## Seminar information archive

Seminar information archive ～12/04｜Today's seminar 12/05 | Future seminars 12/06～

#### Seminar on Mathematics for various disciplines

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

Analysis of growth rates of an ice crystal from supercooled heavy water under microgravity condition in KIBO of International Space Station

--basal plane growth rate and dendritic growth velocity

(JAPANESE)

**Etsuro Yokoyama**(Gakushuin University)Analysis of growth rates of an ice crystal from supercooled heavy water under microgravity condition in KIBO of International Space Station

--basal plane growth rate and dendritic growth velocity

(JAPANESE)

### 2010/04/13

#### Tuesday Seminar on Topology

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

Torsors in non-commutative geometry (ENGLISH)

**Christian Kassel**(CNRS, Univ. de Strasbourg)Torsors in non-commutative geometry (ENGLISH)

[ Abstract ]

G-torsors or principal homogeneous spaces are familiar objects in geometry. I'll present an extension of such objects to "non-commutative geometry". When the structural group G is finite, non-commutative G-torsors are governed by a group that has both an arithmetic component and a geometric one. The arithmetic part is given by a classical Galois cohomology group; the geometric input is encoded in a (not necessarily abelian) group that takes into account all normal abelian subgroups of G of central type. Various examples will be exhibited.

G-torsors or principal homogeneous spaces are familiar objects in geometry. I'll present an extension of such objects to "non-commutative geometry". When the structural group G is finite, non-commutative G-torsors are governed by a group that has both an arithmetic component and a geometric one. The arithmetic part is given by a classical Galois cohomology group; the geometric input is encoded in a (not necessarily abelian) group that takes into account all normal abelian subgroups of G of central type. Various examples will be exhibited.

#### Tuesday Seminar of Analysis

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

Strichartz estimates and the Isozaki-Kitada parametrix

on asymptotically hyperbolic manifolds (ENGLISH)

**Jean-Marc Bouclet**(Toulouse University, France)Strichartz estimates and the Isozaki-Kitada parametrix

on asymptotically hyperbolic manifolds (ENGLISH)

### 2010/04/12

#### Seminar on Geometric Complex Analysis

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

Degeneracy of holomorphic curves into the complements of hypersurfaces in a complex projective space

[ Reference URL ]

http://info.ms.u-tokyo.ac.jp/seminar/geocomp/future.html

**千葉 優作**(東大数理)Degeneracy of holomorphic curves into the complements of hypersurfaces in a complex projective space

[ Reference URL ]

http://info.ms.u-tokyo.ac.jp/seminar/geocomp/future.html

### 2010/04/06

#### Lie Groups and Representation Theory

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

On the characters of tempered modules of affine Hecke

algebras of classical type

**加藤周**(京都大学)On the characters of tempered modules of affine Hecke

algebras of classical type

[ Abstract ]

We present an inductive algorithm to compute the characters

of tempered modules of an affine Hecke algebras of classical

types, based on a new class of representations which we call

"tempered delimits". They have some geometric origin in the

eDL correspondence.

Our new algorithm has some advantage to the Lusztig-Shoji

algorithm (which also describes the characters of tempered

modules via generalized Green functions) in the sense it

enables us to tell how the characters of tempered modules

changes as the parameters vary.

This is a joint work with Dan Ciubotaru at Utah.

We present an inductive algorithm to compute the characters

of tempered modules of an affine Hecke algebras of classical

types, based on a new class of representations which we call

"tempered delimits". They have some geometric origin in the

eDL correspondence.

Our new algorithm has some advantage to the Lusztig-Shoji

algorithm (which also describes the characters of tempered

modules via generalized Green functions) in the sense it

enables us to tell how the characters of tempered modules

changes as the parameters vary.

This is a joint work with Dan Ciubotaru at Utah.

### 2010/04/05

#### Algebraic Geometry Seminar

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

From Lang's Conjecture to finiteness properties of Torelli groups

**Alexandru Dimca**(Université Nice-Sophia Antipolis)From Lang's Conjecture to finiteness properties of Torelli groups

[ Abstract ]

First we recall one of Lang's conjectures in diophantine geometry

on the interplay between subvarieties and translated subgroups in a

commutative algebraic group

(proved by M. Laurent in the case of affine tori in 1984).

Then we present the technique of resonance and characteristic varieties,

a powerful tool in the study of fundamental groups of algebraic varieties.

Finally, using the two ingredients above, we show that the Torelli

groups $T_g$

have some surprising finiteness properties for $g>3$.

In particular, we show that for any subgroup $N$ in $T_g$ containing

the Johnson kernel $K_g$, the complex vector space $N_{ab} \\otimes C$

is finite dimensional.

All the details are available in our joint preprint with S. Papadima

arXiv:1002.0673.

First we recall one of Lang's conjectures in diophantine geometry

on the interplay between subvarieties and translated subgroups in a

commutative algebraic group

(proved by M. Laurent in the case of affine tori in 1984).

Then we present the technique of resonance and characteristic varieties,

a powerful tool in the study of fundamental groups of algebraic varieties.

Finally, using the two ingredients above, we show that the Torelli

groups $T_g$

have some surprising finiteness properties for $g>3$.

In particular, we show that for any subgroup $N$ in $T_g$ containing

the Johnson kernel $K_g$, the complex vector space $N_{ab} \\otimes C$

is finite dimensional.

All the details are available in our joint preprint with S. Papadima

arXiv:1002.0673.

### 2010/03/30

#### GCOE Seminars

10:00-15:00 Room #122 (Graduate School of Math. Sci. Bldg.)

産学連携による新たな数学の課題:非整数階拡散方程式への誘い (JAPANESE)

量子力学のスペクトル・散乱理論における数学的手法 (JAPANESE)

Semismooth Newton法の理論、及び応用 (JAPANESE)

TBA (JAPANESE)

**Masahiro Yamamoto**(University of Tokyo) 10:00-10:50産学連携による新たな数学の課題:非整数階拡散方程式への誘い (JAPANESE)

**Shu Nakamura**(University of Tokyo) 11:00-11:50量子力学のスペクトル・散乱理論における数学的手法 (JAPANESE)

**Kazufumi Ito**(University of Tokyo, North Carolina State University) 13:20-14:10Semismooth Newton法の理論、及び応用 (JAPANESE)

**Georg Weiss**(University of Tokyo) 14:10-15:00TBA (JAPANESE)

### 2010/03/29

#### Seminar on Probability and Statistics

13:00-14:10 Room #002 (Graduate School of Math. Sci. Bldg.)

Inference for partially observed Markov processes and applications

https://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2009/17.html

**Catherine Laredo**(MIA, INRA)Inference for partially observed Markov processes and applications

[ Abstract ]

We present some statistical methods for estimating the param- eters of a population dynamics model of annual plants. It is modelled using multitype branching processes with immigration. The data consist of counts in each type that are measured in several populations for a few consecu- tive years. Parametric inference is first carried out when count data of all types are observed. We prove statistical identifiability for all the parameters ruling the population dynamics model and derive consistent and asymptot- ically Gaussian estimators. However, it often occurs that, in practice, one or more types cannot be observed, leading to partially observed processes. Parametric inference is first studied in the case of Poisson distributions. We characterize the parameter subset where identifiability holds and de- rive consistent and asymptotically normal estimators for this parameter subset. Theses results are then extended to other distributions.

We apply these results to feral oilseed data. The model takes account of reproduction, immigration, and seed survival in a seed bank. The data consist of the number of plants in several developmental stages that were measured in a number of populations for few consecutive years. They are incomplete since seeds could not be counted.

[ Reference URL ]We present some statistical methods for estimating the param- eters of a population dynamics model of annual plants. It is modelled using multitype branching processes with immigration. The data consist of counts in each type that are measured in several populations for a few consecu- tive years. Parametric inference is first carried out when count data of all types are observed. We prove statistical identifiability for all the parameters ruling the population dynamics model and derive consistent and asymptot- ically Gaussian estimators. However, it often occurs that, in practice, one or more types cannot be observed, leading to partially observed processes. Parametric inference is first studied in the case of Poisson distributions. We characterize the parameter subset where identifiability holds and de- rive consistent and asymptotically normal estimators for this parameter subset. Theses results are then extended to other distributions.

We apply these results to feral oilseed data. The model takes account of reproduction, immigration, and seed survival in a seed bank. The data consist of the number of plants in several developmental stages that were measured in a number of populations for few consecutive years. They are incomplete since seeds could not be counted.

https://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2009/17.html

### 2010/03/25

#### Lectures

17:00-18:00 Room #370 (Graduate School of Math. Sci. Bldg.)

Heuristic Choice Rules for Convex Variational Regularization

**Dr Bangti Jin**(Center for Industrial Mathematics University of Bremen, Germany)Heuristic Choice Rules for Convex Variational Regularization

[ Abstract ]

In this talk we shall consider heuristic rules for choosing regularization parameters for general convex variational regularization of linear inverse problems. Several rules of recent origin are described, and some theoretical issues, e.g. existence, convergence, and a posteriori error estimates, are discussed. Numerical examples will be presented to demonstrate their accuracy and practical utility.

In this talk we shall consider heuristic rules for choosing regularization parameters for general convex variational regularization of linear inverse problems. Several rules of recent origin are described, and some theoretical issues, e.g. existence, convergence, and a posteriori error estimates, are discussed. Numerical examples will be presented to demonstrate their accuracy and practical utility.

#### Lectures

16:00-17:00 Room #370 (Graduate School of Math. Sci. Bldg.)

Modern computer architectures for tsunami simulation

**M.M. Lavrentiev, Jr.**(Sobolev Institute of Mathematics, Novosibirsk, Russia)Modern computer architectures for tsunami simulation

[ Abstract ]

Simulation of tsunami wave propagation over the deep water is one of the most time consuming tasks of the tsunami warning system. The authors utilize Method of Splitting Tsunami (MOST) package, accepted by the National Ocean & Atmospheric Administration (NOAA), USA. The software generates calculation of wave propagation at deep water by splitting along coordinate axis. Nonlinear shallow water system is used as the governing equations. Some tasks of the algorithm could be executed in parallel mode, however, direct application of multi processor systems results only in two times performance gain. After a number of optimizations, the authors achieved 16 times performance gain. OpenMP technology was applied. When utilizing Sony PlayStation3 platform (IBM CELL BE architecture) 60 times code acceleration was accomplished. The best result was achieved with modern GPU (GForce 8800 and TESLA), 100 times performance gain.

Simulation of tsunami wave propagation over the deep water is one of the most time consuming tasks of the tsunami warning system. The authors utilize Method of Splitting Tsunami (MOST) package, accepted by the National Ocean & Atmospheric Administration (NOAA), USA. The software generates calculation of wave propagation at deep water by splitting along coordinate axis. Nonlinear shallow water system is used as the governing equations. Some tasks of the algorithm could be executed in parallel mode, however, direct application of multi processor systems results only in two times performance gain. After a number of optimizations, the authors achieved 16 times performance gain. OpenMP technology was applied. When utilizing Sony PlayStation3 platform (IBM CELL BE architecture) 60 times code acceleration was accomplished. The best result was achieved with modern GPU (GForce 8800 and TESLA), 100 times performance gain.

### 2010/03/23

#### GCOE Seminars

15:00-17:15 Room #370 (Graduate School of Math. Sci. Bldg.)

Stability estimates for the anisotropic wave and Schrodinger equations from

the Dirichlet to Neumann map

On uniqueness in inverse elastic obstacle scattering

**Mourad Bellassoued**(Univ. of Bizerte) 15:00-16:00Stability estimates for the anisotropic wave and Schrodinger equations from

the Dirichlet to Neumann map

[ Abstract ]

In this talk we want to obtain stability estimates for the inverse problem of determining the electric potential or the conformal factor in a wave or Schrodinger equations in an anisotropic media with Dirichlet data from measured Neumann boundary observations. This information is enclosed in the dynamical Dirichlet-to-Neumann map associated to the wave equation. We prove in dimension $n\\geq 2$ that the knowledge of the Dirichlet to Neumann map for the wave equation measured on the boundary uniquely determines the electric potential and we prove H\\"older-type stability in determining the potential. We prove similar results for the determination of a conformal factor close to 1.

In this talk we want to obtain stability estimates for the inverse problem of determining the electric potential or the conformal factor in a wave or Schrodinger equations in an anisotropic media with Dirichlet data from measured Neumann boundary observations. This information is enclosed in the dynamical Dirichlet-to-Neumann map associated to the wave equation. We prove in dimension $n\\geq 2$ that the knowledge of the Dirichlet to Neumann map for the wave equation measured on the boundary uniquely determines the electric potential and we prove H\\"older-type stability in determining the potential. We prove similar results for the determination of a conformal factor close to 1.

**Johannes Elschner**(Weierstrass Institute Berlin, Germany) 16:15-17:15On uniqueness in inverse elastic obstacle scattering

[ Abstract ]

The talk is on joint work with M. Yamamoto on the third and fourth exterior boundary value problems of linear isotropic elasticity. We present uniqueness results for the corresponding inverse scattering problems with polyhedral-type obstacles and a finite number of incident plane elastic waves.

Our approach is essentially based on a reflection principle for the Navier equation.

The talk is on joint work with M. Yamamoto on the third and fourth exterior boundary value problems of linear isotropic elasticity. We present uniqueness results for the corresponding inverse scattering problems with polyhedral-type obstacles and a finite number of incident plane elastic waves.

Our approach is essentially based on a reflection principle for the Navier equation.

### 2010/03/19

#### Lectures

11:00-12:00 Room #366 (Graduate School of Math. Sci. Bldg.)

A Regularization Parameter for Nonsmooth Tikhonov Regularization

**竹内 知哉**(North Carolina State University, USA)A Regularization Parameter for Nonsmooth Tikhonov Regularization

[ Abstract ]

We develop a novel criterion for choosing regularization parameters for nonsmooth Tikhonov functionals. The proposed criterion is solely based on the value function, and thus applicable to a broad range of functionals. It is analytically compared with the local minimum criterion, and a posteriori error estimates are derived. An efficient numerical algorithm for computing the minimizer is developed, and its convergence properties are also studied. Numerical results for several common nonsmooth functionals are presented.

We develop a novel criterion for choosing regularization parameters for nonsmooth Tikhonov functionals. The proposed criterion is solely based on the value function, and thus applicable to a broad range of functionals. It is analytically compared with the local minimum criterion, and a posteriori error estimates are derived. An efficient numerical algorithm for computing the minimizer is developed, and its convergence properties are also studied. Numerical results for several common nonsmooth functionals are presented.

### 2010/03/17

#### Lectures

16:30-17:30 Room #128 (Graduate School of Math. Sci. Bldg.)

方向依存性を持つ長距離パーコレーションの臨界曲線

**三角 淳**(東大数理)方向依存性を持つ長距離パーコレーションの臨界曲線

### 2010/03/15

#### Seminar on Probability and Statistics

15:00-16:00 Room #002 (Graduate School of Math. Sci. Bldg.)

BROWNIAN COVARIATION AND CO-JUMPS, GIVEN DISCRETE OBSERVATION

https://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2009/16.html

**Cecilia Mancini**(University of Florence)BROWNIAN COVARIATION AND CO-JUMPS, GIVEN DISCRETE OBSERVATION

[ Abstract ]

We consider two processes driven by Brownian motions plus drift and possibly infinite activity jumps.

Given discrete observations we separately estimate the covariation between the two Brownian parts and the sum of the co-jumps. This allows to gain insight into the dependence structure of the processes and has important applications in finance.

Our estimator is based on a threshold principle allowing to isolate the jumps over a threshold.

The estimator of the continuous covariation is asymptotically Gaussian and converges at speed square root of n when the jump components have finite variation. In presence infinite variation jumps the speed is heavily influenced both by the small jumps dependence structure and by their jump activity indexes.

This talk is based on Mancini and Gobbi (2009), and Mancini (2010).

[ Reference URL ]We consider two processes driven by Brownian motions plus drift and possibly infinite activity jumps.

Given discrete observations we separately estimate the covariation between the two Brownian parts and the sum of the co-jumps. This allows to gain insight into the dependence structure of the processes and has important applications in finance.

Our estimator is based on a threshold principle allowing to isolate the jumps over a threshold.

The estimator of the continuous covariation is asymptotically Gaussian and converges at speed square root of n when the jump components have finite variation. In presence infinite variation jumps the speed is heavily influenced both by the small jumps dependence structure and by their jump activity indexes.

This talk is based on Mancini and Gobbi (2009), and Mancini (2010).

https://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2009/16.html

#### Seminar on Probability and Statistics

14:00-15:00 Room #002 (Graduate School of Math. Sci. Bldg.)

Statistical inference in the partial observation setting, in continuous time

https://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2009/15.html

**Alexandre Brouste**(Université du Maine)Statistical inference in the partial observation setting, in continuous time

[ Abstract ]

In various fields, the {\\it signal} process, whose law depends on an unknown parameter $artheta \\in \\Theta \\subset \\R^p$, can not be observed directly but only through an {\\it observation} process. We will talk about the so called fractional partial observation setting, where the observation process $Y=\\left( Y_t, t \\geq 0 ight)$ is given by a stochastic differential equation: egin{equation} \\label{mod:modelgeneral} Y_t = Y_0 + \\int_0^t h(X_s, artheta) ds + \\sigma W^H_t\\,, \\quad t > 0 \\end{equation} where the function $ h: \\, \\R imes \\Theta \\longrightarrow \\R$ and the constant $\\sigma>0$ are known and the noise $\\left( W^H_t\\,, t\\geq 0 ight)$ is a fractional Brownian motion valued in $\\R$ independent of the signal process $X$ and the initial condition $Y_0$. In this setting, the estimation of the unknown parameter $artheta \\in \\Theta$ given the observation of the continuous sample path $Y^T=\\left( Y_t , 0 \\leq t \\leq T ight)$, $T>0$, naturally arises.

[ Reference URL ]In various fields, the {\\it signal} process, whose law depends on an unknown parameter $artheta \\in \\Theta \\subset \\R^p$, can not be observed directly but only through an {\\it observation} process. We will talk about the so called fractional partial observation setting, where the observation process $Y=\\left( Y_t, t \\geq 0 ight)$ is given by a stochastic differential equation: egin{equation} \\label{mod:modelgeneral} Y_t = Y_0 + \\int_0^t h(X_s, artheta) ds + \\sigma W^H_t\\,, \\quad t > 0 \\end{equation} where the function $ h: \\, \\R imes \\Theta \\longrightarrow \\R$ and the constant $\\sigma>0$ are known and the noise $\\left( W^H_t\\,, t\\geq 0 ight)$ is a fractional Brownian motion valued in $\\R$ independent of the signal process $X$ and the initial condition $Y_0$. In this setting, the estimation of the unknown parameter $artheta \\in \\Theta$ given the observation of the continuous sample path $Y^T=\\left( Y_t , 0 \\leq t \\leq T ight)$, $T>0$, naturally arises.

https://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2009/15.html

### 2010/03/12

#### Colloquium

15:00-17:30 Room #050 (Graduate School of Math. Sci. Bldg.)

ガルニエ系の数理

特性類と不変量を巡る旅

**岡本和夫**(東京大学大学院数理科学研究科) 15:00-16:00ガルニエ系の数理

[ Abstract ]

ガルニエ系は,パンルヴェ方程式の拡張であり,完全積分可能な多時間ハミルトン系として与えられる。これは2階線型常微分方程式のホロノミック変形を与える非線型完全積分可能な偏微分方程式系であり,講演の対象である2次元系では,8つのタイプの基本形がある。ガルニエ系の研究は,初期値空間やソリトン方程式系の相似簡約などの立場から行われているが,材料が揃ってくれば,一般リーマン・ヒルベルト対応を経由して考察することが自然であるし,数学的であるだろう。パンルヴェ方程式の場合もそのような方向に進んでいる。一方,パンルヴェ方程式については,そのハミルトニアンの満足する非線型常微分方程式が,アフィンワイル群の対称性など数学的な材料を与える上で一定の役割を果たした。ガルニエ系についても,そのハミルトニアンについての非線型偏微分方程式系を具体的に書き下すことは,意味のあることと信じているが,未完である。この話題について,部分的な結果を紹介する。

ガルニエ系は,パンルヴェ方程式の拡張であり,完全積分可能な多時間ハミルトン系として与えられる。これは2階線型常微分方程式のホロノミック変形を与える非線型完全積分可能な偏微分方程式系であり,講演の対象である2次元系では,8つのタイプの基本形がある。ガルニエ系の研究は,初期値空間やソリトン方程式系の相似簡約などの立場から行われているが,材料が揃ってくれば,一般リーマン・ヒルベルト対応を経由して考察することが自然であるし,数学的であるだろう。パンルヴェ方程式の場合もそのような方向に進んでいる。一方,パンルヴェ方程式については,そのハミルトニアンの満足する非線型常微分方程式が,アフィンワイル群の対称性など数学的な材料を与える上で一定の役割を果たした。ガルニエ系についても,そのハミルトニアンについての非線型偏微分方程式系を具体的に書き下すことは,意味のあることと信じているが,未完である。この話題について,部分的な結果を紹介する。

**森田茂之**(東京大学大学院数理科学研究科) 16:30-17:30特性類と不変量を巡る旅

[ Abstract ]

40年近くの間,さまざまな幾何構造に関する特性類と不変量の研究を続けてきた.葉層構造やリーマン面のモジュライ空間の特性類,そして3次元多様体の位相不変量等である.これらについて振り返りつつ,これからの目標をいくつかの予想を交えてお話ししたい.

40年近くの間,さまざまな幾何構造に関する特性類と不変量の研究を続けてきた.葉層構造やリーマン面のモジュライ空間の特性類,そして3次元多様体の位相不変量等である.これらについて振り返りつつ,これからの目標をいくつかの予想を交えてお話ししたい.

### 2010/03/09

#### PDE Real Analysis Seminar

10:30-11:30 Room #123 (Graduate School of Math. Sci. Bldg.)

Shallow water waves with singularities

**Joachim Escher**(Leibniz University of Hanover)Shallow water waves with singularities

[ Abstract ]

The Degasperis-Procesi equation is a recently derived shallow water wave equation, which is - similar as the Camassa-Holm equation - embedded in a family of spatially periodic third order dispersive conservation laws.

The coexistence of globally in time defined classical solutions, wave breaking solutions, and spatially periodic peakons and shock waves is evidenced in the talk, and the precise blow-up scenario, including blow-up rates and blow-up sets, is discussed in detail. Finally several conditions on the initial profile are presented, which ensure the occurence of a breaking wave.

The Degasperis-Procesi equation is a recently derived shallow water wave equation, which is - similar as the Camassa-Holm equation - embedded in a family of spatially periodic third order dispersive conservation laws.

The coexistence of globally in time defined classical solutions, wave breaking solutions, and spatially periodic peakons and shock waves is evidenced in the talk, and the precise blow-up scenario, including blow-up rates and blow-up sets, is discussed in detail. Finally several conditions on the initial profile are presented, which ensure the occurence of a breaking wave.

### 2010/02/24

#### Lectures

15:00-16:30 Room #370 (Graduate School of Math. Sci. Bldg.)

Protein Moduli Space

**Robert Penner**(Aarhus University / University of Southern California)Protein Moduli Space

[ Abstract ]

Recent joint works with J. E. Andersen and others

provide explicit discrete and continuous models

of protein geometry. These models are inspired

by corresponding constructions in the study of moduli

spaces of flat G-connections on surfaces, in particular,

for G=PSL(2,R) and G=SO(3). These models can be used

for protein classification as well as for folding prediction,

and computer experiments towards these ends will

be discussed.

Recent joint works with J. E. Andersen and others

provide explicit discrete and continuous models

of protein geometry. These models are inspired

by corresponding constructions in the study of moduli

spaces of flat G-connections on surfaces, in particular,

for G=PSL(2,R) and G=SO(3). These models can be used

for protein classification as well as for folding prediction,

and computer experiments towards these ends will

be discussed.

### 2010/02/23

#### Lectures

14:00-15:00 Room #122 (Graduate School of Math. Sci. Bldg.)

Homogenization Limit and Singular Limit of the Allen-Cahn equation

**Bendong LOU**(同済大学)Homogenization Limit and Singular Limit of the Allen-Cahn equation

[ Abstract ]

We consider the Allen-Cahn equation in a cylinder with periodic undulating boundaries in the plane. Our problem involves two small parameters $\\delta$ and $\\epsilon$, where $\\delta$ appears in the equation to denote the scale of the singular limit and $\\epsilon$ appears in the boundary conditions to denote the scale of the homogenization limit. We consider the following two limiting processes:

(I): taking homogenization limit first and then taking singular limit;

(II) taking singular limit first and then taking homogenization limit.

We formally show that they both result in the same mean curvature flow equation, but with different boundary conditions.

We consider the Allen-Cahn equation in a cylinder with periodic undulating boundaries in the plane. Our problem involves two small parameters $\\delta$ and $\\epsilon$, where $\\delta$ appears in the equation to denote the scale of the singular limit and $\\epsilon$ appears in the boundary conditions to denote the scale of the homogenization limit. We consider the following two limiting processes:

(I): taking homogenization limit first and then taking singular limit;

(II) taking singular limit first and then taking homogenization limit.

We formally show that they both result in the same mean curvature flow equation, but with different boundary conditions.

### 2010/02/19

#### Lie Groups and Representation Theory

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

Discrete groups acting on homogeneous spaces V

**Yves Benoist**(Orsay)Discrete groups acting on homogeneous spaces V

[ Abstract ]

I will focus on recent advances on our understanding of discrete subgroups of Lie groups.

I will first survey how ideas from semisimple algebraic groups, ergodic theory and representation theory help us to understand properties of these discrete subgroups.

I will then focus on a joint work with Jean-Francois Quint studying the dynamics of these discrete subgroups on finite volume homogeneous spaces and proving the following result:

We fix two integral matrices A and B of size d, of determinant 1, and such that no finite union of vector subspaces is invariant by A and B. We fix also an irrational point on the d-dimensional torus. We will then prove that for n large the set of images of this point by the words in A and B of length at most n becomes equidistributed in the torus.

I will focus on recent advances on our understanding of discrete subgroups of Lie groups.

I will first survey how ideas from semisimple algebraic groups, ergodic theory and representation theory help us to understand properties of these discrete subgroups.

I will then focus on a joint work with Jean-Francois Quint studying the dynamics of these discrete subgroups on finite volume homogeneous spaces and proving the following result:

We fix two integral matrices A and B of size d, of determinant 1, and such that no finite union of vector subspaces is invariant by A and B. We fix also an irrational point on the d-dimensional torus. We will then prove that for n large the set of images of this point by the words in A and B of length at most n becomes equidistributed in the torus.

### 2010/02/18

#### GCOE lecture series

10:30-17:00 Room #126 (Graduate School of Math. Sci. Bldg.)

Discrete groups acting on homogeneous spaces III

Discrete groups acting on homogeneous spaces IV

**Yves Benoist**(Pars Sud) 10:30-11:30Discrete groups acting on homogeneous spaces III

[ Abstract ]

In this course I will focus on recent advances

on our understanding of discrete subgroups of Lie groups.

I will first survey how ideas from semisimple algebraic groups,

ergodic theory and representation theory help us to understand properties of these discrete subgroups.

I will then focus on a joint work with Jean-Francois Quint

studying the dynamics of these discrete subgroups on finite volume homogeneous spaces and proving the following result:

We fix two integral matrices A and B of size d, of determinant 1,

and such that no finite union of vector subspaces is invariant by A and B.

We fix also an irrational point on the d-dimensional torus. We will then prove that for n large the set of images of this point by the words in A and B of length at most n becomes equidistributed in the torus.

In this course I will focus on recent advances

on our understanding of discrete subgroups of Lie groups.

I will first survey how ideas from semisimple algebraic groups,

ergodic theory and representation theory help us to understand properties of these discrete subgroups.

I will then focus on a joint work with Jean-Francois Quint

studying the dynamics of these discrete subgroups on finite volume homogeneous spaces and proving the following result:

We fix two integral matrices A and B of size d, of determinant 1,

and such that no finite union of vector subspaces is invariant by A and B.

We fix also an irrational point on the d-dimensional torus. We will then prove that for n large the set of images of this point by the words in A and B of length at most n becomes equidistributed in the torus.

**Yves Benoist**(Paris Sud) 15:00-16:00Discrete groups acting on homogeneous spaces IV

#### Operator Algebra Seminars

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

Von Neumann Algebras and Boundary Quantum Field Theory

**Roberto Longo**(University of Rome, Tor Vergata)Von Neumann Algebras and Boundary Quantum Field Theory

#### Applied Analysis

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

Homogenization limit of a parabolic equation with nonlinear boundary conditions

**Bendong LOU**(同済大学)Homogenization limit of a parabolic equation with nonlinear boundary conditions

[ Abstract ]

We consider a quasilinear parabolic equation with the following nonlinear Neumann boundary condition:

"the slope of the solution on the boundary is a function $g$ of the value of the solution". Here $g$ takes values near its supremum with the frequency of $\\epsilon$. We show that the homogenization limit of the solution, as $\\epsilon$ tends to 0, is the solution satisfying the linear Neumann boundary condition: "the slope of the solution on the boundary is the supremum of $g$".

We consider a quasilinear parabolic equation with the following nonlinear Neumann boundary condition:

"the slope of the solution on the boundary is a function $g$ of the value of the solution". Here $g$ takes values near its supremum with the frequency of $\\epsilon$. We show that the homogenization limit of the solution, as $\\epsilon$ tends to 0, is the solution satisfying the linear Neumann boundary condition: "the slope of the solution on the boundary is the supremum of $g$".

#### GCOE Seminars

10:10-11:00 Room #122 (Graduate School of Math. Sci. Bldg.)

空間的に非一様な場における進行波

**俣野 博**(数理科学)空間的に非一様な場における進行波

#### GCOE Seminars

11:00-11:50 Room #122 (Graduate School of Math. Sci. Bldg.)

岡の連接定理から一致の定理(点分布から分かるもの)まで

**野口 潤次郎**(数理科学)岡の連接定理から一致の定理(点分布から分かるもの)まで

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