

15:00-16:10 Room #128 (Graduate School of Math. Sci. Bldg.)

Inference problems for the standard and geometric telegraph process

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

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

[ Abstract ]

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.

[ Reference 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.

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

### 2006/12/04

#### Algebraic Geometry Seminar

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

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

**Professor Burt Totaro**

(University of Cambridge)When does a curve move on a surface, especially over a finite field?

#### Seminar on Geometric Complex Analysis

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

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

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

### 2006/12/02

#### Infinite Analysis Seminar Tokyo

13:30-14:30 Room #117 (Graduate School of Math. Sci. Bldg.)

Spin Hall effect in metals and in insulators

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

[ Abstract ]

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)

#### Infinite Analysis Seminar Tokyo

15:00-16:00 Room #117 (Graduate School of Math. Sci. Bldg.)

Disorder in Quantum Spin Hall Systems

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

[ Abstract ]

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.

### 2006/12/01

#### Lectures

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

von Neumann 環上の群作用

[ Reference URL ]

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

**竹崎正道**(UCLA)von Neumann 環上の群作用

[ Reference URL ]

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

#### Colloquium

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

Finite generation of the canonical ring

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

[ Abstract ]

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.

### 2006/11/30

#### Lectures

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

von Neumann 環上の群作用

[ Reference URL ]

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

**竹崎正道**(UCLA)von Neumann 環上の群作用

[ Reference URL ]

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

### 2006/11/29

#### Seminar on Mathematics for various disciplines

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

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

[ Abstract ]

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.

[ Reference 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

#### Lectures

16:00-18:00 Room #122 (Graduate School of Math. Sci. Bldg.)

von Neumann 環上の群作用

[ Reference URL ]

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

**竹崎正道**(UCLA)von Neumann 環上の群作用

[ Reference URL ]

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

#### Mathematical Finance

17:30-19:00 Room #118 (Graduate School of Math. Sci. Bldg.)

Gaussian K-Scheme について

**楠岡 成雄**(東京大)Gaussian K-Scheme について

### 2006/11/28

#### Lectures

16:00-18:00 Room #122 (Graduate School of Math. Sci. Bldg.)

von Neumann 環上の群作用

[ Reference URL ]

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

**竹崎正道**(UCLA)von Neumann 環上の群作用

[ Reference URL ]

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

#### Tuesday Seminar on Topology

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

The Yamabe constants of infinite coverings and a positive mass theorem

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

[ Abstract ]

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.

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