## Lectures

Seminar information archive ～12/01｜Next seminar｜Future seminars 12/02～

**Seminar information archive**

### 2008/10/15

15:00-16:00 Room #570 (Graduate School of Math. Sci. Bldg.)

GCOEレクチャー"Holomorphic extensions of unitary representations" その2 "Geometric Background"

http://faculty.ms.u-tokyo.ac.jp/users/gcoe/GCOE_lecture0810Hilgert.html

**Joachim Hilgert**(Paderborn University)GCOEレクチャー"Holomorphic extensions of unitary representations" その2 "Geometric Background"

[ Abstract ]

In this lecture we will explain the complex geometry needed to understand the phenomena described in the first lecture. The key words here are Olshanski semigroups, invariant cones in Lie algebras, Akhiezer-Gindikin domain, and coadjoint orbits of convex type.

[ Reference URL ]In this lecture we will explain the complex geometry needed to understand the phenomena described in the first lecture. The key words here are Olshanski semigroups, invariant cones in Lie algebras, Akhiezer-Gindikin domain, and coadjoint orbits of convex type.

http://faculty.ms.u-tokyo.ac.jp/users/gcoe/GCOE_lecture0810Hilgert.html

### 2008/10/14

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

連続講演 "Thin 3D Navier-Stokes equations" (3次元薄層領域上のナビエストークス方程式) その1

Ultimate boundedness of solutions with large data and global attractors

**George Sell**(ミネソタ大学)連続講演 "Thin 3D Navier-Stokes equations" (3次元薄層領域上のナビエストークス方程式) その1

Ultimate boundedness of solutions with large data and global attractors

[ Abstract ]

In both lectures we will examine a new topic of the thin 3D Navier-Stokes equations with Navier boundary conditions.

In the first lecture we will treat the ultimate boundedness of strong solutions and the related theory of global attractors.

In the second lecture, which will include a brief summary of the first lecture, we will examine the role played by the 2D Limit Problem. These issues are a special challenge for analysis because the 2D Limit Problem is NOT imbedded the 3D problem.

These lectures are based on joint work with Genevieve Raugel, Dragos Iftimie, and Luan Hoang.

In both lectures we will examine a new topic of the thin 3D Navier-Stokes equations with Navier boundary conditions.

In the first lecture we will treat the ultimate boundedness of strong solutions and the related theory of global attractors.

In the second lecture, which will include a brief summary of the first lecture, we will examine the role played by the 2D Limit Problem. These issues are a special challenge for analysis because the 2D Limit Problem is NOT imbedded the 3D problem.

These lectures are based on joint work with Genevieve Raugel, Dragos Iftimie, and Luan Hoang.

### 2008/10/14

15:00-16:00 Room #570 (Graduate School of Math. Sci. Bldg.)

GCOEレクチャー"Holomorphic extensions of unitary representations" その1 "Overview and Examples"

http://faculty.ms.u-tokyo.ac.jp/users/gcoe/GCOE_lecture0810Hilgert.html

**Joachim Hilgert**(Paderborn University)GCOEレクチャー"Holomorphic extensions of unitary representations" その1 "Overview and Examples"

[ Abstract ]

In this lecture we present the Gelfand-Gindikin program of decomposing $L^2$-spaces into families of irreducible representations using complex geometry. We then briefly outline results due to Olshanski, Hilgert-Olafsson-Orsted, Hilgert-Neeb-Orsted, Krotz-Stanton and others in this direction. In particular, we will explain holomorphic extensions of holomorphic discrete series representations and their relation to Hardy and weighted Bergman spaces.

[ Reference URL ]In this lecture we present the Gelfand-Gindikin program of decomposing $L^2$-spaces into families of irreducible representations using complex geometry. We then briefly outline results due to Olshanski, Hilgert-Olafsson-Orsted, Hilgert-Neeb-Orsted, Krotz-Stanton and others in this direction. In particular, we will explain holomorphic extensions of holomorphic discrete series representations and their relation to Hardy and weighted Bergman spaces.

http://faculty.ms.u-tokyo.ac.jp/users/gcoe/GCOE_lecture0810Hilgert.html

### 2008/09/22

14:45-15:45 Room #122 (Graduate School of Math. Sci. Bldg.)

Invariance principle for the random conductance model

with unbounded conductances (a joint work with Martin Barlow)

**Jean-Dominique Deuschel**(TU Berlin)Invariance principle for the random conductance model

with unbounded conductances (a joint work with Martin Barlow)

### 2008/09/22

16:00-17:00 Room #122 (Graduate School of Math. Sci. Bldg.)

Asymptotic expansions of infinite dimensional integrals with applications (quantum mechanics, mathematical finance, biology)

**Sergio Albeverio**(Bonn 大学)Asymptotic expansions of infinite dimensional integrals with applications (quantum mechanics, mathematical finance, biology)

### 2008/09/03

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

Finite time blowup of oscillating solutions to the nonlinear heat equation

**Fred Weissler**(University of Paris 13)Finite time blowup of oscillating solutions to the nonlinear heat equation

[ Abstract ]

(This is joint work with T. Cazenave and F. Dickstein.)

We study finite time blowup properties of solutions of the nonlinear heat equation, both on $R^N$, and on a ball in $R^N$ with Dirichlet boundary conditions. We show, among other results, that the set of initial values producing global solutions is not always star-shaped around the 0 solution. This contrasts with the case where only non-negative solutions are considered.

(This is joint work with T. Cazenave and F. Dickstein.)

We study finite time blowup properties of solutions of the nonlinear heat equation, both on $R^N$, and on a ball in $R^N$ with Dirichlet boundary conditions. We show, among other results, that the set of initial values producing global solutions is not always star-shaped around the 0 solution. This contrasts with the case where only non-negative solutions are considered.

### 2008/08/25

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

Affine Weyl groups, grids, coloured tableaux and characters of affine algebras

**Ronald C. King**(Emeritus Professor, University of Southampton)Affine Weyl groups, grids, coloured tableaux and characters of affine algebras

[ Abstract ]

It is shown that certain coloured Young diagrams serve to specify not

only all the elements of the affine Weyl groups of the classical

affine Lie algebras but also their action on an arbitrary weight

vector. Through a judicious choice of coset representatives with

respect to the finite Weyl groups of the corresponding maximal rank

simple Lie algebras, both denominator and numerator formulae are

derived and exemplified, along with the explicit calculation of

characters of irreducible representations of the affine Lie algebras.

It is shown that certain coloured Young diagrams serve to specify not

only all the elements of the affine Weyl groups of the classical

affine Lie algebras but also their action on an arbitrary weight

vector. Through a judicious choice of coset representatives with

respect to the finite Weyl groups of the corresponding maximal rank

simple Lie algebras, both denominator and numerator formulae are

derived and exemplified, along with the explicit calculation of

characters of irreducible representations of the affine Lie algebras.

### 2008/07/24

16:00-17:00 Room #122 (Graduate School of Math. Sci. Bldg.)

On the modularity of Calabi-Yau varieties over $\\mathbf{Q}$

**Noriko Yui**(Queen's University)On the modularity of Calabi-Yau varieties over $\\mathbf{Q}$

### 2008/06/19

16:20-17:50 Room #052 (Graduate School of Math. Sci. Bldg.)

インターネットにおける数理科学的手法と実際「DNSとWebアクセス」

https://www.ms.u-tokyo.ac.jp/lecture/2008/inet/index.html

**伊藤 高一**(IRI-DNL)インターネットにおける数理科学的手法と実際「DNSとWebアクセス」

[ Abstract ]

テーマごとに個別の企業活動を紹介し、第一線で活躍する企業研究者を招聘し、現場レポートを聴き、議論を行うことで、インターネット数理科学の原理とその応用の実際を紹介する。

[ Reference URL ]テーマごとに個別の企業活動を紹介し、第一線で活躍する企業研究者を招聘し、現場レポートを聴き、議論を行うことで、インターネット数理科学の原理とその応用の実際を紹介する。

https://www.ms.u-tokyo.ac.jp/lecture/2008/inet/index.html

### 2008/06/09

16:30-18:00 Room #470 (Graduate School of Math. Sci. Bldg.)

Some Unsolved Inverse Spectral Problems

**W. Rundell**(Texas A&M Univ.)Some Unsolved Inverse Spectral Problems

[ Abstract ]

Perhaps the first well-studied inverse problem

was the determination of the potential $q(x)$ in

$-u'' + q(x) u = \\lambda_n u$ given the eigenvalues

$\\{\\lambda_n\\}$. Despite its venerable age and

the fact that a considerable literature is still being published,

there are several major outstanding problems;

some are quite simple to state.

This seminar will outline some of these.

We will try to show why the problems are hard,

but leave it to the audience to attempt solutions.

Perhaps the first well-studied inverse problem

was the determination of the potential $q(x)$ in

$-u'' + q(x) u = \\lambda_n u$ given the eigenvalues

$\\{\\lambda_n\\}$. Despite its venerable age and

the fact that a considerable literature is still being published,

there are several major outstanding problems;

some are quite simple to state.

This seminar will outline some of these.

We will try to show why the problems are hard,

but leave it to the audience to attempt solutions.

### 2008/05/19

16:00-17:30 Room #126 (Graduate School of Math. Sci. Bldg.)

)

A non standard unique continuation property related to Schiffer conjecture

https://www.ms.u-tokyo.ac.jp/top/general-access.html

**Jean-Pierre Puel 氏**

(ヴェルサイユ大学 (Universite de Versailles St Quentin))

A non standard unique continuation property related to Schiffer conjecture

[ Abstract ]

Coming from a control problem for a coupled fluid-structure system, we are confronted to the following problem in dimension 2:

\\Delta^2 w = -\\lambda \\Delta w in \\Omega w = {\\partial w}/{\\partial n} = 0 on \\Gamma {\\partial\\Delta w}/{\\partial n}=0 on \\Gamma_0 \\subset \\Gamma.

The question is : do we have w=0?

There is a counterexample when \\Omega is a disc. The analogous of (local) Schiffer's conjecture is : is the disc the only domain for which we can have a non zero solution?

Notice that the term local means that the additional boundary condition occurs only on a part of the boundary and when this boundary is not analytic, this is a major difference. A sub-conjecture would be : when the boundary is not analytic, do we have w=0?

Here we show that when \\Omega has a corner of angle \\theta_{0} with \\theta_{0} \\neq \\pi, 3\\pi/2 and when $\\Gamma_{0}$ is (locally) one edge of this angle then the only solution is w=0.

[ Reference URL ]Coming from a control problem for a coupled fluid-structure system, we are confronted to the following problem in dimension 2:

\\Delta^2 w = -\\lambda \\Delta w in \\Omega w = {\\partial w}/{\\partial n} = 0 on \\Gamma {\\partial\\Delta w}/{\\partial n}=0 on \\Gamma_0 \\subset \\Gamma.

The question is : do we have w=0?

There is a counterexample when \\Omega is a disc. The analogous of (local) Schiffer's conjecture is : is the disc the only domain for which we can have a non zero solution?

Notice that the term local means that the additional boundary condition occurs only on a part of the boundary and when this boundary is not analytic, this is a major difference. A sub-conjecture would be : when the boundary is not analytic, do we have w=0?

Here we show that when \\Omega has a corner of angle \\theta_{0} with \\theta_{0} \\neq \\pi, 3\\pi/2 and when $\\Gamma_{0}$ is (locally) one edge of this angle then the only solution is w=0.

https://www.ms.u-tokyo.ac.jp/top/general-access.html

### 2008/02/20

13:30-17:45 Room #123 (Graduate School of Math. Sci. Bldg.)

Divergence formulae on the space of continuous functions and Malliavin calculus

Ginibre random point field and a notion of convergence of Dirichlet forms

Stochastic PDEs and infinite dimensional integration by parts formulae

ランダム環境下の確率モデルに関連する問題

(A problem arising in stochastic models in random environments)

**乙部厳己**(信州大理) 13:30-14:00Divergence formulae on the space of continuous functions and Malliavin calculus

**長田博文**(九大数理) 14:15-15:15Ginibre random point field and a notion of convergence of Dirichlet forms

**Lorenzo Zambotti**(パリ第6大学) 15:30-16:30Stochastic PDEs and infinite dimensional integration by parts formulae

**志賀徳造**(東工大理工) 16:45-17:45ランダム環境下の確率モデルに関連する問題

(A problem arising in stochastic models in random environments)

### 2008/02/19

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

An overview on archimedean L-factors for G_1xG_2

**Eric Stade**(Colorado University)An overview on archimedean L-factors for G_1xG_2

[ Abstract ]

When G_1xG_2 is one of pairs GL(n)xGL(n), GL(n)xGL(n+1), GL(n)xSO(2n+1), and GL(n+1)xSO(2n+1), we have evaluation of the archimedian L-factors of automorphic L-functions obtained by Rankin-Selberg convolution.

The last two cases are joint works with Taku Ishii (Chiba Inst. of Tech) which are in progress.

When G_1xG_2 is one of pairs GL(n)xGL(n), GL(n)xGL(n+1), GL(n)xSO(2n+1), and GL(n+1)xSO(2n+1), we have evaluation of the archimedian L-factors of automorphic L-functions obtained by Rankin-Selberg convolution.

The last two cases are joint works with Taku Ishii (Chiba Inst. of Tech) which are in progress.

### 2008/02/07

16:30-18:00 Room #002 (Graduate School of Math. Sci. Bldg.)

On Gabber's refined uniformization theorem and applications

**Luc Illusie**(パリ南大学)On Gabber's refined uniformization theorem and applications

[ Abstract ]

Gabber has announced the following theorem : if X is a noetherian quasi-excellent scheme, Z a nowhere dense closed subset, and l a prime number invertible on X, then, locally for the topology on schemes of finite type over X generated, up to thickenings, by proper surjective maps which are generically finite of degree prime to l and by Nisnevich covers, the pair (X,Z) can be uniformized, i. e. replaced by a pair (Y,D), where Y is regular and D a strict normal crossings divisor. The whole proof is not yet written. I will give an overview. The plan is :

1. Statement and reduction to the complete local case (techniques of approximation)

2. Refined partial algebraization of complete local noetherian rings

3. Reduction to the equivariant log regular case (de Jong's techniques)

4. Making actions very tame, end of proof.

If time permits, I will show how the above theorem provides a short proof of Gabber's finiteness theorem for higher direct images of constructible sheaves of torsion prime to the characteristics by morphisms of finite type between quasi-excellent noetherian schemes.

Gabber has announced the following theorem : if X is a noetherian quasi-excellent scheme, Z a nowhere dense closed subset, and l a prime number invertible on X, then, locally for the topology on schemes of finite type over X generated, up to thickenings, by proper surjective maps which are generically finite of degree prime to l and by Nisnevich covers, the pair (X,Z) can be uniformized, i. e. replaced by a pair (Y,D), where Y is regular and D a strict normal crossings divisor. The whole proof is not yet written. I will give an overview. The plan is :

1. Statement and reduction to the complete local case (techniques of approximation)

2. Refined partial algebraization of complete local noetherian rings

3. Reduction to the equivariant log regular case (de Jong's techniques)

4. Making actions very tame, end of proof.

If time permits, I will show how the above theorem provides a short proof of Gabber's finiteness theorem for higher direct images of constructible sheaves of torsion prime to the characteristics by morphisms of finite type between quasi-excellent noetherian schemes.

### 2008/01/31

16:30-18:00 Room #118 (Graduate School of Math. Sci. Bldg.)

On Gabber's refined uniformization theorem and applications

**Luc Illusie**(パリ南大学)On Gabber's refined uniformization theorem and applications

[ Abstract ]

Gabber has announced the following theorem : if X is a noetherian quasi-excellent scheme, Z a nowhere dense closed subset, and l a prime number invertible on X, then, locally for the topology on schemes of finite type over X generated, up to thickenings, by proper surjective maps which are generically finite of degree prime to l and by Nisnevich covers, the pair (X,Z) can be uniformized, i. e. replaced by a pair (Y,D), where Y is regular and D a strict normal crossings divisor. The whole proof is not yet written. I will give an overview. The plan is :

1. Statement and reduction to the complete local case (techniques of approximation)

2. Refined partial algebraization of complete local noetherian rings

3. Reduction to the equivariant log regular case (de Jong's techniques)

4. Making actions very tame, end of proof.

If time permits, I will show how the above theorem provides a short proof of Gabber's finiteness theorem for higher direct images of constructible sheaves of torsion prime to the characteristics by morphisms of finite type between quasi-excellent noetherian schemes.

Gabber has announced the following theorem : if X is a noetherian quasi-excellent scheme, Z a nowhere dense closed subset, and l a prime number invertible on X, then, locally for the topology on schemes of finite type over X generated, up to thickenings, by proper surjective maps which are generically finite of degree prime to l and by Nisnevich covers, the pair (X,Z) can be uniformized, i. e. replaced by a pair (Y,D), where Y is regular and D a strict normal crossings divisor. The whole proof is not yet written. I will give an overview. The plan is :

1. Statement and reduction to the complete local case (techniques of approximation)

2. Refined partial algebraization of complete local noetherian rings

3. Reduction to the equivariant log regular case (de Jong's techniques)

4. Making actions very tame, end of proof.

If time permits, I will show how the above theorem provides a short proof of Gabber's finiteness theorem for higher direct images of constructible sheaves of torsion prime to the characteristics by morphisms of finite type between quasi-excellent noetherian schemes.

### 2008/01/22

16:30-18:00 Room #118 (Graduate School of Math. Sci. Bldg.)

On Gabber's refined uniformization theorem and applications

**Luc Illusie**(パリ南大学)On Gabber's refined uniformization theorem and applications

[ Abstract ]

Gabber has announced the following theorem : if X is a noetherian quasi-excellent scheme, Z a nowhere dense closed subset, and l a prime number invertible on X, then, locally for the topology on schemes of finite type over X generated, up to thickenings, by proper surjective maps which are generically finite of degree prime to l and by Nisnevich covers, the pair (X,Z) can be uniformized, i. e. replaced by a pair (Y,D), where Y is regular and D a strict normal crossings divisor. The whole proof is not yet written. I will give an overview. The plan is :

1. Statement and reduction to the complete local case (techniques of approximation)

2. Refined partial algebraization of complete local noetherian rings

3. Reduction to the equivariant log regular case (de Jong's techniques)

4. Making actions very tame, end of proof.

If time permits, I will show how the above theorem provides a short proof of Gabber's finiteness theorem for higher direct images of constructible sheaves of torsion prime to the characteristics by morphisms of finite type between quasi-excellent noetherian schemes.

Gabber has announced the following theorem : if X is a noetherian quasi-excellent scheme, Z a nowhere dense closed subset, and l a prime number invertible on X, then, locally for the topology on schemes of finite type over X generated, up to thickenings, by proper surjective maps which are generically finite of degree prime to l and by Nisnevich covers, the pair (X,Z) can be uniformized, i. e. replaced by a pair (Y,D), where Y is regular and D a strict normal crossings divisor. The whole proof is not yet written. I will give an overview. The plan is :

1. Statement and reduction to the complete local case (techniques of approximation)

2. Refined partial algebraization of complete local noetherian rings

3. Reduction to the equivariant log regular case (de Jong's techniques)

4. Making actions very tame, end of proof.

If time permits, I will show how the above theorem provides a short proof of Gabber's finiteness theorem for higher direct images of constructible sheaves of torsion prime to the characteristics by morphisms of finite type between quasi-excellent noetherian schemes.

### 2008/01/21

16:00-17:30 Room #126 (Graduate School of Math. Sci. Bldg.)

Potential theory of funnels and wounds

**Torbjorn Lundh**(Chalmers & Göteborg University)Potential theory of funnels and wounds

[ Abstract ]

We will talk about a result concerning Green functions, namely the so called 3G-inequality, which I studied together with H. Aikawa. The focus of the talk will be on the description of the way to that result, where we among other tools used numerical methods to get a better intuitive understanding the situation. We will also discuss a possible potential theoretic view-point of an ancient wound healing question.

We will talk about a result concerning Green functions, namely the so called 3G-inequality, which I studied together with H. Aikawa. The focus of the talk will be on the description of the way to that result, where we among other tools used numerical methods to get a better intuitive understanding the situation. We will also discuss a possible potential theoretic view-point of an ancient wound healing question.

### 2008/01/17

16:30-18:00 Room #118 (Graduate School of Math. Sci. Bldg.)

On Gabber's refined uniformization theorem and applications

[ Abstract ]

Gabber has announced the following theorem : if X is a noetherian quasi-excellent scheme, Z a nowhere dense closed subset, and l a prime number invertible on X, then, locally for the topology on schemes of finite type over X generated, up to thickenings, by proper surjective maps which are generically finite of degree prime to l and by Nisnevich covers, the pair (X,Z) can be uniformized, i. e. replaced by a pair (Y,D), where Y is regular and D a strict normal crossings divisor. The whole proof is not yet written. I will give an overview. The plan is :

1. Statement and reduction to the complete local case (techniques of approximation)

2. Refined partial algebraization of complete local noetherian rings

3. Reduction to the equivariant log regular case (de Jong's techniques)

4. Making actions very tame, end of proof.

If time permits, I will show how the above theorem provides a short proof of Gabber's finiteness theorem for higher direct images of constructible sheaves of torsion prime to the characteristics by morphisms of finite type between quasi-excellent noetherian schemes.

**Luc Illusie**(パリ南大学)On Gabber's refined uniformization theorem and applications

Gabber has announced the following theorem : if X is a noetherian quasi-excellent scheme, Z a nowhere dense closed subset, and l a prime number invertible on X, then, locally for the topology on schemes of finite type over X generated, up to thickenings, by proper surjective maps which are generically finite of degree prime to l and by Nisnevich covers, the pair (X,Z) can be uniformized, i. e. replaced by a pair (Y,D), where Y is regular and D a strict normal crossings divisor. The whole proof is not yet written. I will give an overview. The plan is :

1. Statement and reduction to the complete local case (techniques of approximation)

2. Refined partial algebraization of complete local noetherian rings

3. Reduction to the equivariant log regular case (de Jong's techniques)

4. Making actions very tame, end of proof.

If time permits, I will show how the above theorem provides a short proof of Gabber's finiteness theorem for higher direct images of constructible sheaves of torsion prime to the characteristics by morphisms of finite type between quasi-excellent noetherian schemes.

### 2007/12/20

10:40-12:10 Room #128 (Graduate School of Math. Sci. Bldg.)

Topics in ergodic theory, von Neumann algebras, and rigidity

[ Reference URL ]

https://www.ms.u-tokyo.ac.jp/~yasuyuki/pichot.htm

**Mikael Pichot**(東大数理)Topics in ergodic theory, von Neumann algebras, and rigidity

[ Reference URL ]

https://www.ms.u-tokyo.ac.jp/~yasuyuki/pichot.htm

### 2007/12/13

10:40-12:10 Room #128 (Graduate School of Math. Sci. Bldg.)

Topics in ergodic theory, von Neumann algebras, and rigidity

[ Reference URL ]

https://www.ms.u-tokyo.ac.jp/~yasuyuki/pichot.htm

**Mikael Pichot**(東大数理)Topics in ergodic theory, von Neumann algebras, and rigidity

[ Reference URL ]

https://www.ms.u-tokyo.ac.jp/~yasuyuki/pichot.htm

### 2007/12/06

10:40-12:10 Room #128 (Graduate School of Math. Sci. Bldg.)

Topics in ergodic theory, von Neumann algebras, and rigidity

[ Reference URL ]

https://www.ms.u-tokyo.ac.jp/~yasuyuki/pichot.htm

**Mikael Pichot**(東大数理)Topics in ergodic theory, von Neumann algebras, and rigidity

[ Reference URL ]

https://www.ms.u-tokyo.ac.jp/~yasuyuki/pichot.htm

### 2007/11/29

**Mikael Pichot**(東大数理)

Topics in ergodic theory, von Neumann algebras, and rigidity

[ Reference URL ]

https://www.ms.u-tokyo.ac.jp/~yasuyuki/pichot.htm

### 2007/11/22

**Mikael Pichot**(東大数理)

Topics in ergodic theory, von Neumann algebras, and rigidity

[ Reference URL ]

https://www.ms.u-tokyo.ac.jp/~yasuyuki/pichot.htm

### 2007/11/13

16:00-17:30 Room #052 (Graduate School of Math. Sci. Bldg.)

Modelling the olfactory system: From receptor neuron dynamics over axonal pathfinding and sorting to spatio-temporal activities in the bulb

**Jens Starke**(Technical University of Denmark)Modelling the olfactory system: From receptor neuron dynamics over axonal pathfinding and sorting to spatio-temporal activities in the bulb

[ Abstract ]

The olfactory system of e.g. mice serves as important model case for other brain regions. The odor signals are processed from receptor neurons over the glomeruli level to a neural network of mitral and granular cells while various types of nonlinear behaviour can be observed.

(1) Nonlinear dynamics in receptor neurons:

A mathematical model for Ca oscillations in the cilia of olfactory

sensory neurons is suggested and analyzed. The existence of an oscillatory regime based on a Hopf bifurcation is proven using stoichiometric network analysis where the knowledge of exact parameters is not required. Predictions of the model are in quantitative agreement with experiment, both with respect to oscillations and to fast adaptation.

(2) Sorting by self-organization:

A many particle model with attracting and repulsive interactions is proposed which is able to reproduce the experimental findings of sorting and convergence during axonal pathfinding in the olfactory system. Many axon species, each represented by a huge number of axons, are spatially disordered at the beginning of their growth at the receptor neurons and converge by a self-organized process to a sorted state, i.e. axons of the same receptor type converge to a common position. Under certain model assumptions, it can be proved that the interacting many-particle system of different particle types converges to a sorted state.

(3) Spatio-temporal pattern formation in the olfactory bulb:

Odors evoke a variety of stimulus specific spatio-temporal patterns on the levels of glomeruli and neural network of mitral and granular cells in the olfactory bulb which can be measured in vivo using Ca and voltage sensitive dyes for optical imaging. A spatial independent component analysis of this high-resolution imaging data was used to identify and separate different neuronal populations based on their stimulus specific spatio-temporal activation. Equation-free techniques were used to obtain bifurcation diagramms for the network activity. First, contrast enhancement between several spatially close activations depending on the network topology and second, hysteres effects in recognition of differences between similar odorants depending on the concentration ratios of odorant mixtures.

This is in parts joint work with P. Borowski, M. Eiswirth, C. Ellsaesser, A. Grinvald, N. Hummel, S. Kokkendorff, D. Omer, J. Reidl, H. Spors, J. Strotmann, M. Zapotocky.

The olfactory system of e.g. mice serves as important model case for other brain regions. The odor signals are processed from receptor neurons over the glomeruli level to a neural network of mitral and granular cells while various types of nonlinear behaviour can be observed.

(1) Nonlinear dynamics in receptor neurons:

A mathematical model for Ca oscillations in the cilia of olfactory

sensory neurons is suggested and analyzed. The existence of an oscillatory regime based on a Hopf bifurcation is proven using stoichiometric network analysis where the knowledge of exact parameters is not required. Predictions of the model are in quantitative agreement with experiment, both with respect to oscillations and to fast adaptation.

(2) Sorting by self-organization:

A many particle model with attracting and repulsive interactions is proposed which is able to reproduce the experimental findings of sorting and convergence during axonal pathfinding in the olfactory system. Many axon species, each represented by a huge number of axons, are spatially disordered at the beginning of their growth at the receptor neurons and converge by a self-organized process to a sorted state, i.e. axons of the same receptor type converge to a common position. Under certain model assumptions, it can be proved that the interacting many-particle system of different particle types converges to a sorted state.

(3) Spatio-temporal pattern formation in the olfactory bulb:

Odors evoke a variety of stimulus specific spatio-temporal patterns on the levels of glomeruli and neural network of mitral and granular cells in the olfactory bulb which can be measured in vivo using Ca and voltage sensitive dyes for optical imaging. A spatial independent component analysis of this high-resolution imaging data was used to identify and separate different neuronal populations based on their stimulus specific spatio-temporal activation. Equation-free techniques were used to obtain bifurcation diagramms for the network activity. First, contrast enhancement between several spatially close activations depending on the network topology and second, hysteres effects in recognition of differences between similar odorants depending on the concentration ratios of odorant mixtures.

This is in parts joint work with P. Borowski, M. Eiswirth, C. Ellsaesser, A. Grinvald, N. Hummel, S. Kokkendorff, D. Omer, J. Reidl, H. Spors, J. Strotmann, M. Zapotocky.

### 2007/10/17

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

The method of compensated compactness for

microscopic systems

**J. Fritz**(TU Budapest)The method of compensated compactness for

microscopic systems