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Tuesday Seminar of Analysis
Seminar information archive ～04/19｜Next seminar｜Future seminars 04/20～
| Date, time & place | Tuesday 16:00 - 17:30 156Room #156 (Graduate School of Math. Sci. Bldg.) |
|---|---|
| Organizer(s) | ISHIGE Kazuhiro, SAKAI Hidetaka, ITO Kenichi |
2012/12/04
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)
Group of diffeomorphisms of the unit circle and sub-Riemannian geometry (ENGLISH)
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) 17:30-18:30We 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.
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.
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.
