## Tuesday Seminar of Analysis

Seminar information archive ～09/27｜Next seminar｜Future seminars 09/28～

Date, time & place | Tuesday 16:00 - 17:30 156Room #156 (Graduate School of Math. Sci. Bldg.) |
---|---|

Organizer(s) | ISHIGE Kazuhiro, SAKAI Hidetaka, ITO Kenichi |

**Seminar information archive**

### 2013/05/21

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

Homogenization in a Thin Layer with an Oscillating Interface and Highly Contrast Coefficients (JAPANESE)

**Masaaki Uesaka**(Graduate School of Mathematical Sciences, The University of Tokyo)Homogenization in a Thin Layer with an Oscillating Interface and Highly Contrast Coefficients (JAPANESE)

[ Abstract ]

We consider the homogenization problem of the elliptic boundary value problem in a thin domain which has a high and low conductivity zones. In our model, two media are separated by a highly oscillating interface. The asymptotic behavior is governed by the order of the thickness of the domain, oscillation period of the interface and contrast between two media. In this talk, we show that the limit problem is changed by these parameters. We also introduce the two-scale convergence result in a thin domain which is the key ingredient of the proof.

We consider the homogenization problem of the elliptic boundary value problem in a thin domain which has a high and low conductivity zones. In our model, two media are separated by a highly oscillating interface. The asymptotic behavior is governed by the order of the thickness of the domain, oscillation period of the interface and contrast between two media. In this talk, we show that the limit problem is changed by these parameters. We also introduce the two-scale convergence result in a thin domain which is the key ingredient of the proof.

### 2012/12/11

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

This talk was cancelled! (JAPANESE)

**Rafe Mazzeo**(Stanford University)This talk was cancelled! (JAPANESE)

### 2012/12/04

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.

### 2012/11/06

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

Resonance free domains for homoclinic orbits (ENGLISH)

**Thierry Ramond**(Univ. Paris, Orsay)Resonance free domains for homoclinic orbits (ENGLISH)

### 2012/10/30

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

About the method of characteristics (ENGLISH)

**Francis Nier**(Univ. Rennes 1)About the method of characteristics (ENGLISH)

[ Abstract ]

While studying the mean field dynamics of a systems of bosons, one is led to solve a transport equation for a probability measure in an infinite dimensional phase-space. Since those probability measures are characterized after testing with cylindrical or polynomial observables, which make classes which are not invariant after composing with a nonlinear flow. Thus the standard method of characteristics for transport equations cannot be extended at once to the infinite dimensional case. A solution comes from techniques developed for optimal transport and a probabilistic interpretation of trajectories.

While studying the mean field dynamics of a systems of bosons, one is led to solve a transport equation for a probability measure in an infinite dimensional phase-space. Since those probability measures are characterized after testing with cylindrical or polynomial observables, which make classes which are not invariant after composing with a nonlinear flow. Thus the standard method of characteristics for transport equations cannot be extended at once to the infinite dimensional case. A solution comes from techniques developed for optimal transport and a probabilistic interpretation of trajectories.

### 2012/10/23

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

Topics in quantum entropy and entanglement (ENGLISH)

**Elliott Lieb**(Princeton Univ.)Topics in quantum entropy and entanglement (ENGLISH)

[ Abstract ]

Several recent results on quantum entropy and the uncertainty

principle will be discussed. This is partly joint work with Eric Carlen

on lower bounds for entanglement, which has no classical analog, in terms

of the negative of the conditional entropy, S1 - S12, whose negativity,

when it occurs, also has no classical analog. (see arXiv:1203.4719)

It is also partly joint work with Rupert Frank on the uncertaintly

principle for quantum entropy which compares the quantum von Neumann

entropy with the classical entropies with respect to two different

bases. We prove an extension to the product of two and three spaces, which

has applications in quantum information theory. (see arxiv:1204.0825)

Several recent results on quantum entropy and the uncertainty

principle will be discussed. This is partly joint work with Eric Carlen

on lower bounds for entanglement, which has no classical analog, in terms

of the negative of the conditional entropy, S1 - S12, whose negativity,

when it occurs, also has no classical analog. (see arXiv:1203.4719)

It is also partly joint work with Rupert Frank on the uncertaintly

principle for quantum entropy which compares the quantum von Neumann

entropy with the classical entropies with respect to two different

bases. We prove an extension to the product of two and three spaces, which

has applications in quantum information theory. (see arxiv:1204.0825)

### 2012/07/17

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

On non-radially symmetric solutions of the Liouville-Gel'fand equation on a two-dimensional annular domain (JAPANESE)

**Toru Kan**(Mathematical institute, Tohoku University)On non-radially symmetric solutions of the Liouville-Gel'fand equation on a two-dimensional annular domain (JAPANESE)

[ Abstract ]

指数関数を非線形項に持つ非線形楕円型方程式(Liouville-Gel'fand方程式)について考察する。特に2次元の円環領域では、この方程式の非球対称な解が球対称解から分岐する形で現れる。本講演では、この分岐解の分岐図上での大域的な構造に関して得られた結果を紹介する。

指数関数を非線形項に持つ非線形楕円型方程式(Liouville-Gel'fand方程式)について考察する。特に2次元の円環領域では、この方程式の非球対称な解が球対称解から分岐する形で現れる。本講演では、この分岐解の分岐図上での大域的な構造に関して得られた結果を紹介する。

### 2012/07/10

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

Hadamard variational formula for the Green function

of the Stokes equations with the boundary condition (JAPANESE)

**Erika Ushikoshi**(Mathematical Institute, Tohoku University)Hadamard variational formula for the Green function

of the Stokes equations with the boundary condition (JAPANESE)

### 2012/06/26

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

Absence of embedded eigenvalues for the Schr\\"odinger operator on manifold with ends (JAPANESE)

**Kenichi Ito**(Division of Mathematics, University of Tsukuba)Absence of embedded eigenvalues for the Schr\\"odinger operator on manifold with ends (JAPANESE)

[ Abstract ]

We consider a Riemannian manifold with, at least, one expanding end, and prove the absence of $L^2$-eigenvalues for the Schr\\"odinger operator above some critical value. The critical value is computed from the volume growth rate of the end and the potential behavior at infinity. The end structure is formulated abstractly in terms of some convex function, and the examples include asymptotically Euclidean and hyperbolic ends. The proof consists of a priori superexponential decay estimate for eigenfunctions and the absence of superexponentially decaying eigenfunctions, both of which employs the Mourre-type commutator argument. This talk is based on the recent joint work with E.Skibsted (Aarhus University).

We consider a Riemannian manifold with, at least, one expanding end, and prove the absence of $L^2$-eigenvalues for the Schr\\"odinger operator above some critical value. The critical value is computed from the volume growth rate of the end and the potential behavior at infinity. The end structure is formulated abstractly in terms of some convex function, and the examples include asymptotically Euclidean and hyperbolic ends. The proof consists of a priori superexponential decay estimate for eigenfunctions and the absence of superexponentially decaying eigenfunctions, both of which employs the Mourre-type commutator argument. This talk is based on the recent joint work with E.Skibsted (Aarhus University).

### 2012/05/22

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

Viscosity solutions for nonlinear elliptic-parabolic problems (ENGLISH)

**Norbert Pozar**(Graduate School of Mathematical Sciences, The University of Tokyo)Viscosity solutions for nonlinear elliptic-parabolic problems (ENGLISH)

[ Abstract ]

We introduce a notion of viscosity solutions for a general class of

elliptic-parabolic phase transition problems. These include the

Richards equation, which is a classical model in filtration theory.

Existence and uniqueness results are proved via the comparison

principle. In particular, we show existence and stability properties

of maximal and minimal viscosity solutions for a general class of

initial data. These results are new even in the linear case, where we

also show that viscosity solutions coincide with the regular weak

solutions introduced in [Alt&Luckhaus 1983]. This talk is based on a

recent work with Inwon Kim.

We introduce a notion of viscosity solutions for a general class of

elliptic-parabolic phase transition problems. These include the

Richards equation, which is a classical model in filtration theory.

Existence and uniqueness results are proved via the comparison

principle. In particular, we show existence and stability properties

of maximal and minimal viscosity solutions for a general class of

initial data. These results are new even in the linear case, where we

also show that viscosity solutions coincide with the regular weak

solutions introduced in [Alt&Luckhaus 1983]. This talk is based on a

recent work with Inwon Kim.

### 2012/05/15

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

Strichartz estimates for Schr\\"odinger equations with variable coefficients and unbounded electromagnetic potentials (JAPANESE)

**MIZUTANI, Haruya**(Research Institute for Mathematical Sciences, Kyoto University)Strichartz estimates for Schr\\"odinger equations with variable coefficients and unbounded electromagnetic potentials (JAPANESE)

[ Abstract ]

In this talk we consider the Cauchy problem for Schr\\"odinger equations with variable coefficients and unbounded potentials. Under the assumption that the Hamiltonian is a long-range perturbation of the free Schr\\"odinger operator, we construct an outgoing parametrix for the propagator near infinity, and give applications to sharp Strichartz estimates. The basic idea is to combine the standard approximation by using a time dependent modifier, which is not in the semiclassical regime, with the semiclassical approximation of Isozaki-Kitada type. We also show near sharp Strichartz estimates without asymptotic conditions by using local smoothing effects.

In this talk we consider the Cauchy problem for Schr\\"odinger equations with variable coefficients and unbounded potentials. Under the assumption that the Hamiltonian is a long-range perturbation of the free Schr\\"odinger operator, we construct an outgoing parametrix for the propagator near infinity, and give applications to sharp Strichartz estimates. The basic idea is to combine the standard approximation by using a time dependent modifier, which is not in the semiclassical regime, with the semiclassical approximation of Isozaki-Kitada type. We also show near sharp Strichartz estimates without asymptotic conditions by using local smoothing effects.

### 2012/02/14

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

Symmetry results for Caffarelli-Kohn-Nirenberg inequalities (ENGLISH)

**Michael Loss**(Georgia Institute of Technology)Symmetry results for Caffarelli-Kohn-Nirenberg inequalities (ENGLISH)

### 2012/01/31

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

Obstacle problems in unbounded domains (ENGLISH)

**Michel Chipot**(University of Zurich)Obstacle problems in unbounded domains (ENGLISH)

[ Abstract ]

We will present a formulation of obstacle problems in unbounded

domains when the energy method does not work, i.e. whenthe force does not belong to H^{-1}.

We will present a formulation of obstacle problems in unbounded

domains when the energy method does not work, i.e. whenthe force does not belong to H^{-1}.

### 2011/12/20

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

A trace formula for the perturbed Landau Hamiltonian (ENGLISH)

**Gueorgui Raykov**(Catholic University of Chile)A trace formula for the perturbed Landau Hamiltonian (ENGLISH)

[ Abstract ]

The talk will be based on a joint work with A. Pushnitski

and C. Villegas-Blas, the preprint is available here:

http://arxiv.org/abs/1110.3098 .

The talk will be based on a joint work with A. Pushnitski

and C. Villegas-Blas, the preprint is available here:

http://arxiv.org/abs/1110.3098 .

### 2011/12/13

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

Trivializable subriemannian structures and spectral analysis of associated operators (ENGLISH)

**Wolfram Bauer**(Mathematisches Institut, Georg-August-Universität)Trivializable subriemannian structures and spectral analysis of associated operators (ENGLISH)

### 2011/11/08

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

The University of Tokyo (JSPS Research Fellow))

Stochastic Power-Law Fluid equations: Existence and Uniqueness of weak solutions (joint work with Nobuo Yoshida) (JAPANESE)

**Yutaka Terasawa**(Graduate School of Mathematical Sciences,The University of Tokyo (JSPS Research Fellow))

Stochastic Power-Law Fluid equations: Existence and Uniqueness of weak solutions (joint work with Nobuo Yoshida) (JAPANESE)

### 2011/10/11

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

On the best constant of the weighted Trudinger-Moser

type inequality (JAPANESE)

**Hidemitsu Wadade**(Waseda University (JSPS-PD))On the best constant of the weighted Trudinger-Moser

type inequality (JAPANESE)

### 2011/07/12

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

The inclusion relation between Sobolev and modulation spaces (JAPANESE)

**Masaharau Kobayashi**(Tokyo University of Science)The inclusion relation between Sobolev and modulation spaces (JAPANESE)

[ Abstract ]

In this talk, we consider the inclusion relations between the $L^p$-Sobolev spaces and the modulation spaces. As an application, we give mapping properties of unimodular Fourier multiplier $e^{i|D|^\\alpha}$ between $L^p$-Sobolev spaces and modulation spaces.

Joint work with Mitsuru Sugimoto (Nagoya University).

In this talk, we consider the inclusion relations between the $L^p$-Sobolev spaces and the modulation spaces. As an application, we give mapping properties of unimodular Fourier multiplier $e^{i|D|^\\alpha}$ between $L^p$-Sobolev spaces and modulation spaces.

Joint work with Mitsuru Sugimoto (Nagoya University).

### 2011/04/26

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

On the system of fifth-order differential equations which describes surfaces containing six continuous families of circles (JAPANESE)

**Kiyoomi KATAOKA**(Graduate School of Mathematical Sciences, the University of Tokyo)On the system of fifth-order differential equations which describes surfaces containing six continuous families of circles (JAPANESE)

### 2010/11/16

16:00-18:30 Room #123 (Graduate School of Math. Sci. Bldg.)

Hyperfunctions and vortex sheets (ENGLISH)

Residual trace and equivariant asymptotic trace of Toeplitz operators (ENGLISH)

**Keisuke Uchikoshi**(National Defense Academy of Japan) 16:00-16:45Hyperfunctions and vortex sheets (ENGLISH)

**L. Boutet de Monvel**(University of Paris 6) 17:00-18:30Residual trace and equivariant asymptotic trace of Toeplitz operators (ENGLISH)

### 2010/09/28

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

Some spectral and resonance properties of quantum graphs (ENGLISH)

**Pavel Exner**(Czech Academy of Sciences)Some spectral and resonance properties of quantum graphs (ENGLISH)

[ Abstract ]

In this talk I will discuss three new results about Schr¨odinger operators

on metric graphs obtained in collaboration with Jiri Lipovskyand Brian Davies.

The first one is related to invalidity of the uniform continuation principle for such

operators. One manifestation of this fact are embedded eigenvalues due to

rational relations of graph edge lengths. This effect is non-generic and we show

how geometric perturbations turn these embedded eigenvalues into resonances.

Then second problem is related to high-energy behavior of resonances: we extend

a recent result of Davies and Pushnitski to graphs with general vertex couplings

and find conditions under which the asymptotics does not have Weyl character.

Finally, the last question addressed here concerns the absolutely continuous spectrum

of radial-tree graphs. In a similar vein, we generalize a recent result by Breuer and

Frank that in case of the free (or Kirhhoff) coupling the ac spectrum is absent

provided the edge length are increasing without a bound along the tree.

We show that the result remains valid for a large class of vertex couplings,

but on the other hand, there are nontrivial couplings leading to an ac spectrum.

In this talk I will discuss three new results about Schr¨odinger operators

on metric graphs obtained in collaboration with Jiri Lipovskyand Brian Davies.

The first one is related to invalidity of the uniform continuation principle for such

operators. One manifestation of this fact are embedded eigenvalues due to

rational relations of graph edge lengths. This effect is non-generic and we show

how geometric perturbations turn these embedded eigenvalues into resonances.

Then second problem is related to high-energy behavior of resonances: we extend

a recent result of Davies and Pushnitski to graphs with general vertex couplings

and find conditions under which the asymptotics does not have Weyl character.

Finally, the last question addressed here concerns the absolutely continuous spectrum

of radial-tree graphs. In a similar vein, we generalize a recent result by Breuer and

Frank that in case of the free (or Kirhhoff) coupling the ac spectrum is absent

provided the edge length are increasing without a bound along the tree.

We show that the result remains valid for a large class of vertex couplings,

but on the other hand, there are nontrivial couplings leading to an ac spectrum.

### 2010/07/13

17:00-18:00 Room #128 (Graduate School of Math. Sci. Bldg.)

On a limiting eigenvalue distribution theorem for perturbations of the hydrogen atom (JAPANESE)

**Carlos Villegas Blas**(Universidad Nacional Autonoma de Mexico)On a limiting eigenvalue distribution theorem for perturbations of the hydrogen atom (JAPANESE)

[ Abstract ]

Let H be the hydrogen atom Hamiltonian. We will show that

the operator H+P can have well defined clusters of eigenvalues

for a suitable perturbation P=f(h)Q where Q is a pseudo-differential

operator of order zero and f(h) is a small quantity depending of

the Planck's parameter h. We will show that the distribution of

eigenvalues in those clusters has a semi-classical limit involving

the averages of the principal symbol of Q along the classical orbits

of the Kepler problem.

Let H be the hydrogen atom Hamiltonian. We will show that

the operator H+P can have well defined clusters of eigenvalues

for a suitable perturbation P=f(h)Q where Q is a pseudo-differential

operator of order zero and f(h) is a small quantity depending of

the Planck's parameter h. We will show that the distribution of

eigenvalues in those clusters has a semi-classical limit involving

the averages of the principal symbol of Q along the classical orbits

of the Kepler problem.

### 2010/06/22

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

Resonances for Magnetic Scattering by Two Solenoidal Fields at Large Separation (ENGLISH)

**Ivana Alexandrova**(East Carolina University)Resonances for Magnetic Scattering by Two Solenoidal Fields at Large Separation (ENGLISH)

[ Abstract ]

We consider the problem of quantum resonances in magnetic scattering by two

solenoidal fields at large separation in two dimensions, and we study how a trajectory

oscillating between the two fields gives rise to resonances near the real axis when

the distance between two centers of fields goes to infinity. We give a sharp lower

bound on resonance widths in terms of backward amplitudes calculated explicitly for

scattering by each solenoidal field. The study is based on a new type of complex

scaling method. As an application, we also discuss the relation to semiclassical

resonances in scattering by two solenoidal fields. This is joint work with Hideo Tamura.

We consider the problem of quantum resonances in magnetic scattering by two

solenoidal fields at large separation in two dimensions, and we study how a trajectory

oscillating between the two fields gives rise to resonances near the real axis when

the distance between two centers of fields goes to infinity. We give a sharp lower

bound on resonance widths in terms of backward amplitudes calculated explicitly for

scattering by each solenoidal field. The study is based on a new type of complex

scaling method. As an application, we also discuss the relation to semiclassical

resonances in scattering by two solenoidal fields. This is joint work with Hideo Tamura.

### 2010/06/15

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

Sato's counterexample and the structure of generalized functions (JAPANESE)

**Takashi Takiguchi**(Department of Mathematics, National Defense Academy)Sato's counterexample and the structure of generalized functions (JAPANESE)

[ Abstract ]

In this talk, we discuss the relation between the structure and the microlocal unique continuation property of generalized functions. We also mention some applications of the microlocal unique continuation property.

In this talk, we discuss the relation between the structure and the microlocal unique continuation property of generalized functions. We also mention some applications of the microlocal unique continuation property.

### 2010/04/13

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

Strichartz estimates and the Isozaki-Kitada parametrix

on asymptotically hyperbolic manifolds (ENGLISH)

**Jean-Marc Bouclet**(Toulouse University, France)Strichartz estimates and the Isozaki-Kitada parametrix

on asymptotically hyperbolic manifolds (ENGLISH)