## Seminar information archive

Seminar information archive ～04/19｜Today's seminar 04/20 | Future seminars 04/21～

### 2015/01/10

#### Harmonic Analysis Komaba Seminar

13:00-18:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Scattering theory for the Laplacian on symmetric spaces of noncompact type and its application (JAPANESE)

スケール不変性を持つ臨界Hardyの不等式について (JAPANESE)

**Koichi Kaizuka**(Gakushuin University) 13:30-15:00Scattering theory for the Laplacian on symmetric spaces of noncompact type and its application (JAPANESE)

**Norisuke Ioku**(Ehime University) 15:30-17:00スケール不変性を持つ臨界Hardyの不等式について (JAPANESE)

### 2015/01/07

#### Number Theory Seminar

16:40-17:40 Room #056 (Graduate School of Math. Sci. Bldg.)

Congruences of modular forms modulo p and a variant of the Breuil-Mézard conjecture (English)

**Sandra Rozensztajn**(ENS de Lyon)Congruences of modular forms modulo p and a variant of the Breuil-Mézard conjecture (English)

[ Abstract ]

In this talk I will explain how a problem of congruences modulo p in the space of modular forms $S_k(\Gamma_0(p))$ is related to the geometry of some deformation spaces of Galois representations and can be solved by using a variant of the Breuil-Mézard conjecture.

In this talk I will explain how a problem of congruences modulo p in the space of modular forms $S_k(\Gamma_0(p))$ is related to the geometry of some deformation spaces of Galois representations and can be solved by using a variant of the Breuil-Mézard conjecture.

#### Operator Algebra Seminars

16:30-18:00 Room #122 (Graduate School of Math. Sci. Bldg.)

De Finetti theorems related to Boolean independence (English)

**Tomohiro Hayase**(Univ. Tokyo)De Finetti theorems related to Boolean independence (English)

### 2015/01/06

#### PDE Real Analysis Seminar

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

Global existence and asymptotic behavior for some Keller-Segel systems coupled with Navier-Stokes equations (英語)

**Elio Eduardo Espejo**(National University of Colombia / Osaka University)Global existence and asymptotic behavior for some Keller-Segel systems coupled with Navier-Stokes equations (英語)

[ Abstract ]

There are plenty of examples in nature, where cells move in response to some chemical signal in the environment. Biologists call this phenomenon chemotaxis. In my talk I will approach the problem of describing mathematically the phenomenon of chemotaxis when it happens surrounded by a fluid. This is a new research topic bringing the attention of many scientists because it has given rise to many interesting questions having relevance in both biology and mathematics. In particular, I will present some new mathematical models arising from my current research that have given rise to Keller-Segel type systems coupled with Navier-Stokes systems. I will present some results of global existence and asymptotic behavior. Finally I will discuss some open problems.

There are plenty of examples in nature, where cells move in response to some chemical signal in the environment. Biologists call this phenomenon chemotaxis. In my talk I will approach the problem of describing mathematically the phenomenon of chemotaxis when it happens surrounded by a fluid. This is a new research topic bringing the attention of many scientists because it has given rise to many interesting questions having relevance in both biology and mathematics. In particular, I will present some new mathematical models arising from my current research that have given rise to Keller-Segel type systems coupled with Navier-Stokes systems. I will present some results of global existence and asymptotic behavior. Finally I will discuss some open problems.

### 2014/12/22

#### Mathematical Biology Seminar

15:00-16:20 Room #122 (Graduate School of Math. Sci. Bldg.)

Estimating the seroincidence of pertussis in Japan

**Don Yueping**(Department of Global Health Policy, Graduate School of Medicine, The University of Tokyo)Estimating the seroincidence of pertussis in Japan

[ Abstract ]

Despite relatively high vaccination coverage of pertussis for decades, the disease keeps circulating among both vaccinated and unvaccinated individuals and a periodic large epidemic is observed every 4 years. To understand the transmission dynamics, specific immunoglobulin G (IgG) antibodies against pertussis toxin (PT) have been routinely measured in Japan. Using the cross-sectional serological survey data with a known decay rate of antibody titres as a function of time since infection, we estimate the age-dependent seroincidence of pertussis. The estimated incidence of pertussis declined with age, the shape of which will be extremely useful for reconstructing the transmission dynamics and considering effective countermeasures.

Despite relatively high vaccination coverage of pertussis for decades, the disease keeps circulating among both vaccinated and unvaccinated individuals and a periodic large epidemic is observed every 4 years. To understand the transmission dynamics, specific immunoglobulin G (IgG) antibodies against pertussis toxin (PT) have been routinely measured in Japan. Using the cross-sectional serological survey data with a known decay rate of antibody titres as a function of time since infection, we estimate the age-dependent seroincidence of pertussis. The estimated incidence of pertussis declined with age, the shape of which will be extremely useful for reconstructing the transmission dynamics and considering effective countermeasures.

### 2014/12/19

#### Geometry Colloquium

10:00-11:30 Room #126 (Graduate School of Math. Sci. Bldg.)

Exotic components in linear slices of quasi-Fuchsian groups

**Yuichi KABAYA**(Kyoto University)Exotic components in linear slices of quasi-Fuchsian groups

[ Abstract ]

The linear slice of quasi-Fuchsian punctured torus groups is defined by fixing the length of some simple closed curve to be a fixed positive real number. It is known that the linear slice is a union of disks, and it has one `standard' component containing Fuchsian groups. Komori-Yamashita proved that there exist non-standard components if the length is sufficiently large. In this talk, I give another proof based on the theory of complex projective structures. If time permits, I will talk about a refined statement and a generalization to other surfaces.

The linear slice of quasi-Fuchsian punctured torus groups is defined by fixing the length of some simple closed curve to be a fixed positive real number. It is known that the linear slice is a union of disks, and it has one `standard' component containing Fuchsian groups. Komori-Yamashita proved that there exist non-standard components if the length is sufficiently large. In this talk, I give another proof based on the theory of complex projective structures. If time permits, I will talk about a refined statement and a generalization to other surfaces.

### 2014/12/17

#### Number Theory Seminar

18:00-19:00 Room #117 (Graduate School of Math. Sci. Bldg.)

Equivariant $\wideparen{\mathcal{D}}$ modules on rigid analytic spaces

(English)

**Konstantin Ardakov**(University of Oxford)Equivariant $\wideparen{\mathcal{D}}$ modules on rigid analytic spaces

(English)

[ Abstract ]

Locally analytic representations of p-adic Lie groups are of interest in several branches of arithmetic algebraic geometry, notably the p-adic local Langlands program. I will discuss some work in progress towards a Beilinson-Bernstein style localisation theorem for admissible locally analytic representations of semisimple compact p-adic Lie groups using equivariant formal models of rigid analytic flag varieties.

Locally analytic representations of p-adic Lie groups are of interest in several branches of arithmetic algebraic geometry, notably the p-adic local Langlands program. I will discuss some work in progress towards a Beilinson-Bernstein style localisation theorem for admissible locally analytic representations of semisimple compact p-adic Lie groups using equivariant formal models of rigid analytic flag varieties.

#### Operator Algebra Seminars

16:30-18:00 Room #122 (Graduate School of Math. Sci. Bldg.)

Dynamics of an Open Quantum System with Repeated Harmonic Perturbation (with Hiroshi Tamura) (English)

**Valentin Zagrebnov**(Univ. d'Aix-Marseille)Dynamics of an Open Quantum System with Repeated Harmonic Perturbation (with Hiroshi Tamura) (English)

#### Mathematical Biology Seminar

14:50-16:20 Room #122 (Graduate School of Math. Sci. Bldg.)

A probabilistic interpretation of an evolution model of slime bacteria

(JAPANESE)

**Yumi YAHAGI**(Tokyo City University)A probabilistic interpretation of an evolution model of slime bacteria

(JAPANESE)

### 2014/12/16

#### Tuesday Seminar of Analysis

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

Global Strichartz estimates for Schr¥”odinger equations on

asymptotically conic manifolds (Japanese)

**Haruya Mizutani**(Graduate School of Science, Osaka University)Global Strichartz estimates for Schr¥”odinger equations on

asymptotically conic manifolds (Japanese)

#### Tuesday Seminar on Topology

17:10-18:10 Room #056 (Graduate School of Math. Sci. Bldg.)

Differential forms in diffeological spaces (JAPANESE)

**Norio Iwase**(Kyushu University)Differential forms in diffeological spaces (JAPANESE)

[ Abstract ]

The idea of a space with smooth structure is first introduced by K. T. Chen in his study of a loop space to employ the idea of iterated path integrals.

Following the pattern established by Chen, J. M. Souriau introduced his version of a space with smooth structure which is now called diffeology and become one of the most exciting topics in Algebraic Topology. Following Souriau, P. I.-Zenmour presented de Rham theory associated to a diffeology of a space. However, if one tries to show a version of de Rham theorem for a general diffeological space, he must encounter a difficulty to show the existence of a partition of unity and thus the exactness of the Mayer-Vietoris sequence. To resolve such difficulties, we introduce a new definition of differential forms.

The idea of a space with smooth structure is first introduced by K. T. Chen in his study of a loop space to employ the idea of iterated path integrals.

Following the pattern established by Chen, J. M. Souriau introduced his version of a space with smooth structure which is now called diffeology and become one of the most exciting topics in Algebraic Topology. Following Souriau, P. I.-Zenmour presented de Rham theory associated to a diffeology of a space. However, if one tries to show a version of de Rham theorem for a general diffeological space, he must encounter a difficulty to show the existence of a partition of unity and thus the exactness of the Mayer-Vietoris sequence. To resolve such difficulties, we introduce a new definition of differential forms.

### 2014/12/15

#### Seminar on Geometric Complex Analysis

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

The limits of Kähler-Ricci flows

**Hajime Tsuji**(Sophia University)The limits of Kähler-Ricci flows

#### Algebraic Geometry Seminar

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

A characterization of ordinary abelian varieties in positive characteristic (JAPANESE)

**Akiyoshi Sannai**(University of Tokyo)A characterization of ordinary abelian varieties in positive characteristic (JAPANESE)

[ Abstract ]

This is joint work with Hiromu Tanaka. In this talk, we study F^e_*O_X on a projective variety over the algebraic closed field of positive characteristic. For an ordinary abelian variety X, F^e_*O_X is decomposed into line bundles for every positive integer e. Conversely, if a smooth projective variety X satisfies this property and its Kodaira dimension is non-negative, then X is an ordinary abelian variety.

This is joint work with Hiromu Tanaka. In this talk, we study F^e_*O_X on a projective variety over the algebraic closed field of positive characteristic. For an ordinary abelian variety X, F^e_*O_X is decomposed into line bundles for every positive integer e. Conversely, if a smooth projective variety X satisfies this property and its Kodaira dimension is non-negative, then X is an ordinary abelian variety.

### 2014/12/11

#### Infinite Analysis Seminar Tokyo

15:00-18:30 Room #002 (Graduate School of Math. Sci. Bldg.)

Renormalization group method for many-electron systems (JAPANESE)

Unitary transformations and multivariate special

orthogonal polynomials (JAPANESE)

**Yohei Kashima**(Graduate School of Mathematical Sciences, the University of Tokyo) 15:00-16:30Renormalization group method for many-electron systems (JAPANESE)

[ Abstract ]

We consider quantum many-body systems of electrons

hopping and interacting on a lattice at positive temperature.

As it is possible to write down each order term rigorously in

principle, the perturbation series expansion with the coupling

constant between electrons is thought as a valid method to

compute physical quantities. By directly estimating each term,

one can prove that the perturbation series is convergent if the

coupling constant is less than some power of temperature. This is,

however, a serious constraint for models of interacting electrons

in low temperature. In order to ensure the analyticity of physical

quantities of many-electron systems with the coupling constant in

low temperature, renormalization group methods have been developed

in recent years. As one progress in this direction, we construct a

renormalization group method for the half-filled Hubbard model on a

square lattice, which is a typical model of many-electron, and prove

the following. If the system contains the magnetic flux pi (mod 2 pi)

per plaquette, the free energy density of the system is analytic with

the coupling constant in a neighborhood of the origin and it uniformly

converges to the infinite-volume, zero-temperature limit. It is known

that the flux pi condition is sufficient for the free energy density

to be minimum. Thus, it follows that the same analyticity and the

convergent property hold for the minimum free energy density of the

system.

We consider quantum many-body systems of electrons

hopping and interacting on a lattice at positive temperature.

As it is possible to write down each order term rigorously in

principle, the perturbation series expansion with the coupling

constant between electrons is thought as a valid method to

compute physical quantities. By directly estimating each term,

one can prove that the perturbation series is convergent if the

coupling constant is less than some power of temperature. This is,

however, a serious constraint for models of interacting electrons

in low temperature. In order to ensure the analyticity of physical

quantities of many-electron systems with the coupling constant in

low temperature, renormalization group methods have been developed

in recent years. As one progress in this direction, we construct a

renormalization group method for the half-filled Hubbard model on a

square lattice, which is a typical model of many-electron, and prove

the following. If the system contains the magnetic flux pi (mod 2 pi)

per plaquette, the free energy density of the system is analytic with

the coupling constant in a neighborhood of the origin and it uniformly

converges to the infinite-volume, zero-temperature limit. It is known

that the flux pi condition is sufficient for the free energy density

to be minimum. Thus, it follows that the same analyticity and the

convergent property hold for the minimum free energy density of the

system.

**Genki Shibukawa**(Institute of Mathematics for Industory, Kyushu University) 17:00-18:30Unitary transformations and multivariate special

orthogonal polynomials (JAPANESE)

[ Abstract ]

Investigations into special orthogonal function systems by

using unitary transformations have a long history.

This is, by calculating an image of some unitary transform (e.g. Fourier

trans.) of a known orthogonal system, we derive a new orthogonal system

and obtain its fundamental properties.

This basic concept and technique have been known since ancient times for

a single variable case, and recently these multivariate analogue has

been studied by Davidson, Olafsson, Zhang, Faraut, Wakayama et.al..

In our talk, we introduce the unitary picture for the circular Jacobi

polynomials obtained by Shen, further give a multivariate analogue of

the results of Shen.

These polynomials, which we call multivariate circular Jacobi (MCJ)

polynomials, are generalizations (2-parameter deformation) of the

spherical (zonal) polynomials that are different from the Jack or

Macdonald polynomials, which are well known as an extension of spherical

polynomials.

We also remark that the weight function of their orthogonality relation

coincides with the circular Jacobi ensemble defined by Bourgade,et,al.,

and the modified Cayley transform of the MCJ polynomials satisfy with

some quasi differential equation.

In addition, we can give a generalization of MCJ polynomials as

including the Jack polynomials.

For this generalized MCJ polynomials, we would like to present some

conjectures and problems.

If we have time, we also describe a unitary picture of Meixner,

Charlier and Krawtchouk polynomials which are typical examples of

discrete orthogonal systems, and mention their multivariate analogue.

Investigations into special orthogonal function systems by

using unitary transformations have a long history.

This is, by calculating an image of some unitary transform (e.g. Fourier

trans.) of a known orthogonal system, we derive a new orthogonal system

and obtain its fundamental properties.

This basic concept and technique have been known since ancient times for

a single variable case, and recently these multivariate analogue has

been studied by Davidson, Olafsson, Zhang, Faraut, Wakayama et.al..

In our talk, we introduce the unitary picture for the circular Jacobi

polynomials obtained by Shen, further give a multivariate analogue of

the results of Shen.

These polynomials, which we call multivariate circular Jacobi (MCJ)

polynomials, are generalizations (2-parameter deformation) of the

spherical (zonal) polynomials that are different from the Jack or

Macdonald polynomials, which are well known as an extension of spherical

polynomials.

We also remark that the weight function of their orthogonality relation

coincides with the circular Jacobi ensemble defined by Bourgade,et,al.,

and the modified Cayley transform of the MCJ polynomials satisfy with

some quasi differential equation.

In addition, we can give a generalization of MCJ polynomials as

including the Jack polynomials.

For this generalized MCJ polynomials, we would like to present some

conjectures and problems.

If we have time, we also describe a unitary picture of Meixner,

Charlier and Krawtchouk polynomials which are typical examples of

discrete orthogonal systems, and mention their multivariate analogue.

### 2014/12/10

#### Operator Algebra Seminars

16:30-18:00 Room #122 (Graduate School of Math. Sci. Bldg.)

Approximately inner flows on $C^*$-algebras (English)

**Akitaka Kishimoto**(Hokkaido Univ.)Approximately inner flows on $C^*$-algebras (English)

#### FMSP Lectures

16:00-17:30 Room #118 (Graduate School of Math. Sci. Bldg.)

Singular limit analysis of a reaction-diffusion system with precipitation and dissolution in a porous medium

**Danielle Hilhorst**(CNRS / Univ. Paris-Sud)Singular limit analysis of a reaction-diffusion system with precipitation and dissolution in a porous medium

[ Abstract ]

This talk is concerned with a mathematical model for the storage of radioactive waste. The model which we study deals with the diffusion of chemical species transported by water, with possible dissolution or precipitation and for a rather general kinetics law. In this talk, we consider a three-component reaction-diffusion system with a fast precipitation and dissolution reaction term. We investigate its singular limit as the reaction rate tends to infinity. The limit problem is described by the combination of a Stefan problem and a linear heat equation. The rate of convergence with respect to the reaction rate is established in a specific case. This is joint work with Hideki Murakawa.

This talk is concerned with a mathematical model for the storage of radioactive waste. The model which we study deals with the diffusion of chemical species transported by water, with possible dissolution or precipitation and for a rather general kinetics law. In this talk, we consider a three-component reaction-diffusion system with a fast precipitation and dissolution reaction term. We investigate its singular limit as the reaction rate tends to infinity. The limit problem is described by the combination of a Stefan problem and a linear heat equation. The rate of convergence with respect to the reaction rate is established in a specific case. This is joint work with Hideki Murakawa.

### 2014/12/09

#### Tuesday Seminar on Topology

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

Stable commutator length on mapping class groups (JAPANESE)

**Koji Fujiwara**(Kyoto University)Stable commutator length on mapping class groups (JAPANESE)

[ Abstract ]

Let MCG(S) be the mapping class group of a closed orientable surface S.

We give a precise condition (in terms of the Nielsen-Thurston

decomposition) when an element

in MCG(S) has positive stable commutator length.

Stable commutator length tends to be positive if there is "negative

curvature".

The proofs use our earlier construction in the paper "Constructing group

actions on quasi-trees and applications to mapping class groups" of

group actions on quasi-trees.

This is a joint work with Bestvina and Bromberg.

Let MCG(S) be the mapping class group of a closed orientable surface S.

We give a precise condition (in terms of the Nielsen-Thurston

decomposition) when an element

in MCG(S) has positive stable commutator length.

Stable commutator length tends to be positive if there is "negative

curvature".

The proofs use our earlier construction in the paper "Constructing group

actions on quasi-trees and applications to mapping class groups" of

group actions on quasi-trees.

This is a joint work with Bestvina and Bromberg.

### 2014/12/08

#### Seminar on Geometric Complex Analysis

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

On maximal ideal cycle and fundamental cycle of normal two-dimensional quasi-homogeneous singularities, and singularities with star-shaped resolution (JAPANESE)

**Masatake Tomari**(Nihon University)On maximal ideal cycle and fundamental cycle of normal two-dimensional quasi-homogeneous singularities, and singularities with star-shaped resolution (JAPANESE)

### 2014/12/04

#### Geometry Colloquium

17:00-18:30 Room #123 (Graduate School of Math. Sci. Bldg.)

Deformations and the moduli spaces of generalized complex manifolds (JAPANESE)

**Ryushi Goto**(Osaka University)Deformations and the moduli spaces of generalized complex manifolds (JAPANESE)

### 2014/12/03

#### Operator Algebra Seminars

16:30-18:00 Room #122 (Graduate School of Math. Sci. Bldg.)

Central property (T) for $SU_q(2n+1)$

(English)

**Yuki Arano**(Univ. Tokyo)Central property (T) for $SU_q(2n+1)$

(English)

#### Lectures

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

New isoperimetric inequalities with densities, part II: Detailed proofs and related works (ENGLISH)

**Xavier Cabre**(ICREA and UPC, Barcelona)New isoperimetric inequalities with densities, part II: Detailed proofs and related works (ENGLISH)

[ Abstract ]

This is a sequel to the Tuesday Analysis Seminar on December 2 by the same speaker.

In joint works with X. Ros-Oton and J. Serra, the study of the regularity of stable solutions to reaction-diffusion problems has led us to certain Sobolev and isoperimetric inequalities with weights. We will present our results in these new isoperimetric inequalities with the best constant, that we establish via the ABP method.

More precisely, we obtain a new family of sharp isoperimetric inequalities with weights (or densities) in open convex cones of R^n. Our results apply to all nonnegative homogeneous weights satisfying a concavity condition in the cone. Surprisingly, even that our weights are not radially symmetric, Euclidean balls centered at the origin (intersected with the cone) minimize the weighted isoperimetric quotient. As a particular case of our results, we provide with new proofs of classical results such as the Wulff inequality and the isoperimetric inequality in convex cones of Lions and Pacella. Furthermore, we also study the anisotropic isoperimetric problem for the same class of weights and we prove that the Wulff shape always minimizes the anisotropic weighted perimeter under the weighted volume constraint.

This is a sequel to the Tuesday Analysis Seminar on December 2 by the same speaker.

In joint works with X. Ros-Oton and J. Serra, the study of the regularity of stable solutions to reaction-diffusion problems has led us to certain Sobolev and isoperimetric inequalities with weights. We will present our results in these new isoperimetric inequalities with the best constant, that we establish via the ABP method.

More precisely, we obtain a new family of sharp isoperimetric inequalities with weights (or densities) in open convex cones of R^n. Our results apply to all nonnegative homogeneous weights satisfying a concavity condition in the cone. Surprisingly, even that our weights are not radially symmetric, Euclidean balls centered at the origin (intersected with the cone) minimize the weighted isoperimetric quotient. As a particular case of our results, we provide with new proofs of classical results such as the Wulff inequality and the isoperimetric inequality in convex cones of Lions and Pacella. Furthermore, we also study the anisotropic isoperimetric problem for the same class of weights and we prove that the Wulff shape always minimizes the anisotropic weighted perimeter under the weighted volume constraint.

#### Mathematical Biology Seminar

14:50-16:20 Room #122 (Graduate School of Math. Sci. Bldg.)

Global analysis of age-structured SIS epidemic models with spatial

heterogeneity (JAPANESE)

**Toshikazu Kuniya**(Graduate School of System Informatics, Kobe University)Global analysis of age-structured SIS epidemic models with spatial

heterogeneity (JAPANESE)

### 2014/12/02

#### Tuesday Seminar on Topology

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

The Atiyah-Segal completion theorem in noncommutative topology (JAPANESE)

**Yosuke Kubota**(The University of Tokyo)The Atiyah-Segal completion theorem in noncommutative topology (JAPANESE)

[ Abstract ]

We introduce a new perspevtive on the Atiyah-Segal completion

theorem applying the "noncommutative" topology, which deals with

topological properties of C*-algebras. The homological algebra of the

Kasparov category as a triangulated category, which is developed by R.

Meyer and R. Nest, plays a central role. It contains the Atiyah-Segal

type completion theorems for equivariant K-homology and twisted K-theory.

This is a joint work with Yuki Arano.

We introduce a new perspevtive on the Atiyah-Segal completion

theorem applying the "noncommutative" topology, which deals with

topological properties of C*-algebras. The homological algebra of the

Kasparov category as a triangulated category, which is developed by R.

Meyer and R. Nest, plays a central role. It contains the Atiyah-Segal

type completion theorems for equivariant K-homology and twisted K-theory.

This is a joint work with Yuki Arano.

#### PDE Real Analysis Seminar

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

Remark on single exponential bound of the vorticity gradient for the two-dimensional Euler flow around a corner (JAPANESE)

**Tsubasa Itoh**(Tokyo Institute of Technology)Remark on single exponential bound of the vorticity gradient for the two-dimensional Euler flow around a corner (JAPANESE)

[ Abstract ]

In this talk, the two dimensional Euler flow under a simple symmetry condition with hyperbolic structure in a unit square $D=\{(x_{1}, x_{2}): 0 < x_{1} + x_{2} < \sqrt{2},\ 0<-x_{1} + x_{2} < \sqrt{2}\}$ is considered.

It is shown that the Lipschitz estimate of the vorticity on the boundary is at most single exponential growth near the stagnation point.

(Joint work with Tsuyoshi Yoneda and Hideyuki Miura.)

In this talk, the two dimensional Euler flow under a simple symmetry condition with hyperbolic structure in a unit square $D=\{(x_{1}, x_{2}): 0 < x_{1} + x_{2} < \sqrt{2},\ 0<-x_{1} + x_{2} < \sqrt{2}\}$ is considered.

It is shown that the Lipschitz estimate of the vorticity on the boundary is at most single exponential growth near the stagnation point.

(Joint work with Tsuyoshi Yoneda and Hideyuki Miura.)

#### Tuesday Seminar of Analysis

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

New isoperimetric inequalities with densities arising in reaction-diffusion problems (English)

**Xavier Cabre**(ICREA and UPC, Barcelona)New isoperimetric inequalities with densities arising in reaction-diffusion problems (English)

[ Abstract ]

In joint works with X. Ros-Oton and J. Serra, the study of the

regularity of stable solutions to reaction-diffusion problems

has led us to certain Sobolev and isoperimetric inequalities

with weights. We will present our results in these new

isoperimetric inequalities with the best constant, that we

establish via the ABP method. More precisely, we obtain

a new family of sharp isoperimetric inequalities with weights

(or densities) in open convex cones of R^n. Our results apply

to all nonnegative homogeneous weights satisfying a concavity

condition in the cone. Surprisingly, even that our weights are

not radially symmetric, Euclidean balls centered at the origin

(intersected with the cone) minimize the weighted isoperimetric

quotient. As a particular case of our results, we provide with

new proofs of classical results such as the Wulff inequality and

the isoperimetric inequality in convex cones of Lions and Pacella.

Furthermore, we also study the anisotropic isoperimetric problem

for the same class of weights and we prove that the Wulff shape

always minimizes the anisotropic weighted perimeter under the

weighted volume constraint.

In joint works with X. Ros-Oton and J. Serra, the study of the

regularity of stable solutions to reaction-diffusion problems

has led us to certain Sobolev and isoperimetric inequalities

with weights. We will present our results in these new

isoperimetric inequalities with the best constant, that we

establish via the ABP method. More precisely, we obtain

a new family of sharp isoperimetric inequalities with weights

(or densities) in open convex cones of R^n. Our results apply

to all nonnegative homogeneous weights satisfying a concavity

condition in the cone. Surprisingly, even that our weights are

not radially symmetric, Euclidean balls centered at the origin

(intersected with the cone) minimize the weighted isoperimetric

quotient. As a particular case of our results, we provide with

new proofs of classical results such as the Wulff inequality and

the isoperimetric inequality in convex cones of Lions and Pacella.

Furthermore, we also study the anisotropic isoperimetric problem

for the same class of weights and we prove that the Wulff shape

always minimizes the anisotropic weighted perimeter under the

weighted volume constraint.

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