## Seminar information archive

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

### 2009/10/14

#### GCOE lecture series

15:30-17:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Macroscopic fluctuation theory for nonequilibrium stationary states, Ⅳ

**Claudio Landim**(IMPA, Brazil)Macroscopic fluctuation theory for nonequilibrium stationary states, Ⅳ

#### GCOE lecture series

13:30-15:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Mini course on the gradient models, Ⅱ: Convex interaction potential

**Jean-Dominique Deuschel**(TU Berlin)Mini course on the gradient models, Ⅱ: Convex interaction potential

[ Abstract ]

Much is known for strictly convex interactions, which under rescaling behave much like the harmonic model. In particular the unicity of the ergodic component have been established by Funaki and Spohn, and the scaling limit to the gradient of the continuous gaussian free field by Naddaf and Spencer. The results are based on special analytical and probabilistic tools such as the Brascamp-Lieb inequality and the Hellfer-Sj\\"osstrand random walk representation. These techniques rely on the strict convexity of the interaction potential.

Much is known for strictly convex interactions, which under rescaling behave much like the harmonic model. In particular the unicity of the ergodic component have been established by Funaki and Spohn, and the scaling limit to the gradient of the continuous gaussian free field by Naddaf and Spencer. The results are based on special analytical and probabilistic tools such as the Brascamp-Lieb inequality and the Hellfer-Sj\\"osstrand random walk representation. These techniques rely on the strict convexity of the interaction potential.

#### Geometry Seminar

14:45-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Fixed point theorems for non-positively curved spaces and random groups

Lagrangian mean curvature flow and symplectic area

**近藤剛史 (Kondo Takefumi)**(神戸大学大学院理学研究科) 14:45-16:15Fixed point theorems for non-positively curved spaces and random groups

[ Abstract ]

It is not easy to construct a finitely generated group with a fixed point property for non-positively curved spaces. However, if we randomly choose relators, then we can get examples of such groups. To show this, we need a criterion for deducing a fixed point property from a local property of a group. In this talk, we will introduce one such criterion, and our approach is via a scaling limit argument.

It is not easy to construct a finitely generated group with a fixed point property for non-positively curved spaces. However, if we randomly choose relators, then we can get examples of such groups. To show this, we need a criterion for deducing a fixed point property from a local property of a group. In this talk, we will introduce one such criterion, and our approach is via a scaling limit argument.

**赤穂まなぶ (Akaho Manabu)**(首都大学東京大学院理工学研究科) 16:30-18:00Lagrangian mean curvature flow and symplectic area

[ Abstract ]

In this talk, we consider symplectic area of smooth maps from a Riemann surface with boundary on embedded Lagrangian mean curvature flow in Kahler-Einstein manifolds. As an application, we observe a relation between embedded Lagrangian mean curvature flow and Floer theory of monotone Lagrangian submanifolds in Kahler-Einstein manifolds; in this case non-trivial holomorphic discs turn out to be an obstruction to the existence of long time solution of the flow.

In this talk, we consider symplectic area of smooth maps from a Riemann surface with boundary on embedded Lagrangian mean curvature flow in Kahler-Einstein manifolds. As an application, we observe a relation between embedded Lagrangian mean curvature flow and Floer theory of monotone Lagrangian submanifolds in Kahler-Einstein manifolds; in this case non-trivial holomorphic discs turn out to be an obstruction to the existence of long time solution of the flow.

#### GCOE Seminars

16:30-17:30 Room #370 (Graduate School of Math. Sci. Bldg.)

Partial Cauchy data for general second order elliptic operators in two dimensions

**O. Emanouilov**(Colorado State University)Partial Cauchy data for general second order elliptic operators in two dimensions

[ Abstract ]

We consider the problem of determining the coefficients of a first-order perturbation of the Laplacian in two dimensions by measuring the corresponding Cauchy data on an arbitrary open subset of the boundary. From this information we obtained a coupled PDE system of first order which the coefficients satisfy. As a corollary we show for the magnetic Schr"odinger equation that the magnetic field and the electric potential are uniquely determined by measuring the partial Cauchy data on an arbitrary part of the boundary. We also show that the coefficients of any real vector field perturbation of the Laplacian, the convection terms, are uniquely determined by their partial Cauchy data.

We consider the problem of determining the coefficients of a first-order perturbation of the Laplacian in two dimensions by measuring the corresponding Cauchy data on an arbitrary open subset of the boundary. From this information we obtained a coupled PDE system of first order which the coefficients satisfy. As a corollary we show for the magnetic Schr"odinger equation that the magnetic field and the electric potential are uniquely determined by measuring the partial Cauchy data on an arbitrary part of the boundary. We also show that the coefficients of any real vector field perturbation of the Laplacian, the convection terms, are uniquely determined by their partial Cauchy data.

### 2009/10/13

#### Tuesday Seminar on Topology

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

Instanton Floer homology for lens spaces

**笹平 裕史**(東京大学大学院数理科学研究科)Instanton Floer homology for lens spaces

[ Abstract ]

Let Y be an oriented closed 3-manifold and P be an SU(2)-bundle on Y. Under a certain condition, instanton Floer homology for Y can be defined as the Morse homology of the Chern-Simons functional. The condition is that all flat connections on P are irreducible. When there is a reducible flat connection on P, instanton Floer homology is not defined in general.

Since the fundamental group of a lens sapce is commutative, all flat connections on the lens space are reducible. In this talk I will introduce instanton Floer homology for lens spaces. I also show calculations for some lens spaces.

Let Y be an oriented closed 3-manifold and P be an SU(2)-bundle on Y. Under a certain condition, instanton Floer homology for Y can be defined as the Morse homology of the Chern-Simons functional. The condition is that all flat connections on P are irreducible. When there is a reducible flat connection on P, instanton Floer homology is not defined in general.

Since the fundamental group of a lens sapce is commutative, all flat connections on the lens space are reducible. In this talk I will introduce instanton Floer homology for lens spaces. I also show calculations for some lens spaces.

#### Lie Groups and Representation Theory

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

Extensions between finite-dimensional simple modules over a generalized current Lie algebra

https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar.html

**小寺諒介**(東京大学)Extensions between finite-dimensional simple modules over a generalized current Lie algebra

[ Abstract ]

$\\mathfrak{g}$を$\\mathbb{C}$上の有限次元半単純Lie代数,$A$を有限生成可換$\\mathbb {C}$代数とする.

テンソル積$A \\otimes \\mathfrak{g}$に自然にLie代数の構造を与えたものを一般化されたカレントLie代数と呼ぶ.

一般化されたカレントLie代数の任意の2つの有限次元既約表現に対して,その1次のExt群を完全に決定することができたので,その結果について発表する.

[ Reference URL ]$\\mathfrak{g}$を$\\mathbb{C}$上の有限次元半単純Lie代数,$A$を有限生成可換$\\mathbb {C}$代数とする.

テンソル積$A \\otimes \\mathfrak{g}$に自然にLie代数の構造を与えたものを一般化されたカレントLie代数と呼ぶ.

一般化されたカレントLie代数の任意の2つの有限次元既約表現に対して,その1次のExt群を完全に決定することができたので,その結果について発表する.

https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar.html

### 2009/10/09

#### Lecture Series on Mathematical Sciences in Soceity

16:20-17:50 Room #117 (Graduate School of Math. Sci. Bldg.)

暗号の実践編

**岡本龍明**(NTT 情報流通プラットフォーム研究所 岡本特別研究室長)暗号の実践編

#### GCOE lecture series

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

Associated varieties for Representations of classical Lie

super-algebras

https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar.html

**Michel Duflo**(Paris 7)Associated varieties for Representations of classical Lie

super-algebras

[ Abstract ]

In this lecture, I'll discuss the notion of "Associated

varieties for Representations of classical Lie super-algebras (joint work with Vera Serganova)" and the relation with the degree of atypicality. This is related to a conjecture of Kac and Wakimoto.

[ Reference URL ]In this lecture, I'll discuss the notion of "Associated

varieties for Representations of classical Lie super-algebras (joint work with Vera Serganova)" and the relation with the degree of atypicality. This is related to a conjecture of Kac and Wakimoto.

https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar.html

### 2009/10/07

#### PDE Real Analysis Seminar

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

Wiener measure and Feynman-Kac formula on the Heisenberg group

**劉和平(Liu Heping)**(Beijing University)Wiener measure and Feynman-Kac formula on the Heisenberg group

[ Abstract ]

It is well known that the Feynman-Kac formula on the Euclidean space gives the solution of Schrodinger equation by the Wiener integral. We will discuss the Wiener measure and Feynman-Kac formula on the Heisenberg group. The results hold on the H-type groups.

It is well known that the Feynman-Kac formula on the Euclidean space gives the solution of Schrodinger equation by the Wiener integral. We will discuss the Wiener measure and Feynman-Kac formula on the Heisenberg group. The results hold on the H-type groups.

#### GCOE lecture series

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

Representations of classical Lie super-algebras

https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar.html

**Michel Duflo**(Paris 7)Representations of classical Lie super-algebras

[ Abstract ]

In this lecture, I'll survey classical topics on finite dimensional representations of classical Lie super-algebras, in particular the notion of the degree of atypicality.

[ Reference URL ]In this lecture, I'll survey classical topics on finite dimensional representations of classical Lie super-algebras, in particular the notion of the degree of atypicality.

https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar.html

#### Number Theory Seminar

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

On GAGA theorems for the rigide-étale topology

**Ahmed Abbes**(Université de Rennes 1)On GAGA theorems for the rigide-étale topology

[ Abstract ]

Last year, I finished my course in Todai on "Rigide Geometry following M. Raynaud" by stating a GAGA theorem for the rigide-étale topology, due to Gabber and Fujiwara. I will give a new proof of this theorem, inspired by another theorem of Gabber, namely the Affine analog of the proper base change theorem.

Last year, I finished my course in Todai on "Rigide Geometry following M. Raynaud" by stating a GAGA theorem for the rigide-étale topology, due to Gabber and Fujiwara. I will give a new proof of this theorem, inspired by another theorem of Gabber, namely the Affine analog of the proper base change theorem.

### 2009/10/05

#### GCOE lecture series

15:30-17:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Macroscopic fluctuation theory for nonequilibrium stationary states, Ⅲ

**Claudio Landim**(IMPA, Brazil)Macroscopic fluctuation theory for nonequilibrium stationary states, Ⅲ

#### GCOE lecture series

13:30-15:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Mini course on the gradient models, I: Effective gradient models, definitions and examples

**Jean-Dominique Deuschel**(TU Berlin)Mini course on the gradient models, I: Effective gradient models, definitions and examples

[ Abstract ]

We describe a phase separation in $R^{d+1}$ by an effective interface model with basis in $Z^d$ and height in $R$. We assume that the interaction potential depends only on the discrete gradient and that the a priori measure is the product Lebesgue measure. Note that this is an unbounded massless model with continuous symmetry and this implies that the interface is delocalized for the infinite model in lower lattice dimensions $d=1,2$. Instead of looking at the distribution of the height of the interface itself, we consider the measure on the height differences the so called gradient Gibbs measure, which exists in any dimensions. The gradient field must satisfy the loop condition, that is the sum of the gradient along any closed loop is zero, this implies a long range interaction with a slow decay of the correlations. We are interested in characterizing the ergodic components of this gradient field, in the decay of correlations, large deviations and continuous scaling limits. As an example we consider the harmonic or discrete gaussian free field with quadratic interactions.

We describe a phase separation in $R^{d+1}$ by an effective interface model with basis in $Z^d$ and height in $R$. We assume that the interaction potential depends only on the discrete gradient and that the a priori measure is the product Lebesgue measure. Note that this is an unbounded massless model with continuous symmetry and this implies that the interface is delocalized for the infinite model in lower lattice dimensions $d=1,2$. Instead of looking at the distribution of the height of the interface itself, we consider the measure on the height differences the so called gradient Gibbs measure, which exists in any dimensions. The gradient field must satisfy the loop condition, that is the sum of the gradient along any closed loop is zero, this implies a long range interaction with a slow decay of the correlations. We are interested in characterizing the ergodic components of this gradient field, in the decay of correlations, large deviations and continuous scaling limits. As an example we consider the harmonic or discrete gaussian free field with quadratic interactions.

#### Algebraic Geometry Seminar

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

代数曲面上の随伴束の基底点集合について

**伊藤 敦**(東大数理)代数曲面上の随伴束の基底点集合について

[ Abstract ]

偏極付き代数多様体上(X,L)は、Lに数値的な条件を付け加えると

その随伴束が自由になったり、基底点集合が具体的にかけることがある。しかし

、曲線の場合は簡単であるが高次元の場合は難しい。今回の講演では主に代数曲

面の場合について解説する。

偏極付き代数多様体上(X,L)は、Lに数値的な条件を付け加えると

その随伴束が自由になったり、基底点集合が具体的にかけることがある。しかし

、曲線の場合は簡単であるが高次元の場合は難しい。今回の講演では主に代数曲

面の場合について解説する。

### 2009/10/02

#### Lecture Series on Mathematical Sciences in Soceity

16:20-17:50 Room #117 (Graduate School of Math. Sci. Bldg.)

暗号の基礎編

**岡本龍明**(NTT 情報流通プラットフォーム研究所 岡本特別研究室長)暗号の基礎編

### 2009/09/30

#### GCOE lecture series

15:30-17:00 Room #123 (Graduate School of Math. Sci. Bldg.)

Macroscopic fluctuation theory for nonequilibrium stationary states, Ⅱ

**Claudio Landim**(IMPA, Brazil)Macroscopic fluctuation theory for nonequilibrium stationary states, Ⅱ

#### Seminar on Probability and Statistics

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

HDLSSデータにおけるPCAについて

https://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2009/05.html

**矢田 和善**(筑波大学大学院数理物質科学研究科)HDLSSデータにおけるPCAについて

[ Abstract ]

マイクロアレイデータなどに見られるように,データの次元数dが標本数nよりも遥かに大きな高次元小標本(HDLSS)データが,解析対象になる場面が増えてきている.

HDLSSデータに対して従来の統計手法を用いると,次元の呪いによって解析が上手くいかない.解決策の一つとして次元縮約法があり, その一つにPCAがある.高次元における従来型のPCAの漸近的性質は,正規性もしくは同等な仮定のもとで,先行研究が多数存在する. しかしながら,これら仮定は,HDLSSを研究する上で,厳しい制約にもなっている. Yata and Aoshimaの一連の研究は,この制約条件の枠を外すことから始まった.HDLSSにおける従来型PCAの限界は何か?推測が一致性をもつための標本数nと 次元数dの関係が,オーダー条件として明らかにされる.従来型PCAの限界を超える手法は何か?一つの実用的な方法として,クロス行列と呼ばれるデータの変換行列が導入され, この行列の特異値分解に基づいた新しいPCAが提案される.

当日は,マイクロアレイデータによる実例と,シミュレーション結果も交えながら,お話します.本研究は,筑波大学数理物質科学研究科の青嶋誠先生との共同研究です.

[ Reference URL ]マイクロアレイデータなどに見られるように,データの次元数dが標本数nよりも遥かに大きな高次元小標本(HDLSS)データが,解析対象になる場面が増えてきている.

HDLSSデータに対して従来の統計手法を用いると,次元の呪いによって解析が上手くいかない.解決策の一つとして次元縮約法があり, その一つにPCAがある.高次元における従来型のPCAの漸近的性質は,正規性もしくは同等な仮定のもとで,先行研究が多数存在する. しかしながら,これら仮定は,HDLSSを研究する上で,厳しい制約にもなっている. Yata and Aoshimaの一連の研究は,この制約条件の枠を外すことから始まった.HDLSSにおける従来型PCAの限界は何か?推測が一致性をもつための標本数nと 次元数dの関係が,オーダー条件として明らかにされる.従来型PCAの限界を超える手法は何か?一つの実用的な方法として,クロス行列と呼ばれるデータの変換行列が導入され, この行列の特異値分解に基づいた新しいPCAが提案される.

当日は,マイクロアレイデータによる実例と,シミュレーション結果も交えながら,お話します.本研究は,筑波大学数理物質科学研究科の青嶋誠先生との共同研究です.

https://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2009/05.html

### 2009/09/29

#### Tuesday Seminar on Topology

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

Symbol of the Conway polynomial and Drinfeld associator

**Sergei Duzhin**(Steklov Mathematical Institute, Petersburg Division)Symbol of the Conway polynomial and Drinfeld associator

[ Abstract ]

The Magnus expansion is a universal finite type invariant of pure braids

with values in the space of horizontal chord diagrams. The Conway polynomial

composed with the short circuit map from braids to knots gives rise to a

series of finite type invariants of pure braids and thus factors through

the Magnus map. We describe explicitly the resulting mapping from horizontal

chord diagrams on 3 strands to univariante polynomials and evaluate it on

the Drinfeld associator obtaining a beautiful generating function whose

coefficients are integer combinations of multple zeta values.

The Magnus expansion is a universal finite type invariant of pure braids

with values in the space of horizontal chord diagrams. The Conway polynomial

composed with the short circuit map from braids to knots gives rise to a

series of finite type invariants of pure braids and thus factors through

the Magnus map. We describe explicitly the resulting mapping from horizontal

chord diagrams on 3 strands to univariante polynomials and evaluate it on

the Drinfeld associator obtaining a beautiful generating function whose

coefficients are integer combinations of multple zeta values.

### 2009/09/28

#### GCOE lecture series

15:30-17:00 Room #123 (Graduate School of Math. Sci. Bldg.)

Macroscopic fluctuation theory for nonequilibrium stationary states, I

**Claudio Landim**(IMPA, Brazil)Macroscopic fluctuation theory for nonequilibrium stationary states, I

[ Abstract ]

We present a review of recent work on the statistical mechanics of nonequilibrium processes based on the analysis of large deviations properties of microscopic systems. Stochastic lattice gases are non trivial models of such phenomena and can be studied rigorously providing a source of challenging mathematical problems. In this way, some principles of wide validity have been obtained leading to interesting physical consequences.

We present a review of recent work on the statistical mechanics of nonequilibrium processes based on the analysis of large deviations properties of microscopic systems. Stochastic lattice gases are non trivial models of such phenomena and can be studied rigorously providing a source of challenging mathematical problems. In this way, some principles of wide validity have been obtained leading to interesting physical consequences.

### 2009/09/17

#### Applied Analysis

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

On a parabolic free boundary problem modelling price formation

**Norayr MATEVOSYAN**(ケンブリッジ大学・数理)On a parabolic free boundary problem modelling price formation

[ Abstract ]

We will discuss existence and uniqueness of solutions for a one dimensional parabolic evolution equation with a free boundary. This problem was introduced by J.-M. Lasry and P.-L. Lions as description of the dynamical formation of the price of a trading good. Short time existence and uniqueness is established by a contraction argument. Then we discuss the issue of global-in time-extension of the local solution which is intimately connected to the regularity of the free boundary.

We also present numerical results.

We will discuss existence and uniqueness of solutions for a one dimensional parabolic evolution equation with a free boundary. This problem was introduced by J.-M. Lasry and P.-L. Lions as description of the dynamical formation of the price of a trading good. Short time existence and uniqueness is established by a contraction argument. Then we discuss the issue of global-in time-extension of the local solution which is intimately connected to the regularity of the free boundary.

We also present numerical results.

### 2009/09/15

#### Tuesday Seminar of Analysis

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

渦層の超局所解析

**打越 敬祐**(防衛大学校数学教育室)渦層の超局所解析

[ Abstract ]

渦層とは,2次元流体が界面を境に2層に分かれて流れる状態で,今井功氏は「佐藤超関数は渦層である」と言っている.この考え方を用いると,

界面の時間変化を記述するBirkoff-Rott方程式を,擬微分方程式に書き直して解けることを説明する.

渦層とは,2次元流体が界面を境に2層に分かれて流れる状態で,今井功氏は「佐藤超関数は渦層である」と言っている.この考え方を用いると,

界面の時間変化を記述するBirkoff-Rott方程式を,擬微分方程式に書き直して解けることを説明する.

### 2009/09/14

#### Number Theory Seminar

11:00-12:00 Room #123 (Graduate School of Math. Sci. Bldg.)

Modular forms and Calabi-Yau varieties

**Dinakar Ramakrishnan**(カリフォルニア工科大学)Modular forms and Calabi-Yau varieties

### 2009/09/10

#### Applied Analysis

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

A two phase free boundary problem with applications in potential theory

**Henrik SHAHGHOLIAN**(王立工科大学・ストックホルム)A two phase free boundary problem with applications in potential theory

[ Abstract ]

In this talk I will present some recent directions, still to be developed, in potential theory, that are connected to a two-phase free boundary problems. The potential theoretic topic that I will discuss is the so called Quadrature Domains.

The most simple free boundary/potential problem that we can present is the following. Given constants $a_\\pm, \\lambda_\\pm >0$ and two points $x^\\pm$ in ${\\bf R}^n$. Find a function $u$ such that

$$\\Delta u = \\left( \\lambda_+ \\chi_{\\{u>0 \\}} - a_+\\delta_{x^+}\\right) - \\left( \\lambda_- \\chi_{\\{u<0 \\}} - a_-\\delta_{x^-}\\right),$$

where $\\delta$ is the Dirac mass.

In general this problem is solvable for two Dirac masses. The requirement, somehow implicit in the above equation, is that the support of the measures (in this case the Dirac masses) is to be in included in the positivity and the negativity set (respectively).

In general this problem does not have a solution, and there some strong restrictions on the measures, in order to have some partial results.

In this talk I will present some recent directions, still to be developed, in potential theory, that are connected to a two-phase free boundary problems. The potential theoretic topic that I will discuss is the so called Quadrature Domains.

The most simple free boundary/potential problem that we can present is the following. Given constants $a_\\pm, \\lambda_\\pm >0$ and two points $x^\\pm$ in ${\\bf R}^n$. Find a function $u$ such that

$$\\Delta u = \\left( \\lambda_+ \\chi_{\\{u>0 \\}} - a_+\\delta_{x^+}\\right) - \\left( \\lambda_- \\chi_{\\{u<0 \\}} - a_-\\delta_{x^-}\\right),$$

where $\\delta$ is the Dirac mass.

In general this problem is solvable for two Dirac masses. The requirement, somehow implicit in the above equation, is that the support of the measures (in this case the Dirac masses) is to be in included in the positivity and the negativity set (respectively).

In general this problem does not have a solution, and there some strong restrictions on the measures, in order to have some partial results.

### 2009/09/08

#### GCOE Seminars

15:00-16:00 Room #123 (Graduate School of Math. Sci. Bldg.)

Global compact attractors and their tripartition under persistence (ENGLISH)

**H.R.Thieme**(Arizona State University)Global compact attractors and their tripartition under persistence (ENGLISH)

[ Abstract ]

The study of the dynamics of a semiflow (inertial manifolds, persistence) is largely facilitated if there is a global compact attractor, i.e. a compact invariant subset which attracts a sufficiently broad class of subsets of the state space.

Unfortunately, there in no uniform use of the concept of a global compact attractor in the literature: it has been used for a compact attractor of points, compact attractor of neighborhoods of compact sets, and compact attractor of bounded sets.

Persistence theory allows to discuss the long-term survival of populations in a dynamical systems framework. There is a two-way interaction between persistence and global compact attractors. On the one hand, the existence of a compact attractor of points helps to establish the persistence of the semiflow. On the other hand, the global attractor of a uniformly persistent semiflow divides into three invariant parts: an extinction attractor, a persistence attractor, and a set of orbits that connect the extinction to the persistence attractor. The persistence attractor has further interesting properties like local stability and connectedness. Examples are presented where the persistence attractor can be used to prove the global stability of the persistence equilibrium. (joint work with Hal L. Smith)

The study of the dynamics of a semiflow (inertial manifolds, persistence) is largely facilitated if there is a global compact attractor, i.e. a compact invariant subset which attracts a sufficiently broad class of subsets of the state space.

Unfortunately, there in no uniform use of the concept of a global compact attractor in the literature: it has been used for a compact attractor of points, compact attractor of neighborhoods of compact sets, and compact attractor of bounded sets.

Persistence theory allows to discuss the long-term survival of populations in a dynamical systems framework. There is a two-way interaction between persistence and global compact attractors. On the one hand, the existence of a compact attractor of points helps to establish the persistence of the semiflow. On the other hand, the global attractor of a uniformly persistent semiflow divides into three invariant parts: an extinction attractor, a persistence attractor, and a set of orbits that connect the extinction to the persistence attractor. The persistence attractor has further interesting properties like local stability and connectedness. Examples are presented where the persistence attractor can be used to prove the global stability of the persistence equilibrium. (joint work with Hal L. Smith)

#### GCOE Seminars

16:15-17:15 Room #123 (Graduate School of Math. Sci. Bldg.)

Analysis of a Model for Transfer Phenomena in Biological Populations (ENGLISH)

**Glenn Webb**(Vanderbilt University)Analysis of a Model for Transfer Phenomena in Biological Populations (ENGLISH)

[ Abstract ]

We study the problem of transfer in a population structured by a continuum variable corresponding to the quantity being transferred. The transfer of the quantity occurs between individuals according to specified rules. The model is of Boltzmann type with kernel corresponding to the transfer process. We prove that the transfer process preserves total mass of the transferred quantity and the solutions of the simple model converge weakly to Radon measures. We generalize the model by introducing proliferation of individuals and production and diffusion of the transferable quantity. It is shown that the generalized model admits a globally asymptotically stable steady state, provided that transfer is sufficiently small. We discuss an application of our model to cancer cell populations, in which individual cells exchange the surface protein P-glycoprotein, an important factor in acquired multidrug resistance against cancer chemotherapy.

We study the problem of transfer in a population structured by a continuum variable corresponding to the quantity being transferred. The transfer of the quantity occurs between individuals according to specified rules. The model is of Boltzmann type with kernel corresponding to the transfer process. We prove that the transfer process preserves total mass of the transferred quantity and the solutions of the simple model converge weakly to Radon measures. We generalize the model by introducing proliferation of individuals and production and diffusion of the transferable quantity. It is shown that the generalized model admits a globally asymptotically stable steady state, provided that transfer is sufficiently small. We discuss an application of our model to cancer cell populations, in which individual cells exchange the surface protein P-glycoprotein, an important factor in acquired multidrug resistance against cancer chemotherapy.

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