## Seminar information archive

Seminar information archive ～01/18｜Today's seminar 01/19 | Future seminars 01/20～

#### Tuesday Seminar on Topology

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

Conformal field theory for C2-cofinite vertex algebras (JAPANESE)

**Yoshitake Hashimoto**(Tokyo City University)Conformal field theory for C2-cofinite vertex algebras (JAPANESE)

[ Abstract ]

This is a jount work with Akihiro Tsuchiya (Kavli IPMU).

We consider sheaves of covacua and conformal blocks over parameter spaces of n-pointed Riemann surfaces

for a vertex algebra of which the category of modules is not necessarily semi-simple.

We assume the C2-cofiniteness condition for vertex algebras.

We define "tensor product" of two modules over a C2-cofinite vertex algebra.

This is a jount work with Akihiro Tsuchiya (Kavli IPMU).

We consider sheaves of covacua and conformal blocks over parameter spaces of n-pointed Riemann surfaces

for a vertex algebra of which the category of modules is not necessarily semi-simple.

We assume the C2-cofiniteness condition for vertex algebras.

We define "tensor product" of two modules over a C2-cofinite vertex algebra.

#### Tuesday Seminar of Analysis

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

Evolution of smooth shapes and the KP hierarchy (ENGLISH)

Group of diffeomorphisms of the unit circle and sub-Riemannian geometry (ENGLISH)

**Alexander Vasiliev**(Department of Mathematics, University of Bergen, Norway) 16:30-17:30Evolution of smooth shapes and the KP hierarchy (ENGLISH)

[ Abstract ]

We consider a homotopic evolution in the space of smooth

shapes starting from the unit circle. Based on the Loewner-Kufarev

equation we give a Hamiltonian formulation of this evolution and

provide conservation laws. The symmetries of the evolution are given

by the Virasoro algebra. The 'positive' Virasoro generators span the

holomorphic part of the complexified vector bundle over the space of

conformal embeddings of the unit disk into the complex plane and

smooth on the boundary. In the covariant formulation they are

conserved along the Hamiltonian flow. The 'negative' Virasoro

generators can be recovered by an iterative method making use of the

canonical Poisson structure. We study an embedding of the

Loewner-Kufarev trajectories into the Segal-Wilson Grassmannian,

construct the tau-function, the Baker-Akhiezer function, and finally,

give a class of solutions to the KP hierarchy, which are invariant on

Loewner-Kufarev trajectories.

We consider a homotopic evolution in the space of smooth

shapes starting from the unit circle. Based on the Loewner-Kufarev

equation we give a Hamiltonian formulation of this evolution and

provide conservation laws. The symmetries of the evolution are given

by the Virasoro algebra. The 'positive' Virasoro generators span the

holomorphic part of the complexified vector bundle over the space of

conformal embeddings of the unit disk into the complex plane and

smooth on the boundary. In the covariant formulation they are

conserved along the Hamiltonian flow. The 'negative' Virasoro

generators can be recovered by an iterative method making use of the

canonical Poisson structure. We study an embedding of the

Loewner-Kufarev trajectories into the Segal-Wilson Grassmannian,

construct the tau-function, the Baker-Akhiezer function, and finally,

give a class of solutions to the KP hierarchy, which are invariant on

Loewner-Kufarev trajectories.

**Irina Markina**(Department of Mathematics, University of Bergen, Norway) 17:30-18:30Group of diffeomorphisms of the unit circle and sub-Riemannian geometry (ENGLISH)

[ Abstract ]

We consider the group of sense-preserving diffeomorphisms of the unit

circle and its central extension - the Virasoro-Bott group as

sub-Riemannian manifolds. Shortly, a sub-Riemannian manifold is a

smooth manifold M with a given sub-bundle D of the tangent bundle, and

with a metric defined on the sub-bundle D. The different sub-bundles

on considered groups are related to some spaces of normalized

univalent functions. We present formulas for geodesics for different

choices of metrics. The geodesic equations are generalizations of

Camassa-Holm, Huter-Saxton, KdV, and other known non-linear PDEs. We

show that any two points in these groups can be connected by a curve

tangent to the chosen sub-bundle. We also discuss the similarities and

peculiarities of the structure of sub-Riemannian geodesics on infinite

and finite dimensional manifolds.

We consider the group of sense-preserving diffeomorphisms of the unit

circle and its central extension - the Virasoro-Bott group as

sub-Riemannian manifolds. Shortly, a sub-Riemannian manifold is a

smooth manifold M with a given sub-bundle D of the tangent bundle, and

with a metric defined on the sub-bundle D. The different sub-bundles

on considered groups are related to some spaces of normalized

univalent functions. We present formulas for geodesics for different

choices of metrics. The geodesic equations are generalizations of

Camassa-Holm, Huter-Saxton, KdV, and other known non-linear PDEs. We

show that any two points in these groups can be connected by a curve

tangent to the chosen sub-bundle. We also discuss the similarities and

peculiarities of the structure of sub-Riemannian geodesics on infinite

and finite dimensional manifolds.

### 2012/12/03

#### Seminar on Geometric Complex Analysis

10:30-12:00 Room #126 (Graduate School of Math. Sci. Bldg.)

On the geometric meaning of the maximal number

of exceptional values of Gauss maps for immersed surfaces in space forms

(JAPANESE)

**Yu Kawakami**(Yamaguchi University)On the geometric meaning of the maximal number

of exceptional values of Gauss maps for immersed surfaces in space forms

(JAPANESE)

### 2012/12/01

#### Infinite Analysis Seminar Tokyo

13:30-15:00 Room #117 (Graduate School of Math. Sci. Bldg.)

Manin matrices and quantum integrable systems (ENGLISH)

**Alexey Silantyev**(Univ. Tokyo)Manin matrices and quantum integrable systems (ENGLISH)

[ Abstract ]

Manin matrices (known also as right quantum matrices) is a class of

matrices with non-commutative entries. The natural generalization of the

usual determinant for these matrices is so-called column determinant.

Manin matrices, their determinants and minors have the most part of the

properties possessed by the usual number matrices. Manin matrices arise

from the RLL-relations and help to find quantum analogues of Poisson

commuting traces of powers of Lax operators and to establish relations

between different types of quantum commuting families. The RLL-relations

also give us q-analogues of Manin matrices in the case of trigonometric

R-matrix (which define commutation relations for the quantum affine

algebra).

Manin matrices (known also as right quantum matrices) is a class of

matrices with non-commutative entries. The natural generalization of the

usual determinant for these matrices is so-called column determinant.

Manin matrices, their determinants and minors have the most part of the

properties possessed by the usual number matrices. Manin matrices arise

from the RLL-relations and help to find quantum analogues of Poisson

commuting traces of powers of Lax operators and to establish relations

between different types of quantum commuting families. The RLL-relations

also give us q-analogues of Manin matrices in the case of trigonometric

R-matrix (which define commutation relations for the quantum affine

algebra).

### 2012/11/30

#### Colloquium

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

What do Siegel Eisenstein series know about all modular forms? (ENGLISH)

**Siegfried BOECHERER**(University of Tokyo)What do Siegel Eisenstein series know about all modular forms? (ENGLISH)

[ Abstract ]

Eisenstein series came up in C.L.Siegel's famous work on quadratic forms. The main properties of such Eisensetin series such as analytic continuation and explict form of Fourier expansion are well understood. Nowadays, we use Eisenstein series of higher rank symplectic groups and their restrictions to study properties of all modular forms. I will try to survey the use of “pullbacks of Eisenstein series”: Basis problem, L-functions, p-adic properties, rationality and integrality questions.

Eisenstein series came up in C.L.Siegel's famous work on quadratic forms. The main properties of such Eisensetin series such as analytic continuation and explict form of Fourier expansion are well understood. Nowadays, we use Eisenstein series of higher rank symplectic groups and their restrictions to study properties of all modular forms. I will try to survey the use of “pullbacks of Eisenstein series”: Basis problem, L-functions, p-adic properties, rationality and integrality questions.

#### Mathematical Biology Seminar

14:30-15:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Targeting control in the presence of uncertainty (ENGLISH)

**Michael Tildesley**( Infectious Disease Epidemiology (Modelling) at the University of Warwick)Targeting control in the presence of uncertainty (ENGLISH)

[ Abstract ]

The availability of epidemiological data in the early stages of an outbreak of an infectious disease is vital to enable modellers to make accurate predictions regarding the likely spread of disease and preferred intervention strategies. However, in some countries, epidemic data are not available whilst necessary demographic data are only available at an aggregate scale. Here we investigate the ability of models of livestock infectious diseases to predict epidemic spread and optimal control policies in the event of uncertainty. We focus on investigating predictions in the presence of uncertainty regarding contact networks, demographic data and epidemiological parameters. Our results indicate that mathematical models could be utilized in regions where individual farm-level data are not available, to allow predictive analyses to be carried out regarding the likely spread of disease. This method can also be used for contingency planning in collaboration with policy makers to determine preferred control strategies in the event of a future outbreak of infectious disease in livestock.

The availability of epidemiological data in the early stages of an outbreak of an infectious disease is vital to enable modellers to make accurate predictions regarding the likely spread of disease and preferred intervention strategies. However, in some countries, epidemic data are not available whilst necessary demographic data are only available at an aggregate scale. Here we investigate the ability of models of livestock infectious diseases to predict epidemic spread and optimal control policies in the event of uncertainty. We focus on investigating predictions in the presence of uncertainty regarding contact networks, demographic data and epidemiological parameters. Our results indicate that mathematical models could be utilized in regions where individual farm-level data are not available, to allow predictive analyses to be carried out regarding the likely spread of disease. This method can also be used for contingency planning in collaboration with policy makers to determine preferred control strategies in the event of a future outbreak of infectious disease in livestock.

#### Seminar on Probability and Statistics

14:50-16:00 Room #006 (Graduate School of Math. Sci. Bldg.)

Effective PCA for high-dimensional, non-Gaussian data under power spiked model (JAPANESE)

http://www.sigmath.es.osaka-u.ac.jp/~kamatani/statseminar/2012/11.html

**YATA, Kazuyoshi**(Institute of Mathematics, University of Tsukuba)Effective PCA for high-dimensional, non-Gaussian data under power spiked model (JAPANESE)

[ Abstract ]

In this talk, we introduce a general spiked model called the power spiked model in high-dimensional settings. We first consider asymptotic properties of the conventional estimator of eigenvalues under the power spiked model. We give several conditions on the dimension $p$, the sample size $n$ and the high-dimensional noise structure in order to hold several consistency properties of the estimator. We show that the estimator is affected by the noise structure, directly, so that the estimator becomes inconsistent for such cases. In order to overcome such difficulties in a high-dimensional situation, we develop new PCAs called the noise-reduction methodology and the cross-data-matrix methodology under the power spiked model. This is a joint work with Prof. Aoshima (University of Tsukuba).

[ Reference URL ]In this talk, we introduce a general spiked model called the power spiked model in high-dimensional settings. We first consider asymptotic properties of the conventional estimator of eigenvalues under the power spiked model. We give several conditions on the dimension $p$, the sample size $n$ and the high-dimensional noise structure in order to hold several consistency properties of the estimator. We show that the estimator is affected by the noise structure, directly, so that the estimator becomes inconsistent for such cases. In order to overcome such difficulties in a high-dimensional situation, we develop new PCAs called the noise-reduction methodology and the cross-data-matrix methodology under the power spiked model. This is a joint work with Prof. Aoshima (University of Tsukuba).

http://www.sigmath.es.osaka-u.ac.jp/~kamatani/statseminar/2012/11.html

### 2012/11/29

#### Lie Groups and Representation Theory

16:30-17:30 Room #122 (Graduate School of Math. Sci. Bldg.)

On a relation between certain character values of symmetric groups (JAPANESE)

**Masaki Watanabe**(the University of Tokyo)On a relation between certain character values of symmetric groups (JAPANESE)

[ Abstract ]

We present a relation of new kind between character values of

symmetric groups which explains a curious phenomenon in character

tables of symmetric groups. Similar relations for characters of

Brauer and walled Brauer algebras and projective characters of

symmetric groups are also presented.

We present a relation of new kind between character values of

symmetric groups which explains a curious phenomenon in character

tables of symmetric groups. Similar relations for characters of

Brauer and walled Brauer algebras and projective characters of

symmetric groups are also presented.

#### GCOE lecture series

10:00-12:10 Room #123 (Graduate School of Math. Sci. Bldg.)

Sobolev maps with values into the circle (ENGLISH)

**Haim Brezis**(Rutgers University / Technion)Sobolev maps with values into the circle (ENGLISH)

[ Abstract ]

Sobolev functions with values into R are very well understood and play an immense role in many branches of Mathematics. By contrast, the theory of Sobolev maps with values into the unit circle is still under construction. Such maps occur e.g. in the asymptotic analysis of the Ginzburg-Landau model. The reason one is interested in Sobolev maps, rather than smooth maps is to allow singularities such as x/|x| in 2D or line singularities 3D which appear in physical problems. Our focus in these lectures is not the Ginzburg-Landau equation per se, but rather the intrinsic study of the function space W^{1,p} of maps from a smooth domain in R^N taking their values into the unit circle. Such classes of maps have an amazingly rich structure. Geometrical and Topological effects are already noticeable in this simple framework, since S^1 has nontrivial topology. Moreover the fact that the target space is the circle (as opposed to higher-dimensional manifolds) offers the option to introduce a lifting. We'll see that "optimal liftings" are in one-to-one correspondence with minimal connections (resp. minimal surfaces) spanned by the topological singularities of u.

I will also discuss the question of uniqueness of lifting . A key ingredient in some of the proofs is a formula (due to myself, Bourgain and Mironescu) which provides an original way of approximating Sobolev norms (or the total variation) by nonlocal functionals. Nonconvex versions of these functionals raise very challenging questions recently tackled together with H.-M. Nguyen. Comparable functionals also occur in Image Processing and suggest exciting interactions with this field.

Sobolev functions with values into R are very well understood and play an immense role in many branches of Mathematics. By contrast, the theory of Sobolev maps with values into the unit circle is still under construction. Such maps occur e.g. in the asymptotic analysis of the Ginzburg-Landau model. The reason one is interested in Sobolev maps, rather than smooth maps is to allow singularities such as x/|x| in 2D or line singularities 3D which appear in physical problems. Our focus in these lectures is not the Ginzburg-Landau equation per se, but rather the intrinsic study of the function space W^{1,p} of maps from a smooth domain in R^N taking their values into the unit circle. Such classes of maps have an amazingly rich structure. Geometrical and Topological effects are already noticeable in this simple framework, since S^1 has nontrivial topology. Moreover the fact that the target space is the circle (as opposed to higher-dimensional manifolds) offers the option to introduce a lifting. We'll see that "optimal liftings" are in one-to-one correspondence with minimal connections (resp. minimal surfaces) spanned by the topological singularities of u.

I will also discuss the question of uniqueness of lifting . A key ingredient in some of the proofs is a formula (due to myself, Bourgain and Mironescu) which provides an original way of approximating Sobolev norms (or the total variation) by nonlocal functionals. Nonconvex versions of these functionals raise very challenging questions recently tackled together with H.-M. Nguyen. Comparable functionals also occur in Image Processing and suggest exciting interactions with this field.

### 2012/11/28

#### Geometry Colloquium

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

Ricci curvature and angles (JAPANESE)

**Shouhei Honda**(Kyushu University)Ricci curvature and angles (JAPANESE)

[ Abstract ]

Let X be the Gromov-Hausdorff limit space of a sequence of Riemannian manifolds with a lower Ricci curvature bound. In this talk we will give the definition of angles between geodesics on X. We apply this to prove there is a weakly twice differentiable structure on X and prove there is a unique Levi-Civita connection allowing us to define the Hessian of a twice differentiable function.

Let X be the Gromov-Hausdorff limit space of a sequence of Riemannian manifolds with a lower Ricci curvature bound. In this talk we will give the definition of angles between geodesics on X. We apply this to prove there is a weakly twice differentiable structure on X and prove there is a unique Levi-Civita connection allowing us to define the Hessian of a twice differentiable function.

#### Operator Algebra Seminars

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

Fundamental group of simple $C^*$-algebras with unique trace (JAPANESE)

**Norio Nawata**(Chiba University)Fundamental group of simple $C^*$-algebras with unique trace (JAPANESE)

#### Lectures

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.

#### GCOE lecture series

13:30-14:30 Room #123 (Graduate School of Math. Sci. Bldg.)

How Poincare became my hero (ENGLISH)

**Haim Brezis**(Rutgers University / Technion)How Poincare became my hero (ENGLISH)

[ Abstract ]

I recently discovered little-known texts of Poincare which include fundamental results on PDEs together with prophetic insights into their future impact on various branches of modern mathematics.

I recently discovered little-known texts of Poincare which include fundamental results on PDEs together with prophetic insights into their future impact on various branches of modern mathematics.

#### GCOE lecture series

14:50-17:30 Room #123 (Graduate School of Math. Sci. Bldg.)

Can you hear the degree of a map from the circle into itself? An intriguing story which is not yet finished (ENGLISH)

**Haim Brezis**(Rutgers University / Technion)Can you hear the degree of a map from the circle into itself? An intriguing story which is not yet finished (ENGLISH)

[ Abstract ]

A few years ago - following a suggestion by I. M. Gelfand - I discovered an intriguing connection between the topological degree of a map from the circle into itself and its Fourier coefficients. This relation is easily justified when the map is smooth. However, the situation turns out to be much more delicate if one assumes only continuity, or even Holder continuity. I will present recent developments and open problems. The initial motivation for this direction of research came from the analysis of the Ginzburg-Landau model.

A few years ago - following a suggestion by I. M. Gelfand - I discovered an intriguing connection between the topological degree of a map from the circle into itself and its Fourier coefficients. This relation is easily justified when the map is smooth. However, the situation turns out to be much more delicate if one assumes only continuity, or even Holder continuity. I will present recent developments and open problems. The initial motivation for this direction of research came from the analysis of the Ginzburg-Landau model.

### 2012/11/27

#### Tuesday Seminar on Topology

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

On a finite aspect of characteristic classes of foliations (JAPANESE)

**Hiraku Nozawa**(JSPS-IHES fellow)On a finite aspect of characteristic classes of foliations (JAPANESE)

[ Abstract ]

Characteristic classes of foliations are not bounded due to Thurston.

In this talk, we will explain finiteness of characteristic classes for

foliations with certain transverse structures (e.g. transverse

conformally flat structure) and its relation to unboundedness and

rigidity of foliations.

(This talk is based on a joint work with Jesús Antonio

Álvarez López at University of Santiago de Compostela,

which is available as arXiv:1205.3375.)

Characteristic classes of foliations are not bounded due to Thurston.

In this talk, we will explain finiteness of characteristic classes for

foliations with certain transverse structures (e.g. transverse

conformally flat structure) and its relation to unboundedness and

rigidity of foliations.

(This talk is based on a joint work with Jesús Antonio

Álvarez López at University of Santiago de Compostela,

which is available as arXiv:1205.3375.)

#### Lie Groups and Representation Theory

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

Convergence of Kahler to real polarizations on flag manifolds (JAPANESE)

**Hiroshi Konno**(the University of Tokyo)Convergence of Kahler to real polarizations on flag manifolds (JAPANESE)

[ Abstract ]

In this talk we will discuss geometric quantization of a flag manifold. In particular, we construct a family of complex structures on a flag manifold that converge 'at the quantum level' to the real polarization coming from the Gelfand-Cetlin integrable system.

Our construction is based on a toric degeneration of flag varieties and a deformation of K¥"ahler structure on toric varieties by symplectic potentials.

This is a joint work with Mark Hamilton.

In this talk we will discuss geometric quantization of a flag manifold. In particular, we construct a family of complex structures on a flag manifold that converge 'at the quantum level' to the real polarization coming from the Gelfand-Cetlin integrable system.

Our construction is based on a toric degeneration of flag varieties and a deformation of K¥"ahler structure on toric varieties by symplectic potentials.

This is a joint work with Mark Hamilton.

### 2012/11/26

#### Algebraic Geometry Seminar

15:30-17:00 Room #122 (Graduate School of Math. Sci. Bldg.)

A configuration of rational curves on the superspecial K3 surface (JAPANESE)

**Toshiyuki Katsura**(Hosei University)A configuration of rational curves on the superspecial K3 surface (JAPANESE)

#### Seminar on Geometric Complex Analysis

10:30-12:00 Room #126 (Graduate School of Math. Sci. Bldg.)

Quaternionic CR geometry (JAPANESE)

**Shin Nayatani**(Nagoya University)Quaternionic CR geometry (JAPANESE)

### 2012/11/22

#### Lectures

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)

http://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.

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

#### Lectures

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

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

http://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.

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

#### Lectures

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

Gelfand type problem for two phase porous media (ENGLISH)

http://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).

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

#### Lectures

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)

http://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.

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

### 2012/11/21

#### PDE Real Analysis Seminar

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

Shape Optimization And Asymptotic For The Twisted Dirichlet Eigenvalue (ENGLISH)

**Giovanni Pisante**(Seconda Università degli Studi di Napoli)Shape Optimization And Asymptotic For The Twisted Dirichlet Eigenvalue (ENGLISH)

[ Abstract ]

Aim of the talk is to discuss some recent results obtained with G. Croce and A. Henrot on a generalization of the functional defining the first twisted eigenvalue.

We look at the set functional defined by minimizing a Rayleigh quotient involving Lebesgue norms with different exponents p and q among functions satisfying a zero boundary condition as well as a zero mean condition of order q.

First under suitable conditions on p and q, that ensure the existence of a minimizing function, we investigate the validity of an isoperimetric type inequality of the Reyleigh-Faber-Krahn type.

Then we study the limit of the functional for p and q tending to 1 and to infinity and discuss the relation with the limits of the second eigenvalues of the p-laplacian operator.

Aim of the talk is to discuss some recent results obtained with G. Croce and A. Henrot on a generalization of the functional defining the first twisted eigenvalue.

We look at the set functional defined by minimizing a Rayleigh quotient involving Lebesgue norms with different exponents p and q among functions satisfying a zero boundary condition as well as a zero mean condition of order q.

First under suitable conditions on p and q, that ensure the existence of a minimizing function, we investigate the validity of an isoperimetric type inequality of the Reyleigh-Faber-Krahn type.

Then we study the limit of the functional for p and q tending to 1 and to infinity and discuss the relation with the limits of the second eigenvalues of the p-laplacian operator.

#### Classical Analysis

16:00-17:30 Room #128 (Graduate School of Math. Sci. Bldg.)

Beyond the fundamental group (ENGLISH)

**Philip Boalch**(ENS-DMA & CNRS Paris)Beyond the fundamental group (ENGLISH)

[ Abstract ]

Moduli spaces of representations of the fundamental group of a Riemann surface have been studied from numerous points of view and appear in many parts of mathematics and theoretical physics. They form an interesting class of symplectic manifolds, they often have Kahler or hyperkahler metrics (in which case they are diffeomorphic to spaces of Higgs bundles, i.e. Hitchin integrable systems), and they admit nonlinear actions of braid groups and mapping class groups with fascinating dynamical properties. The aim of this talk is to describe some aspects of this story and sketch their extension to the context of the "wild fundamental group", which naturally appears when one considers {\\em meromorphic} connections on Riemann surfaces. In particular some new examples of hyperkahler manifolds appear in this way, some of which are familiar from classical work on the Painleve equations.

Moduli spaces of representations of the fundamental group of a Riemann surface have been studied from numerous points of view and appear in many parts of mathematics and theoretical physics. They form an interesting class of symplectic manifolds, they often have Kahler or hyperkahler metrics (in which case they are diffeomorphic to spaces of Higgs bundles, i.e. Hitchin integrable systems), and they admit nonlinear actions of braid groups and mapping class groups with fascinating dynamical properties. The aim of this talk is to describe some aspects of this story and sketch their extension to the context of the "wild fundamental group", which naturally appears when one considers {\\em meromorphic} connections on Riemann surfaces. In particular some new examples of hyperkahler manifolds appear in this way, some of which are familiar from classical work on the Painleve equations.

### 2012/11/20

#### Tuesday Seminar on Topology

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

3 dimensional hyperbolic geometry and cluster algebras (JAPANESE)

**Kentaro Nagao**(Nagoya University)3 dimensional hyperbolic geometry and cluster algebras (JAPANESE)

[ Abstract ]

The cluster algebra was discovered by Fomin-Zelevinsky in 2000.

Recently, the structures of cluster algebras are recovered in

many areas including the theory of quantum groups, low

dimensional topology, discrete integrable systems, Donaldson-Thomas

theory, and string theory and there is dynamic development in the

research on these subjects. In this talk I introduce a relation between

3 dimensional hyperbolic geometry and cluster algebras motivated

by some duality in string theory.

The cluster algebra was discovered by Fomin-Zelevinsky in 2000.

Recently, the structures of cluster algebras are recovered in

many areas including the theory of quantum groups, low

dimensional topology, discrete integrable systems, Donaldson-Thomas

theory, and string theory and there is dynamic development in the

research on these subjects. In this talk I introduce a relation between

3 dimensional hyperbolic geometry and cluster algebras motivated

by some duality in string theory.

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