## Lectures

Seminar information archive ～06/12｜Next seminar｜Future seminars 06/13～

**Seminar information archive**

### 2013/01/30

17:15-18:15 Room #118 (Graduate School of Math. Sci. Bldg.)

Group actions with Rohlin property (ENGLISH)

**Hiroki Asano**(Univ. Tokyo)Group actions with Rohlin property (ENGLISH)

### 2013/01/16

10:00-11:00 Room #123 (Graduate School of Math. Sci. Bldg.)

$W^*$-superrigidity of mixing Gaussian actions of rigid groups (ENGLISH)

**R\'emi Boutonnet**(ENS Lyon)$W^*$-superrigidity of mixing Gaussian actions of rigid groups (ENGLISH)

### 2013/01/16

11:30-12:30 Room #123 (Graduate School of Math. Sci. Bldg.)

The Approximation Property for Lie groups (ENGLISH)

**Tim de Laat**(University of Copenhagen)The Approximation Property for Lie groups (ENGLISH)

### 2013/01/16

14:40-15:40 Room #118 (Graduate School of Math. Sci. Bldg.)

Unique Cartan decomposition for II$_1$ factors arising from cross section equivalence relations (ENGLISH)

**Arnaud Brothier**(KU Leuven)Unique Cartan decomposition for II$_1$ factors arising from cross section equivalence relations (ENGLISH)

### 2013/01/16

15:55-16:55 Room #118 (Graduate School of Math. Sci. Bldg.)

Rigid $C^*$ tensor categories of bimodules over interpolated

free group factors (ENGLISH)

**Michael Hartglass**(UC Berkeley)Rigid $C^*$ tensor categories of bimodules over interpolated

free group factors (ENGLISH)

### 2013/01/16

17:10-18:10 Room #118 (Graduate School of Math. Sci. Bldg.)

Manifestly unitary conformal field theory (ENGLISH)

**James Tener**(UC Berkeley)Manifestly unitary conformal field theory (ENGLISH)

### 2013/01/11

10:00-11:00 Room #123 (Graduate School of Math. Sci. Bldg.)

A duality between easy quantum groups and reflection groups (ENGLISH)

**Sven Raum**(KU Leuven)A duality between easy quantum groups and reflection groups (ENGLISH)

### 2013/01/11

11:15-12:15 Room #123 (Graduate School of Math. Sci. Bldg.)

Some non-uniqueness results for Cartan subalgebras in II$_1$ factors (ENGLISH)

**An Speelman**(KU Leuven)Some non-uniqueness results for Cartan subalgebras in II$_1$ factors (ENGLISH)

### 2013/01/11

14:00-15:00 Room #123 (Graduate School of Math. Sci. Bldg.)

Structural results for II$_1$ factors of negatively curved groups (ENGLISH)

**Ionut Chifan**(University of Iowa)Structural results for II$_1$ factors of negatively curved groups (ENGLISH)

### 2013/01/11

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

UHF slicing and classification of nuclear $C^*$-algebras (ENGLISH)

**Karen Strung**(Universit\"at M\"unster)UHF slicing and classification of nuclear $C^*$-algebras (ENGLISH)

### 2013/01/11

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

The generator problem for $C^*$-algebras (ENGLISH)

**Hannes Thiel**(University of Copenhagen)The generator problem for $C^*$-algebras (ENGLISH)

### 2012/11/28

10:45-11:45 Room #002 (Graduate School of Math. Sci. Bldg.)

Pattern formation in the hyperbolic plane (ENGLISH)

**Pascal Chossat**(CNRS / University of Nice)Pattern formation in the hyperbolic plane (ENGLISH)

[ Abstract ]

Initially motivated by a model for the visual perception of textures by the cortex, the problem of pattern formation in the hyperbolic plane, or equivalently the Poincaré disc D, shows some similar but mostly quite different features from the same problem posed on the Euclidean plane. The hyperbolic structure induces a large variety of possible periodic patterns and even the bifurcation of "hyperbolic" traveling waves. We call these patterns "H-planforms". I shall show how H-planforms are determined by the means of equivariant bifurcation theory and Helgason-Fourier analysis in D. However the question of their observability is still open. The talk will be illustrated with pictures of H-planforms that have been computed using non trivial algorithms based on harmonic analysis in D.

Initially motivated by a model for the visual perception of textures by the cortex, the problem of pattern formation in the hyperbolic plane, or equivalently the Poincaré disc D, shows some similar but mostly quite different features from the same problem posed on the Euclidean plane. The hyperbolic structure induces a large variety of possible periodic patterns and even the bifurcation of "hyperbolic" traveling waves. We call these patterns "H-planforms". I shall show how H-planforms are determined by the means of equivariant bifurcation theory and Helgason-Fourier analysis in D. However the question of their observability is still open. The talk will be illustrated with pictures of H-planforms that have been computed using non trivial algorithms based on harmonic analysis in D.

### 2012/11/22

13:30-14:15 Room #002 (Graduate School of Math. Sci. Bldg.)

A multiple scale pattern formation cascade in reaction-diffusion systems of activator-inhibitor type (ENGLISH)

https://www.ms.u-tokyo.ac.jp/gcoe/index_007.html

**Danielle Hilhorst**(CNRS / Univ. Paris-Sud)A multiple scale pattern formation cascade in reaction-diffusion systems of activator-inhibitor type (ENGLISH)

[ Abstract ]

A family of singular limits of reaction-diffusion systems of activator-inhibitor type in which stable stationary sharp-interface patterns may form is investigated. For concreteness, the analysis is performed for the FitzHugh-Nagumo model on a suitably rescaled bounded domain in $\\R^N$, with $N \\geq 2$. It is proved that when the system is sufficiently close to the limit the dynamics starting from the appropriate smooth initial data breaks down into five distinct stages on well-separated time scales, each of which can be approximated by a suitable reduced problem. The analysis allows to follow fully the progressive refinement of spatiotemporal patterns forming in the systems under consideration and provides a framework for understanding the pattern formation scenarios in a large class of physical, chemical, and biological systems modeled by the class of reaction-diffusion equations, which we consider. This is joint work with Marie Henry and Cyrill Muratov.

[ Reference URL ]A family of singular limits of reaction-diffusion systems of activator-inhibitor type in which stable stationary sharp-interface patterns may form is investigated. For concreteness, the analysis is performed for the FitzHugh-Nagumo model on a suitably rescaled bounded domain in $\\R^N$, with $N \\geq 2$. It is proved that when the system is sufficiently close to the limit the dynamics starting from the appropriate smooth initial data breaks down into five distinct stages on well-separated time scales, each of which can be approximated by a suitable reduced problem. The analysis allows to follow fully the progressive refinement of spatiotemporal patterns forming in the systems under consideration and provides a framework for understanding the pattern formation scenarios in a large class of physical, chemical, and biological systems modeled by the class of reaction-diffusion equations, which we consider. This is joint work with Marie Henry and Cyrill Muratov.

https://www.ms.u-tokyo.ac.jp/gcoe/index_007.html

### 2012/11/22

14:25-15:10 Room #002 (Graduate School of Math. Sci. Bldg.)

Formal asymptotic limit of a diffuse interface tumor-growth model (ENGLISH)

https://www.ms.u-tokyo.ac.jp/gcoe/index_007.html

**Thanh Nam Ngyuen**(University of Paris-Sud)Formal asymptotic limit of a diffuse interface tumor-growth model (ENGLISH)

[ Abstract ]

We consider a diffuse interface tumor-growth model, which has the form of a phase-field system. We discuss the singular limit of this problem. More precisely, we formally prove that as the reaction coefficient tends to zero, the solution converges to the solution of a free boundary problem.

This is a joint work with Danielle Hilhorst, Johannes Kampmann and Kristoffer G. van der Zee.

[ Reference URL ]We consider a diffuse interface tumor-growth model, which has the form of a phase-field system. We discuss the singular limit of this problem. More precisely, we formally prove that as the reaction coefficient tends to zero, the solution converges to the solution of a free boundary problem.

This is a joint work with Danielle Hilhorst, Johannes Kampmann and Kristoffer G. van der Zee.

https://www.ms.u-tokyo.ac.jp/gcoe/index_007.html

### 2012/11/22

15:30-16:15 Room #002 (Graduate School of Math. Sci. Bldg.)

Gelfand type problem for two phase porous media (ENGLISH)

https://www.ms.u-tokyo.ac.jp/gcoe/index_007.html

**Peter Gordon**(Akron University)Gelfand type problem for two phase porous media (ENGLISH)

[ Abstract ]

In this talk I will introduce a generalization of well known Gelfand problem arising in a Frank-Kamenetskii theory of thermal explosion. This generalization is a natural extension of the Gelfand problem to two phase materials, where, in contrast to classical Gelfand problem which utilizes single temperature approach, the state of the system is described by two different temperatures. As a result the problem is modeled by a system of two coupled nonlinear heat equations. The new ingredient in such a generalized Gelfand problem is a presence of inter-phase heat exchange which can be viewed as a strength of coupling for the system.

I will show that similar to classical Gelfand problem the thermal explosion (blow up of solution) for generalized Gelfand problem occurs exclusively due to the absence of stationary temperature distribution, that is non-existence of solution of corresponding elliptic problem. I also will show that the presence of inter-phase heat exchange delays a thermal explosion. Moreover, in the limit of infinite heat exchange between phases the problem of thermal explosion in two phase porous media reduces to classical Gelfand problem with re-normalized constants. The latter result partially justifies a single temperature approach to two phase systems often used in a physical literature.

This is a joint work with Vitaly Moroz (Swansea University).

[ Reference URL ]In this talk I will introduce a generalization of well known Gelfand problem arising in a Frank-Kamenetskii theory of thermal explosion. This generalization is a natural extension of the Gelfand problem to two phase materials, where, in contrast to classical Gelfand problem which utilizes single temperature approach, the state of the system is described by two different temperatures. As a result the problem is modeled by a system of two coupled nonlinear heat equations. The new ingredient in such a generalized Gelfand problem is a presence of inter-phase heat exchange which can be viewed as a strength of coupling for the system.

I will show that similar to classical Gelfand problem the thermal explosion (blow up of solution) for generalized Gelfand problem occurs exclusively due to the absence of stationary temperature distribution, that is non-existence of solution of corresponding elliptic problem. I also will show that the presence of inter-phase heat exchange delays a thermal explosion. Moreover, in the limit of infinite heat exchange between phases the problem of thermal explosion in two phase porous media reduces to classical Gelfand problem with re-normalized constants. The latter result partially justifies a single temperature approach to two phase systems often used in a physical literature.

This is a joint work with Vitaly Moroz (Swansea University).

https://www.ms.u-tokyo.ac.jp/gcoe/index_007.html

### 2012/11/22

16:25-17:10 Room #002 (Graduate School of Math. Sci. Bldg.)

On the shape of charged drops: an isoperimetric problem with a competing non-local term (ENGLISH)

https://www.ms.u-tokyo.ac.jp/gcoe/index_007.html

**Cyrill Muratov**(New Jersey Institute of Technology)On the shape of charged drops: an isoperimetric problem with a competing non-local term (ENGLISH)

[ Abstract ]

In this talk I will give an overview of my recent work with H. Knuepfer on the analysis of a class of geometric problems in the calculus of variations. I will discuss the basic questions of existence and non-existence of energy minimizers for the isoperimetric problem with a competing non-local term. A complete answer will be given for the case of slowly decaying kernels in two space dimensions, and qualitative properties of the minimizers will be established for general Riesz kernels.

[ Reference URL ]In this talk I will give an overview of my recent work with H. Knuepfer on the analysis of a class of geometric problems in the calculus of variations. I will discuss the basic questions of existence and non-existence of energy minimizers for the isoperimetric problem with a competing non-local term. A complete answer will be given for the case of slowly decaying kernels in two space dimensions, and qualitative properties of the minimizers will be established for general Riesz kernels.

https://www.ms.u-tokyo.ac.jp/gcoe/index_007.html

### 2012/11/19

16:45-17:45 Room #126 (Graduate School of Math. Sci. Bldg.)

Invariant measure of the stochastic Allen-Cahn equation: the regime of small noise and large system size (ENGLISH)

**Hendrik Weber**(University of Warwick)Invariant measure of the stochastic Allen-Cahn equation: the regime of small noise and large system size (ENGLISH)

[ Abstract ]

We study the invariant measure of the one-dimensional stochastic Allen-Cahn equation for a small noise strength and a large but finite system. We endow the system with inhomogeneous Dirichlet boundary conditions that enforce at least one transition from -1 to 1. We are interested in the competition between the ``energy'' that should be minimized due to the small noise strength and the ``entropy'' that is induced by the large system size.

Our methods handle system sizes that are exponential with respect to the inverse noise strength, up to the ``critical'' exponential size predicted by the heuristics. We capture the competition between energy and entropy through upper and lower bounds on the probability of extra transitions between $\\pm 1$. These bounds are sharp on the exponential scale and imply in particular that the probability of having one and only one transition from -1 to +1 is exponentially close to one. In addition, we show that the position of the transition layer is uniformly distributed over the system on scales larger than the logarithm of the inverse noise strength.

Our arguments rely on local large deviation bounds, the strong Markov property, the symmetry of the potential, and measure-preserving reflections.

This is a joint work with Felix Otto and Maria Westdickenberg.

We study the invariant measure of the one-dimensional stochastic Allen-Cahn equation for a small noise strength and a large but finite system. We endow the system with inhomogeneous Dirichlet boundary conditions that enforce at least one transition from -1 to 1. We are interested in the competition between the ``energy'' that should be minimized due to the small noise strength and the ``entropy'' that is induced by the large system size.

Our methods handle system sizes that are exponential with respect to the inverse noise strength, up to the ``critical'' exponential size predicted by the heuristics. We capture the competition between energy and entropy through upper and lower bounds on the probability of extra transitions between $\\pm 1$. These bounds are sharp on the exponential scale and imply in particular that the probability of having one and only one transition from -1 to +1 is exponentially close to one. In addition, we show that the position of the transition layer is uniformly distributed over the system on scales larger than the logarithm of the inverse noise strength.

Our arguments rely on local large deviation bounds, the strong Markov property, the symmetry of the potential, and measure-preserving reflections.

This is a joint work with Felix Otto and Maria Westdickenberg.

### 2012/10/30

17:00-18:00 Room #470 (Graduate School of Math. Sci. Bldg.)

Discrete Topologgy of Cellular Microstructures

and Complicatedness Measurements for Cell Complexes (JAPANESE)

**Frank Lutz**(Technische Universität Berlin)Discrete Topologgy of Cellular Microstructures

and Complicatedness Measurements for Cell Complexes (JAPANESE)

[ Abstract ]

Our first aim is to use methods from discrete and geometric topology

to recover structural information from the composition of

monocrystalline materials that have a periodic foam structure

(such as gas hydrates and transition metal alloys) and also of

polycrystalline materials (such as metals and certain ceramics).

For more general complexes, even with a billion of faces, homological

information can be obtained with computational homology packages

such as CHomP or RedHom. These packages extensively use discrete Morse

theory as a preprocessing step. Although it is NP-hard to find optimal

discrete Morse functions, most data appears to be easy and it is

in fact hard to construct ``complicated'' examples. As we will see,

random discrete Morse theory will allow us to measure the

``complicatedness'' of complexes.

Our first aim is to use methods from discrete and geometric topology

to recover structural information from the composition of

monocrystalline materials that have a periodic foam structure

(such as gas hydrates and transition metal alloys) and also of

polycrystalline materials (such as metals and certain ceramics).

For more general complexes, even with a billion of faces, homological

information can be obtained with computational homology packages

such as CHomP or RedHom. These packages extensively use discrete Morse

theory as a preprocessing step. Although it is NP-hard to find optimal

discrete Morse functions, most data appears to be easy and it is

in fact hard to construct ``complicated'' examples. As we will see,

random discrete Morse theory will allow us to measure the

``complicatedness'' of complexes.

### 2012/10/18

16:00-17:00 Room #470 (Graduate School of Math. Sci. Bldg.)

Multidimensional ill-posed problems (ENGLISH)

**Anatoly Yagola**(Lomonosov Moscow State University)Multidimensional ill-posed problems (ENGLISH)

[ Abstract ]

It is very important now to develop methods of solving multidimensional ill-posed problems using regularization procedures and parallel computers. The main purpose of the talk is to show how 2D and 3D Fredholm integral equations of the 1st kind can be effectively solved.

We will consider ill-posed problems on compact sets of convex functions [1] and functions convex along lines parallel to coordinate axes [2].

Recovery of magnetic target parameters from magnetic sensor measurements has attracted wide interests and found many practical applications. However, difficulties present in identifying the permanent magnetization due to the complications of magnetization distributions over the ship body, and errors and noises of measurement data degrade the accuracy and quality of the parameter identification. In this paper, we use a two step sequential solutions to solve the inversion problem. In the first step, a numerical model is built and used to determine the induced magnetization of the ship. In the second step, we solve a type of continuous magnetization inversion problem by solving 2D and 3D Fredholm integral

equations of the 1st kind. We use parallel computing which allows solve the inverse problem with high accuracy. Tikhonov regularization has been applied in solving the inversion problems. The proposed methods have been validated using simulation data with added noises [4, 6].

2D and 3D inverse problems also could be found in tomography [3] and electron microscopy [5]. We will demonstrate examples of applied problems and discuss methods of numerical solving.

This paper was supported by the Visby program and RFBR grants 11-01-00040–а and 12-01-91153-NSFC-a.

It is very important now to develop methods of solving multidimensional ill-posed problems using regularization procedures and parallel computers. The main purpose of the talk is to show how 2D and 3D Fredholm integral equations of the 1st kind can be effectively solved.

We will consider ill-posed problems on compact sets of convex functions [1] and functions convex along lines parallel to coordinate axes [2].

Recovery of magnetic target parameters from magnetic sensor measurements has attracted wide interests and found many practical applications. However, difficulties present in identifying the permanent magnetization due to the complications of magnetization distributions over the ship body, and errors and noises of measurement data degrade the accuracy and quality of the parameter identification. In this paper, we use a two step sequential solutions to solve the inversion problem. In the first step, a numerical model is built and used to determine the induced magnetization of the ship. In the second step, we solve a type of continuous magnetization inversion problem by solving 2D and 3D Fredholm integral

equations of the 1st kind. We use parallel computing which allows solve the inverse problem with high accuracy. Tikhonov regularization has been applied in solving the inversion problems. The proposed methods have been validated using simulation data with added noises [4, 6].

2D and 3D inverse problems also could be found in tomography [3] and electron microscopy [5]. We will demonstrate examples of applied problems and discuss methods of numerical solving.

This paper was supported by the Visby program and RFBR grants 11-01-00040–а and 12-01-91153-NSFC-a.

### 2012/10/17

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

Inverse problems for differential operators on spatial networks (ENGLISH)

**Vjacheslav Yurko**(Saratov University)Inverse problems for differential operators on spatial networks (ENGLISH)

### 2012/07/23

16:30-17:30 Room #126 (Graduate School of Math. Sci. Bldg.)

Combinatorial Ergodicity (ENGLISH)

**Thomas W. Roby**(University of Connecticut)Combinatorial Ergodicity (ENGLISH)

[ Abstract ]

Many cyclic actions $\\tau$ on a finite set $S$ of

combinatorial objects, along with many natural

statistics $\\phi$ on $S$, exhibit``combinatorial ergodicity'':

the average of $\\phi$ over each $\\tau$-orbit in $S$ is

the same as the average of $\\phi$ over the whole set $S$.

One example is the case where $S$ is the set of

length $n$ binary strings $a_{1}\\dots a_{n}$

with exactly $k$ 1's,

$\\tau$ is the map that cyclically rotates them,

and $\\phi$ is the number of \\textit{inversions}

(i.e, pairs $(a_{i},a_{j})=(1,0)$ for $iJ$ less than $j$).

This phenomenon was first noticed by Panyushev

in 2007 in the context of antichains in root posets;

Armstrong, Stump, and Thomas proved his

conjecture in 2011.

We describe a theoretical framework for results of this kind,

and discuss old and new results for products of two chains.

This is joint work with Jim Propp.

Many cyclic actions $\\tau$ on a finite set $S$ of

combinatorial objects, along with many natural

statistics $\\phi$ on $S$, exhibit``combinatorial ergodicity'':

the average of $\\phi$ over each $\\tau$-orbit in $S$ is

the same as the average of $\\phi$ over the whole set $S$.

One example is the case where $S$ is the set of

length $n$ binary strings $a_{1}\\dots a_{n}$

with exactly $k$ 1's,

$\\tau$ is the map that cyclically rotates them,

and $\\phi$ is the number of \\textit{inversions}

(i.e, pairs $(a_{i},a_{j})=(1,0)$ for $iJ$ less than $j$).

This phenomenon was first noticed by Panyushev

in 2007 in the context of antichains in root posets;

Armstrong, Stump, and Thomas proved his

conjecture in 2011.

We describe a theoretical framework for results of this kind,

and discuss old and new results for products of two chains.

This is joint work with Jim Propp.

### 2012/07/12

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

Real time tsunami parameters evaluation (ENGLISH)

**M. Lavrentiev**(Sobolev Institute of Mathematics)Real time tsunami parameters evaluation (ENGLISH)

[ Abstract ]

We would like to propose several improvements to the existing software tools for tsunami modeling. Combination of optimaly located system of sensors with advantages of modern hardware architectures will make it possible to deliver calculated parameters of tsunami wave in 12-15 minutes after seismic event.

We would like to propose several improvements to the existing software tools for tsunami modeling. Combination of optimaly located system of sensors with advantages of modern hardware architectures will make it possible to deliver calculated parameters of tsunami wave in 12-15 minutes after seismic event.

### 2012/06/18

09:45-12:30 Room #002 (Graduate School of Math. Sci. Bldg.)

Fluctuation of solutions to PDEs with random coefficients (Part 2) (ENGLISH)

https://www.ms.u-tokyo.ac.jp/~matano/SDE2012/

Weak coupling limits for particles and PDEs (Part 2) (ENGLISH)

https://www.ms.u-tokyo.ac.jp/~matano/SDE2012/

**James Nolen**(Duke University) 09:45-10:45Fluctuation of solutions to PDEs with random coefficients (Part 2) (ENGLISH)

[ Abstract ]

This is continuation of the previous day's lecture.

[ Reference URL ]This is continuation of the previous day's lecture.

https://www.ms.u-tokyo.ac.jp/~matano/SDE2012/

**Leonid Ryzhik**(Stanford University) 11:00-12:30Weak coupling limits for particles and PDEs (Part 2) (ENGLISH)

[ Abstract ]

This is continuation of the previous day's lecture.

[ Reference URL ]This is continuation of the previous day's lecture.

https://www.ms.u-tokyo.ac.jp/~matano/SDE2012/

### 2012/06/17

09:45-17:30 Room #002 (Graduate School of Math. Sci. Bldg.)

Fluctuation of solutions to PDEs with random coefficients (Part 1) (JAPANESE)

https://www.ms.u-tokyo.ac.jp/~matano/SDE2012/

Weak coupling limits for particles and PDEs (Part 1) (JAPANESE)

https://www.ms.u-tokyo.ac.jp/~matano/SDE2012/

Asymptotic spreading for heterogeneous Fisher-KPP reaction-diffusion equations (JAPANESE)

https://www.ms.u-tokyo.ac.jp/~matano/SDE2012/

**James Nolen**(Duke University) 09:45-17:30Fluctuation of solutions to PDEs with random coefficients (Part 1) (JAPANESE)

[ Abstract ]

For PDEs with random coefficients, it is interesting to understand whether the solutions exhibit some universal statistical behavior that is independent of the details of the coefficients. In particular, how do solutions fluctuate around the mean behavior? We will discuss this issue in the context of three examples:

(1) Traveling fronts in random media in one dimension.

(2) Elliptic homogenization problems.

(3) Random Hamilton-Jacobi equations.

The relation between PDE tools and probabilistic ideas will be

explained.

[ Reference URL ]For PDEs with random coefficients, it is interesting to understand whether the solutions exhibit some universal statistical behavior that is independent of the details of the coefficients. In particular, how do solutions fluctuate around the mean behavior? We will discuss this issue in the context of three examples:

(1) Traveling fronts in random media in one dimension.

(2) Elliptic homogenization problems.

(3) Random Hamilton-Jacobi equations.

The relation between PDE tools and probabilistic ideas will be

explained.

https://www.ms.u-tokyo.ac.jp/~matano/SDE2012/

**Leonid Ryzhik**(Stanford Univeristy) 13:00-14:45Weak coupling limits for particles and PDEs (Part 1) (JAPANESE)

[ Abstract ]

Weak random fluctuations in medium parameters may lead to a non-trivial effect after large times and propagation over long distances. We will consider several examples when the large time limit can be treated:

(1) a particle in a weakly random velocity field.

(2) weak random fluctuations of Hamilton equations, and

(3) the linear Scrhoedinger equation with a weak random potential.

The role of long range correlation of the random fluctuations will also be discussed.

[ Reference URL ]Weak random fluctuations in medium parameters may lead to a non-trivial effect after large times and propagation over long distances. We will consider several examples when the large time limit can be treated:

(1) a particle in a weakly random velocity field.

(2) weak random fluctuations of Hamilton equations, and

(3) the linear Scrhoedinger equation with a weak random potential.

The role of long range correlation of the random fluctuations will also be discussed.

https://www.ms.u-tokyo.ac.jp/~matano/SDE2012/

**Gregoire Nadin**(CNRS / Paris 6) 15:15-17:15Asymptotic spreading for heterogeneous Fisher-KPP reaction-diffusion equations (JAPANESE)

[ Abstract ]

The solutions of the heterogeneous Fisher-KPP equation associated with compactly supported initial data are known to take off from the unstable steady state 0 and to converge to the steady state 1 for large times. The aim of this lecture is to estimate the speed at which the interface between 0 and 1 spreads.

Using the new notion of generalized principal eigenvalues for non-compact elliptic operators, we will derive such estimates which will be proved to be optimal for several classes of heterogeneity such as periodic, almost periodic or random stationary ergodic ones.

[ Reference URL ]The solutions of the heterogeneous Fisher-KPP equation associated with compactly supported initial data are known to take off from the unstable steady state 0 and to converge to the steady state 1 for large times. The aim of this lecture is to estimate the speed at which the interface between 0 and 1 spreads.

Using the new notion of generalized principal eigenvalues for non-compact elliptic operators, we will derive such estimates which will be proved to be optimal for several classes of heterogeneity such as periodic, almost periodic or random stationary ergodic ones.

https://www.ms.u-tokyo.ac.jp/~matano/SDE2012/

### 2012/06/16

13:15-17:00 Room #002 (Graduate School of Math. Sci. Bldg.)

A viewpoint of the stochastic analysis in differential equations (JAPANESE)

[ Reference URL ]

https://www.ms.u-tokyo.ac.jp/~matano/SDE2012/

Stochastic (partial) differential equations from a functional analytic point of view (JAPANESE)

[ Reference URL ]

https://www.ms.u-tokyo.ac.jp/~matano/SDE2012/

**Tadahisa Funaki**(University of Tokyo) 13:15-14:45A viewpoint of the stochastic analysis in differential equations (JAPANESE)

[ Reference URL ]

https://www.ms.u-tokyo.ac.jp/~matano/SDE2012/

**Yoshiki Otobe**(Shinshu University) 15:00-17:00Stochastic (partial) differential equations from a functional analytic point of view (JAPANESE)

[ Reference URL ]

https://www.ms.u-tokyo.ac.jp/~matano/SDE2012/