## Seminar information archive

Seminar information archive ～05/28｜Today's seminar 05/29 | Future seminars 05/30～

### 2006/10/10

#### Tuesday Seminar on Topology

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

Estimating Lusternik-Schnirelmann Category for Foliations:A Survey of Available Techniques

**Elmar Vogt**(Frie Universitat Berlin)Estimating Lusternik-Schnirelmann Category for Foliations:A Survey of Available Techniques

[ Abstract ]

The Lusternik-Schnirelmann category of a space $X$ is the smallest number $r$ such that $X$ can be covered by $r + 1$ open sets which are contractible in $X$.For foliated manifolds there are several notions generalizing this concept, all of them due

to Helen Colman. We are mostly concerned with the concept of tangential Lusternik-Schnirelmann category (tangential LS-category). Here one requires a covering by open sets $U$ with the following property. There is a leafwise homotopy starting with the inclusion of $U$ and ending in a map that throws for each leaf $F$ of the foliation each component of $U \\cap F$ onto a single point. A leafwise homotopy is a homotopy moving points only inside leaves. Rather than presenting the still very few results obtained about the LS category of foliations, we survey techniques, mostly quite elementary, to estimate the tangential LS-category from below and above.

The Lusternik-Schnirelmann category of a space $X$ is the smallest number $r$ such that $X$ can be covered by $r + 1$ open sets which are contractible in $X$.For foliated manifolds there are several notions generalizing this concept, all of them due

to Helen Colman. We are mostly concerned with the concept of tangential Lusternik-Schnirelmann category (tangential LS-category). Here one requires a covering by open sets $U$ with the following property. There is a leafwise homotopy starting with the inclusion of $U$ and ending in a map that throws for each leaf $F$ of the foliation each component of $U \\cap F$ onto a single point. A leafwise homotopy is a homotopy moving points only inside leaves. Rather than presenting the still very few results obtained about the LS category of foliations, we survey techniques, mostly quite elementary, to estimate the tangential LS-category from below and above.

### 2006/10/06

#### Mathematical Biology Seminar

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

人口指標の精度について

**石井 太**(国立社会保障・人口問題研究所)人口指標の精度について

[ Abstract ]

近年の急速な少子化の進行に伴い、出生率の動向への関心が高まっている。しかしながら、その指標が注目され、様々に論じられる中に、指標の持つ精度を度外視した議論なども見受けられる。人口統計は人口分析の中心となるデータソースであり、人口指標の精度は重要な問題であるが、わが国においてはこれまで比較的精度の高い人口統計が取得されてきたこともあり、それほど重視せず、確定的なものと捉えがちな傾向があったように思われる。一方では、近年、統計調査環境の悪化などもあり、各々の人口指標についてどこまで詳細な議論が可能なのか、指標の精度について理論的・実務的な観点からより深い認識を持つことが必要となってきている。

本報告では、出生率での具体例を中心に、さまざまな誤差の発生要因に応じた人口指標の評価について提示することとしたい。

近年の急速な少子化の進行に伴い、出生率の動向への関心が高まっている。しかしながら、その指標が注目され、様々に論じられる中に、指標の持つ精度を度外視した議論なども見受けられる。人口統計は人口分析の中心となるデータソースであり、人口指標の精度は重要な問題であるが、わが国においてはこれまで比較的精度の高い人口統計が取得されてきたこともあり、それほど重視せず、確定的なものと捉えがちな傾向があったように思われる。一方では、近年、統計調査環境の悪化などもあり、各々の人口指標についてどこまで詳細な議論が可能なのか、指標の精度について理論的・実務的な観点からより深い認識を持つことが必要となってきている。

本報告では、出生率での具体例を中心に、さまざまな誤差の発生要因に応じた人口指標の評価について提示することとしたい。

### 2006/10/05

#### Seminar on Mathematics for various disciplines

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

Global existence and asymptotic behavior of Gowdy symmetric spacetimes with nonlinear scalar field

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

**成田誠**(Department of Mathematics, National Taiwan University)Global existence and asymptotic behavior of Gowdy symmetric spacetimes with nonlinear scalar field

[ Abstract ]

We study global properties of Gowdy symmetric (the existence of a symmetry group with two-dimensional spacelike orbits) spacetimes with nonlinear scalar field, which naturally arises in modern cosmology based on superstring theory.

The system of the Einstein and scalar field equations becomes a system consisting of wave map and nonlinear wave equations in one space dimension. We prove a global existence theorem for this system. Also, asymptotic energy decay will be discussed.

[ Reference URL ]We study global properties of Gowdy symmetric (the existence of a symmetry group with two-dimensional spacelike orbits) spacetimes with nonlinear scalar field, which naturally arises in modern cosmology based on superstring theory.

The system of the Einstein and scalar field equations becomes a system consisting of wave map and nonlinear wave equations in one space dimension. We prove a global existence theorem for this system. Also, asymptotic energy decay will be discussed.

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

### 2006/09/27

#### PDE Real Analysis Seminar

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

Integral operators in the weighted Lebesgue spaces with a variable exponent

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

**Professor Vakhtang Kokilashvili**(A. Razmadze Mathematical Institute, Georgian Academy of Science)Integral operators in the weighted Lebesgue spaces with a variable exponent

[ Abstract ]

We present a boundedness criteria of the maximal functions and the singular integral operators defined on Carleson curves in the weighted Lebesgue spaces with a variable exponent. There are also given the weighted estimates for the generalized singular integrals raised in the theory of generalized analytic functions of I.N.Vekua and the weighted Sobolev theorems for potentials on Carleson curves. The weight functions may be of power function type as well as oscillating type. The certain version of a Muckenhoupt-type condition for a variable exponent will be considered.

We also expect to treat two-weight problems for the classical integral operators in the variable Lebesgue spaces and to give some applications of the obtained results to the summability problems of Fourier series in two-weighted setting.

[ Reference URL ]We present a boundedness criteria of the maximal functions and the singular integral operators defined on Carleson curves in the weighted Lebesgue spaces with a variable exponent. There are also given the weighted estimates for the generalized singular integrals raised in the theory of generalized analytic functions of I.N.Vekua and the weighted Sobolev theorems for potentials on Carleson curves. The weight functions may be of power function type as well as oscillating type. The certain version of a Muckenhoupt-type condition for a variable exponent will be considered.

We also expect to treat two-weight problems for the classical integral operators in the variable Lebesgue spaces and to give some applications of the obtained results to the summability problems of Fourier series in two-weighted setting.

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

#### Mathematical Biology Seminar

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

The coevolution of altruism and punishment:role of the selfish punisher

**中丸麻由子**(東京工業大学)The coevolution of altruism and punishment:role of the selfish punisher

[ Abstract ]

Punishment is an important mechanism promoting the evolution of altruism among nonrelatives. We investigate the coevolution of altruism and punitive behavior, considering four strategies: a cooperator who punishes defectors (AP), a pure cooperator (AN), a defector who punishes defectors (selfish punisher or SP), and a pure defector (SN). We especially focus on the effects of SP on the coevolution of altruism and punishment, studying both the score-dependent viability model (whereby the game's score affects survivorship only) and the score-dependent fertility model (whereby the score affects fertility only). In the viability model of a completely mixed population, SP helps cooperators to evolve, but SP does not in the fertility model. In both models of a lattice-structured population, SP promotes the spread of AP, but AN discourages it. These results indicate that punishment is a form of spite behavior and that different models give different magnitude of advantage to spite behavior.

Punishment is an important mechanism promoting the evolution of altruism among nonrelatives. We investigate the coevolution of altruism and punitive behavior, considering four strategies: a cooperator who punishes defectors (AP), a pure cooperator (AN), a defector who punishes defectors (selfish punisher or SP), and a pure defector (SN). We especially focus on the effects of SP on the coevolution of altruism and punishment, studying both the score-dependent viability model (whereby the game's score affects survivorship only) and the score-dependent fertility model (whereby the score affects fertility only). In the viability model of a completely mixed population, SP helps cooperators to evolve, but SP does not in the fertility model. In both models of a lattice-structured population, SP promotes the spread of AP, but AN discourages it. These results indicate that punishment is a form of spite behavior and that different models give different magnitude of advantage to spite behavior.

### 2006/09/25

#### Seminar on Probability and Statistics

13:00-14:10 Room #128 (Graduate School of Math. Sci. Bldg.)

Nonparametric testing time-homogeneity for L'evy processes

https://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2006/09.html

**西山 陽一**(統計数理研究所)Nonparametric testing time-homogeneity for L'evy processes

[ Abstract ]

First, a review about uniform central limit theorems for martingales is given. The main part of the talk is concerned with a change point problem for L'evy processes. The null hypothesis is that the L'evy process is time-homogeneous, and the alternative is that the L'evy measure changes at a certain time point of the observation period. We propose an empirical process type statistics, and derive its asymptotic behaviour under the null and the alternative hypotheses. The limiting distribution under the null hypothesis is a functional of the standard Brownian motion.

[ Reference URL ]First, a review about uniform central limit theorems for martingales is given. The main part of the talk is concerned with a change point problem for L'evy processes. The null hypothesis is that the L'evy process is time-homogeneous, and the alternative is that the L'evy measure changes at a certain time point of the observation period. We propose an empirical process type statistics, and derive its asymptotic behaviour under the null and the alternative hypotheses. The limiting distribution under the null hypothesis is a functional of the standard Brownian motion.

https://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2006/09.html

### 2006/09/13

#### Functional Analysis Seminar

13:00-16:45 Room #123 (Graduate School of Math. Sci. Bldg.)

On the determination of non-analytic resonances

(joint work with T.Ramond and J. Sjostrand)

Global existence for energy critical waves in 3-d domains

(joint work with G. Lebeau and F. Planchon)

On the Born-Oppenheimer approximation of wave-operators

**Andre' Martinez**(Bologna University) 13:00-14:00On the determination of non-analytic resonances

(joint work with T.Ramond and J. Sjostrand)

**Nicolas Burq**(Université de Paris Sud) 14:15-15:15Global existence for energy critical waves in 3-d domains

(joint work with G. Lebeau and F. Planchon)

[ Abstract ]

I will present some recent results obtained recently with G. Lebeau and F. Planchon. We prove that the energy critical (quintic) non linear wave equation in 3-d domains with Dirichlet boundary conditions is globally well posed for any initial data (with finite energy). I will give some hints about the proof of this result which is based on some recent results by Smith and Sogge on $L^p$ estimates for spectral projectors and a carefull study of the boundary value problem.

I will present some recent results obtained recently with G. Lebeau and F. Planchon. We prove that the energy critical (quintic) non linear wave equation in 3-d domains with Dirichlet boundary conditions is globally well posed for any initial data (with finite energy). I will give some hints about the proof of this result which is based on some recent results by Smith and Sogge on $L^p$ estimates for spectral projectors and a carefull study of the boundary value problem.

**Vania Sordoni**(Bologna University) 15:45-16:45On the Born-Oppenheimer approximation of wave-operators

### 2006/09/11

#### Operator Algebra Seminars

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

Operator-algebraic techniques in nonequilibrium statistical mechanics

**Claude-Alain Pillet**(Univ. de Toulon et du Var)Operator-algebraic techniques in nonequilibrium statistical mechanics

### 2006/09/06

#### Number Theory Seminar

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

Computation of the mod l Galois representations associated to Delta

**Bas Edixhoven**(Univ. of Leiden)Computation of the mod l Galois representations associated to Delta

### 2006/09/04

#### Mathematical Finance

17:00-18:30 Room #117 (Graduate School of Math. Sci. Bldg.)

Dynamic Risk Measures and Backward Stochastic Differential Equation

**Freddy Delbaen**(ETH)Dynamic Risk Measures and Backward Stochastic Differential Equation

#### Mathematical Finance

15:45-16:45 Room #117 (Graduate School of Math. Sci. Bldg.)

リスク尺度入門及び概説

**楠岡成雄氏・梅澤祐二**(東京大)リスク尺度入門及び概説

### 2006/08/25

#### Number Theory Seminar

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

p-adic local constants

**A. Marmora**(パリ北大・東大/学振)p-adic local constants

### 2006/08/22

#### Seminar on Probability and Statistics

15:30-16:40 Room #128 (Graduate School of Math. Sci. Bldg.)

A unifying approach to inference in semimartingale and long-memory models

https://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2006/07.html

**Jeannette H.C. WOERNER**(University of Gottingen)A unifying approach to inference in semimartingale and long-memory models

[ Abstract ]

Over the recent years classical stochastic volatility models based on Brownian motion have been generalized in different ways, either replacing the Brownian motion by a pure jump Levy process, which leads to a pure jump model, or by a fractional Brownian motion, which makes it possible to model both long memory or turbulent behaviour. We consider robust and easily computable estimators for the inte- grated volatility, providing insight in the level of volatility, as needed for risk measurement and pricing of variance and volatility swaps. We discuss consistency and distributional results for the power and multi- power variation estimates based on high frequency data. Furthermore, we consider robustness against additive components and compare the results for the different classes of semimartingale and fractional Brow- nian motion models.

[ Reference URL ]Over the recent years classical stochastic volatility models based on Brownian motion have been generalized in different ways, either replacing the Brownian motion by a pure jump Levy process, which leads to a pure jump model, or by a fractional Brownian motion, which makes it possible to model both long memory or turbulent behaviour. We consider robust and easily computable estimators for the inte- grated volatility, providing insight in the level of volatility, as needed for risk measurement and pricing of variance and volatility swaps. We discuss consistency and distributional results for the power and multi- power variation estimates based on high frequency data. Furthermore, we consider robustness against additive components and compare the results for the different classes of semimartingale and fractional Brow- nian motion models.

https://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2006/07.html

#### Seminar on Probability and Statistics

16:50-18:00 Room #128 (Graduate School of Math. Sci. Bldg.)

A computation of Theta in a jump diffusion model by integration by parts

https://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2006/08.html

**Delphine DAVID**(Departement de Mathematiques, Universite de La Rochelle)A computation of Theta in a jump diffusion model by integration by parts

[ Abstract ]

Using Malliavin weights in a jump-diffusion model we obtain an expression for Theta (the sensitivity of an option price with respect to the time remaining until exercise), with application to non-smooth payoff functions. Optimal weights are computed by minimization of variance and numerical simulations are presented for digital and European options. Some results are also presented for Asian options. Our representation formula for Theta differs in general from the one obtained from the Black-Scholes PDE in terms of Delta and Gamma.

[ Reference URL ]Using Malliavin weights in a jump-diffusion model we obtain an expression for Theta (the sensitivity of an option price with respect to the time remaining until exercise), with application to non-smooth payoff functions. Optimal weights are computed by minimization of variance and numerical simulations are presented for digital and European options. Some results are also presented for Asian options. Our representation formula for Theta differs in general from the one obtained from the Black-Scholes PDE in terms of Delta and Gamma.

https://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2006/08.html

### 2006/08/03

#### Operator Algebra Seminars

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

The Cuntz semigroup as an invariant for $C^*$-algebras

**George Elliott**(University of Toronto)The Cuntz semigroup as an invariant for $C^*$-algebras

### 2006/07/31

#### Lie Groups and Representation Theory

15:00-17:30 Room #126 (Graduate School of Math. Sci. Bldg.)

The Heat equation, the Segal-Bargmann transform and generalizations - II

[ Reference URL ]

http://akagi.ms.u-tokyo.ac.jp/seminar.html

Radon transforms on Grassmannians and Matrix Spaces

http://akagi.ms.u-tokyo.ac.jp/seminar.html

**Guster Olafsson**(Louisiana State University) 15:00-16:00The Heat equation, the Segal-Bargmann transform and generalizations - II

[ Reference URL ]

http://akagi.ms.u-tokyo.ac.jp/seminar.html

**Boris Rubin**(Louisiana State University) 16:30-17:30Radon transforms on Grassmannians and Matrix Spaces

[ Abstract ]

Diverse geometric problems in $R^N$ get a new flavor if a generic point $x=(x_1,...,x_N)$ is regarded as a matrix with appropriately organized entries (set, e.g., $x=(x_{i,j})_{n \\times m}$ for $N=nm$). This well known observation has led to a series of breakthrough achievements in mathematics. In integral geometry it suggests a number of the so-called ``higher-rank" problems when such traditional scalar notions as ``distance", ``angle", and ``scaling" become matrix-valued. I will be speaking about Radon transforms on Grassmann manifolds and matrix spaces and some related problems of harmonic analysis where these phenomena come into play.

[ Reference URL ]Diverse geometric problems in $R^N$ get a new flavor if a generic point $x=(x_1,...,x_N)$ is regarded as a matrix with appropriately organized entries (set, e.g., $x=(x_{i,j})_{n \\times m}$ for $N=nm$). This well known observation has led to a series of breakthrough achievements in mathematics. In integral geometry it suggests a number of the so-called ``higher-rank" problems when such traditional scalar notions as ``distance", ``angle", and ``scaling" become matrix-valued. I will be speaking about Radon transforms on Grassmann manifolds and matrix spaces and some related problems of harmonic analysis where these phenomena come into play.

http://akagi.ms.u-tokyo.ac.jp/seminar.html

### 2006/07/25

#### Lie Groups and Representation Theory

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

Radon Transforms: Basic Concepts

http://akagi.ms.u-tokyo.ac.jp/seminar.html

**Boris Rubin**(Louisiana State University)Radon Transforms: Basic Concepts

[ Abstract ]

How can we reconstruct a function on a manifold from integrals of this function over certain submanifolds?

This is one of the central problems in integral geometry and tomography, which leads to the notion of the Radon transform.

The first talk is of introductory character.

We discuss basic ideas of the original Radon's paper (1917), then proceed to the Minkowski-Funk transform and more general totally geodesic Radon transforms on the $n$-dimensional unit sphere.

The main emphasis is an intimate connection of these transforms with the relevant harmonic analysis.

We will see that Radon transforms of this type and their inverses can be regarded as members of analytic families of suitable convolution operators and successfully studied in the framework of these families.

I also plan to discuss an open problem of small divisors on the unit sphere, which arises in studying injectivity of generalized Minkowski-Funk transforms for non-central spherical sections.

[ Reference URL ]How can we reconstruct a function on a manifold from integrals of this function over certain submanifolds?

This is one of the central problems in integral geometry and tomography, which leads to the notion of the Radon transform.

The first talk is of introductory character.

We discuss basic ideas of the original Radon's paper (1917), then proceed to the Minkowski-Funk transform and more general totally geodesic Radon transforms on the $n$-dimensional unit sphere.

The main emphasis is an intimate connection of these transforms with the relevant harmonic analysis.

We will see that Radon transforms of this type and their inverses can be regarded as members of analytic families of suitable convolution operators and successfully studied in the framework of these families.

I also plan to discuss an open problem of small divisors on the unit sphere, which arises in studying injectivity of generalized Minkowski-Funk transforms for non-central spherical sections.

http://akagi.ms.u-tokyo.ac.jp/seminar.html

### 2006/07/24

#### Tuesday Seminar on Topology

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

Invariant foliations in hyperbolic dynamics:

Smoothness and smooth equivalence

http://faculty.ms.u-tokyo.ac.jp/~topology/

**Boris Hasselblatt**(Tufts University)Invariant foliations in hyperbolic dynamics:

Smoothness and smooth equivalence

[ Abstract ]

The stable and unstable leaves of a hyperbolic dynamical system are smooth and form a continuous foliation. Smoothness of this foliation sometimes constrains the topological type of the foliation, other times restricts at least the smooth equivalence class of the dynamical system, or for geodesic flows, the type of the underlying manifold. I will present a broad introduction as well as recent work, work in progress, and open problems.

[ Reference URL ]The stable and unstable leaves of a hyperbolic dynamical system are smooth and form a continuous foliation. Smoothness of this foliation sometimes constrains the topological type of the foliation, other times restricts at least the smooth equivalence class of the dynamical system, or for geodesic flows, the type of the underlying manifold. I will present a broad introduction as well as recent work, work in progress, and open problems.

http://faculty.ms.u-tokyo.ac.jp/~topology/

### 2006/07/20

#### Operator Algebra Seminars

16:30-18:00 Room #052 (Graduate School of Math. Sci. Bldg.)

Linear response theory in quantum statistical mechanics

**緒方芳子**(東大数理・学振)Linear response theory in quantum statistical mechanics

#### Lie Groups and Representation Theory

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

The Heat equation, the Segal-Bargmann transform and generalizations - I

[ Reference URL ]

http://akagi.ms.u-tokyo.ac.jp/seminar.html

**Guster Olafsson**(Louisiana State University)The Heat equation, the Segal-Bargmann transform and generalizations - I

[ Reference URL ]

http://akagi.ms.u-tokyo.ac.jp/seminar.html

### 2006/07/19

#### Seminar on Probability and Statistics

16:20-17:30 Room #128 (Graduate School of Math. Sci. Bldg.)

Edgeworth Expansion for Likelihood Analysis on Ergodic Diffusions with applications to Bootstrap

https://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2006/06.html

**深澤 正彰**(東京大学大学院数理科学研究科)Edgeworth Expansion for Likelihood Analysis on Ergodic Diffusions with applications to Bootstrap

[ Abstract ]

We shall consider the maximal lilelihood estimator for the drift coefficient of a given one-dimensional diffusion. An Edgeworth expansion formula will be presented and verify a second-order correct confidence interval we shall newly propose. We are also going to mention the likelihood ratio statistic, which enjoys second-order correctness. There are Bootstrap methods closely related to the subject and introduced recently by the author. Some generalized results on those methods will be also introduced in this talk.

[ Reference URL ]We shall consider the maximal lilelihood estimator for the drift coefficient of a given one-dimensional diffusion. An Edgeworth expansion formula will be presented and verify a second-order correct confidence interval we shall newly propose. We are also going to mention the likelihood ratio statistic, which enjoys second-order correctness. There are Bootstrap methods closely related to the subject and introduced recently by the author. Some generalized results on those methods will be also introduced in this talk.

https://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2006/06.html

### 2006/07/13

#### Operator Algebra Seminars

16:30-18:00 Room #052 (Graduate School of Math. Sci. Bldg.)

Property (T) for universal lattices, after Y. Shalom

**小沢登高**(東大数理)Property (T) for universal lattices, after Y. Shalom

[ Abstract ]

I will talk on Shalom's recent result that

$SL_n(Z[X])$ ($n\\geq 3$) has the property (T).

The talk should be elementary.

I will talk on Shalom's recent result that

$SL_n(Z[X])$ ($n\\geq 3$) has the property (T).

The talk should be elementary.

### 2006/07/12

#### Number Theory Seminar

16:30-17:30 Room #117 (Graduate School of Math. Sci. Bldg.)

Beilinson-Drinfeld chiral algebra, geometric Langlands program and open Gromov-Witten invariants

**桜井 真**(東京大学理学系研究科)Beilinson-Drinfeld chiral algebra, geometric Langlands program and open Gromov-Witten invariants

[ Abstract ]

都合により、とりやめになりました。

都合により、とりやめになりました。

#### Mathematical Finance

18:30-20:00 Room #118 (Graduate School of Math. Sci. Bldg.)

A complete-market generalization of the Black-Scholes model

**高岡 浩一郎**(一橋大)A complete-market generalization of the Black-Scholes model

#### PDE Real Analysis Seminar

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

Analysis of a crystal growth model

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

**Piotr Rybka**(Warsaw University)Analysis of a crystal growth model

[ Abstract ]

We are concerned with mathematical model of a single crystal growing from vapor. Mathematically this is an exterior, one-phase Stefan-type problem with Gibbs-Thomson law. We restrict our attention to an idealization of a ice crystal, i.e. our evolving free boundary is a circular cylinder. The system under consideration consists of an equation for the motion of the free boundary (the crystal surface) coupled to the quasi-steady approximation of the diffusion equation for the supersaturation of vapor. We present analysis of the system, we show well-posedness and draw the phase portrait, we use here the fact that we need just to variable to describe evolution of a cylinder.

We are mostly concerned with the shape-persitency problem of the

evolution. The problem is, the Gibbs-Thomson relation is in fact a

driven, weighted, mean, singular curvature flow and it is not obvious that the shape of the initial interface will persists throughout the evolution or even for some time. In order to solve this problem we show existence of the region in the phase plane which is a neighborhood of a unique steady state, such that in this region the shape of the cylinder is preserved. However, this set is not invariant with respect to dynamics of the problem.

It is a very interesting question what happens to surface of our crystal at the boundary of the shape-persitency (or shape stability) region. This problem in its full generality is open. However, we give some insight by studying the Gibbs-Thomson relation with a given driving, which inherits properties of the coupling to the diffusion field. We study the resulting driven weighted mean curvature flow for graphs and some special closed Lipschitz curves. We show well-posedness of the problem, but mainly we exhibit the phenomenon of bending flat parts of the curve, which grow ``too big''.

[ Reference URL ]We are concerned with mathematical model of a single crystal growing from vapor. Mathematically this is an exterior, one-phase Stefan-type problem with Gibbs-Thomson law. We restrict our attention to an idealization of a ice crystal, i.e. our evolving free boundary is a circular cylinder. The system under consideration consists of an equation for the motion of the free boundary (the crystal surface) coupled to the quasi-steady approximation of the diffusion equation for the supersaturation of vapor. We present analysis of the system, we show well-posedness and draw the phase portrait, we use here the fact that we need just to variable to describe evolution of a cylinder.

We are mostly concerned with the shape-persitency problem of the

evolution. The problem is, the Gibbs-Thomson relation is in fact a

driven, weighted, mean, singular curvature flow and it is not obvious that the shape of the initial interface will persists throughout the evolution or even for some time. In order to solve this problem we show existence of the region in the phase plane which is a neighborhood of a unique steady state, such that in this region the shape of the cylinder is preserved. However, this set is not invariant with respect to dynamics of the problem.

It is a very interesting question what happens to surface of our crystal at the boundary of the shape-persitency (or shape stability) region. This problem in its full generality is open. However, we give some insight by studying the Gibbs-Thomson relation with a given driving, which inherits properties of the coupling to the diffusion field. We study the resulting driven weighted mean curvature flow for graphs and some special closed Lipschitz curves. We show well-posedness of the problem, but mainly we exhibit the phenomenon of bending flat parts of the curve, which grow ``too big''.

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

< Previous 123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960616263646566676869707172737475767778798081828384858687888990919293949596979899100101102103104105106107108109110111112113114115116117118119120121122123124125126127128129130131132133134135136137138139140141142143144145146147148149150151152153154155156157158159160161162163164165166167168169170171172173174175176177178179180181182183184185186187 Next >