## Seminar information archive

Seminar information archive ～07/23｜Today's seminar 07/24 | Future seminars 07/25～

### 2007/01/10

#### Lectures

16:00-17:30 Room #118 (Graduate School of Math. Sci. Bldg.)

Some Problems of Global Controllability of Burgers Equation and Navier-Stokes system.

**Oleg Yu. Emanouilov**(Colorado State University)Some Problems of Global Controllability of Burgers Equation and Navier-Stokes system.

### 2007/01/09

#### Lectures

16:00-17:30 Room #118 (Graduate School of Math. Sci. Bldg.)

Some Problems of Global Controllability of Burgers Equation and Navier-Stokes system.

**Oleg Yu. Emanouilov**(Colorado State University)Some Problems of Global Controllability of Burgers Equation and Navier-Stokes system.

[ Abstract ]

We show that 1-D Burgers equation is globally uncontrollable with control acting at two endpoints. Then we establish the global controllability of the 2-D Burgers equation. Finally we show that for 2-D Navier-Stokes system the problem of global exact controllability is solvable for the dense set of the initial data with a control acting on part of the boundary.

We show that 1-D Burgers equation is globally uncontrollable with control acting at two endpoints. Then we establish the global controllability of the 2-D Burgers equation. Finally we show that for 2-D Navier-Stokes system the problem of global exact controllability is solvable for the dense set of the initial data with a control acting on part of the boundary.

### 2006/12/28

#### Operator Algebra Seminars

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

Operator Algebras and Conformal Field Theory II

**Roberto Longo**(University of Rome)Operator Algebras and Conformal Field Theory II

### 2006/12/25

#### Monthly Seminar on Arithmetic of Automorphic Forms

13:30-16:00 Room #123 (Graduate School of Math. Sci. Bldg.)

なし

**研究集会の情報**(なし)なし

[ Abstract ]

秋から、少しお休みしていますので、替わりにまとめて集会をします。

12月25日午後から27日午後3時くらいまでです。詳細はURL:

https://www.ms.u-tokyo.ac.jp/activity/meeting061225.htm

をご覧下さい。織田孝幸

秋から、少しお休みしていますので、替わりにまとめて集会をします。

12月25日午後から27日午後3時くらいまでです。詳細はURL:

https://www.ms.u-tokyo.ac.jp/activity/meeting061225.htm

をご覧下さい。織田孝幸

### 2006/12/21

#### Seminar for Mathematical Past of Asia

17:00-18:30 Room #123 (Graduate School of Math. Sci. Bldg.)

インド数学における証明

[ Reference URL ]

https://www.ms.u-tokyo.ac.jp/~kawazumi/asia.html

**楠葉隆徳**(大阪経済大学人間科学部)インド数学における証明

[ Reference URL ]

https://www.ms.u-tokyo.ac.jp/~kawazumi/asia.html

#### Applied Analysis

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

An Inviscid Dyadic Model For Turbulence

**Susan Friedlander**(University of Illinois-Chicago)An Inviscid Dyadic Model For Turbulence

[ Abstract ]

We discuss properties of a GOY type model for the inviscid fluid equations. We prove that the forced system has a unique equilibrium which a an exponential global attractor. Every solution blows up in H^5/6 in finite time . After this time, all solutions stay in H^s, s<5/6, and "turbulent" dissipation occurs. Onsager's conjecture is confirmed for the model system.

This is joint work with Alexey Cheskidov and Natasa Pavlovic.

We discuss properties of a GOY type model for the inviscid fluid equations. We prove that the forced system has a unique equilibrium which a an exponential global attractor. Every solution blows up in H^5/6 in finite time . After this time, all solutions stay in H^s, s<5/6, and "turbulent" dissipation occurs. Onsager's conjecture is confirmed for the model system.

This is joint work with Alexey Cheskidov and Natasa Pavlovic.

#### Operator Algebra Seminars

14:45-18:00 Room #126 (Graduate School of Math. Sci. Bldg.)

Convergence of unitary matrix integrals and free probability

Operator Algebras and Conformal Field Theory

**Benoit Collins**(Univ. Claude Bernard Lyon 1) 14:45-16:15Convergence of unitary matrix integrals and free probability

**Roberto Longo**(University of Rome) 16:30-18:00Operator Algebras and Conformal Field Theory

### 2006/12/20

#### Number Theory Seminar

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

On the profinite regular inverse Galois problem

An elementary perspective on modular representation theory

**Anna Cadoret**(RIMS/JSPS) 16:30-17:30On the profinite regular inverse Galois problem

[ Abstract ]

Given a field $k$ and a (pro)finite group $G$, consider the

following weak version of the regular inverse Galois problem:

(WRIGP/$G$/$k$) \\textit{there exists a smooth geometrically

irreducible curve $X_{G}/k$ and a Galois extension $E/k(X_{G})$

regular over $k$ with group $G$.} (the regular inverse Galois

problem (RIGP/$G$/$k$) corresponding to the case

$X_{G}=\\mathbb{P}^{1}_{k}$). A standard descent argument shows that

for a finite group $G$ the (WRIGP/$G$/$k$) can be deduced from the

(RIGP/$G$/$k((T))$). For

profinite groups $G$, the (WRIGP/$G$/$k((T))$) has been proved for

lots of fields (including the cyclotomic closure of characteristic $0$

fields) but the descent argument no longer works.\\\\

\\indent Let $p\\geq 2$ be a prime, then a profinite group

$G$ is said to be \\textit{$p$-obstructed} if it fits in a profinite group extension

$$1\\rightarrow K\\rightarrow G\\rightarrow G_{0}\\rightarrow 1$$

with $G_{0}$ a finite group and $K\\twoheadrightarrow

\\mathbb{Z}_{p}$. Typical examples of such profinite groups $G$ are

universal $p$-Frattini covers of finite $p$-perfect groups or

pronilpotent projective groups.\\\\

\\indent I will show that the (WRIGP/$G$/$k$) - even under

its weaker formulation: (WWRIGP/$G$/$k$) \\textit{there exists a

smooth geometrically irreducible curve $X_{G}/k$ and a Galois

extension $E/k(X_{G}).\\overline{k}$ with group $G$ and field of

moduli $k$.} - fails for the whole class of $p$-obstructed profinite

groups $G$ and any field $k$ which is either a finitely generated

field of characteristic $0$ or a finite field of characteristic

$\\not= p$.\\\\

\\indent The proof uses a profinite generalization of the cohomological obstruction

for a G-cover to be defined over its field of moduli and an analysis of the constrainsts

imposed on a smooth geometrically irreducible curve $X$ by a degree $p^{n}$

cyclic G-cover $X_{n}\\rightarrow X$, constrainsts which are too rigid to allow the

existence of projective systems $(X_{n}\\rightarrow

X_{G})_{n\\geq 0}$ of degree $p^{n}$ cyclic G-covers

defined over $k$. I will also discuss other implicsations of these constrainsts

for the (RIGP).

Given a field $k$ and a (pro)finite group $G$, consider the

following weak version of the regular inverse Galois problem:

(WRIGP/$G$/$k$) \\textit{there exists a smooth geometrically

irreducible curve $X_{G}/k$ and a Galois extension $E/k(X_{G})$

regular over $k$ with group $G$.} (the regular inverse Galois

problem (RIGP/$G$/$k$) corresponding to the case

$X_{G}=\\mathbb{P}^{1}_{k}$). A standard descent argument shows that

for a finite group $G$ the (WRIGP/$G$/$k$) can be deduced from the

(RIGP/$G$/$k((T))$). For

profinite groups $G$, the (WRIGP/$G$/$k((T))$) has been proved for

lots of fields (including the cyclotomic closure of characteristic $0$

fields) but the descent argument no longer works.\\\\

\\indent Let $p\\geq 2$ be a prime, then a profinite group

$G$ is said to be \\textit{$p$-obstructed} if it fits in a profinite group extension

$$1\\rightarrow K\\rightarrow G\\rightarrow G_{0}\\rightarrow 1$$

with $G_{0}$ a finite group and $K\\twoheadrightarrow

\\mathbb{Z}_{p}$. Typical examples of such profinite groups $G$ are

universal $p$-Frattini covers of finite $p$-perfect groups or

pronilpotent projective groups.\\\\

\\indent I will show that the (WRIGP/$G$/$k$) - even under

its weaker formulation: (WWRIGP/$G$/$k$) \\textit{there exists a

smooth geometrically irreducible curve $X_{G}/k$ and a Galois

extension $E/k(X_{G}).\\overline{k}$ with group $G$ and field of

moduli $k$.} - fails for the whole class of $p$-obstructed profinite

groups $G$ and any field $k$ which is either a finitely generated

field of characteristic $0$ or a finite field of characteristic

$\\not= p$.\\\\

\\indent The proof uses a profinite generalization of the cohomological obstruction

for a G-cover to be defined over its field of moduli and an analysis of the constrainsts

imposed on a smooth geometrically irreducible curve $X$ by a degree $p^{n}$

cyclic G-cover $X_{n}\\rightarrow X$, constrainsts which are too rigid to allow the

existence of projective systems $(X_{n}\\rightarrow

X_{G})_{n\\geq 0}$ of degree $p^{n}$ cyclic G-covers

defined over $k$. I will also discuss other implicsations of these constrainsts

for the (RIGP).

**Eric Friedlander**(Northwestern) 17:45-18:45An elementary perspective on modular representation theory

### 2006/12/19

#### Tuesday Seminar on Topology

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

Poisson structures on the homology of the spaces of knots

On projections of pseudo-ribbon sphere-links

**境 圭一**(東京大学大学院数理科学研究科) 16:30-17:30Poisson structures on the homology of the spaces of knots

[ Abstract ]

We study the homological properties of the space $K$ of (framed) long knots in $\\R^n$, $n>3$, in particular its Poisson algebra structures.

We had known two kinds of Poisson structures, both of which are based on the action of little disks operad. One definition is via the action on the space $K$. Another comes from the action of chains of little disks on the Hochschild complex of an operad, which appears as $E^1$-term of certain spectral sequence converging to $H_* (K)$. The main result is that these two Poisson structures are the same.

We compute the first non-trivial example of the Poisson bracket. We show that this gives a first example of the homology class of $K$ which does not directly correspond to any chord diagrams.

We study the homological properties of the space $K$ of (framed) long knots in $\\R^n$, $n>3$, in particular its Poisson algebra structures.

We had known two kinds of Poisson structures, both of which are based on the action of little disks operad. One definition is via the action on the space $K$. Another comes from the action of chains of little disks on the Hochschild complex of an operad, which appears as $E^1$-term of certain spectral sequence converging to $H_* (K)$. The main result is that these two Poisson structures are the same.

We compute the first non-trivial example of the Poisson bracket. We show that this gives a first example of the homology class of $K$ which does not directly correspond to any chord diagrams.

**吉田 享平**(東京大学大学院数理科学研究科) 17:30-18:30On projections of pseudo-ribbon sphere-links

[ Abstract ]

Suppose $F$ is an embedded closed surface in $R^4$.

We call $F$ a pseudo-ribbon surface link

if its projection is an immersion of $F$ into $R^3$

whose self-intersection set $\\Gamma(F)$ consists of disjointly embedded circles.

H. Aiso classified pseudo-ribbon sphere-knots ($F$ is a sphere.)

when $\\Gamma(F)$ consists of less than 6 circles.

We classify pseudo-ribbon sphere-links

when $F$ is two spheres and $\\Gamma(F)$ consists of less than 7 circles.

Suppose $F$ is an embedded closed surface in $R^4$.

We call $F$ a pseudo-ribbon surface link

if its projection is an immersion of $F$ into $R^3$

whose self-intersection set $\\Gamma(F)$ consists of disjointly embedded circles.

H. Aiso classified pseudo-ribbon sphere-knots ($F$ is a sphere.)

when $\\Gamma(F)$ consists of less than 6 circles.

We classify pseudo-ribbon sphere-links

when $F$ is two spheres and $\\Gamma(F)$ consists of less than 7 circles.

### 2006/12/18

#### Seminar on Geometric Complex Analysis

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

Height functions and affine space regular automorphisms

**川口 周**(京都大学大学院理学研究科)Height functions and affine space regular automorphisms

### 2006/12/14

#### Applied Analysis

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

数学専攻)

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・大学院理学研究科・理学部数学専攻)

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。

#### 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

https://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).

https://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

https://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.

https://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?

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