Tuesday Seminar of Analysis

Seminar information archive ~09/27Next seminarFuture 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


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


16:30-18:00   Room #128 (Graduate School of Math. Sci. Bldg.)
Rafe Mazzeo (Stanford University)
This talk was cancelled! (JAPANESE)


16:30-18:30   Room #128 (Graduate School of Math. Sci. Bldg.)
Alexander Vasiliev (Department of Mathematics, University of Bergen, Norway) 16:30-17:30
Evolution 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.
Irina Markina (Department of Mathematics, University of Bergen, Norway) 17:30-18:30
Group 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.


16:30-18:00   Room #128 (Graduate School of Math. Sci. Bldg.)
Thierry Ramond (Univ. Paris, Orsay)
Resonance free domains for homoclinic orbits (ENGLISH)


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


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


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


16:30-18:00   Room #128 (Graduate School of Math. Sci. Bldg.)
Erika Ushikoshi (Mathematical Institute, Tohoku University)
Hadamard variational formula for the Green function
of the Stokes equations with the boundary condition (JAPANESE)


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


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


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


16:30-18:00   Room #128 (Graduate School of Math. Sci. Bldg.)
Michael Loss (Georgia Institute of Technology)
Symmetry results for Caffarelli-Kohn-Nirenberg inequalities (ENGLISH)


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


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


16:30-18:00   Room #128 (Graduate School of Math. Sci. Bldg.)
Wolfram Bauer (Mathematisches Institut, Georg-August-Universität)
Trivializable subriemannian structures and spectral analysis of associated operators (ENGLISH)


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


16:30-18:00   Room #128 (Graduate School of Math. Sci. Bldg.)
Hidemitsu Wadade (Waseda University (JSPS-PD))
On the best constant of the weighted Trudinger-Moser
type inequality (JAPANESE)


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


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


16:00-18:30   Room #123 (Graduate School of Math. Sci. Bldg.)
Keisuke Uchikoshi (National Defense Academy of Japan) 16:00-16:45
Hyperfunctions and vortex sheets (ENGLISH)
L. Boutet de Monvel (University of Paris 6) 17:00-18:30
Residual trace and equivariant asymptotic trace of Toeplitz operators (ENGLISH)


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


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


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


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


16:30-18:00   Room #128 (Graduate School of Math. Sci. Bldg.)
Jean-Marc Bouclet (Toulouse University, France)
Strichartz estimates and the Isozaki-Kitada parametrix
on asymptotically hyperbolic manifolds (ENGLISH)

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