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Analysis-Geometry
Seminar
Meets
Fridays 2-3 pm in 509 Lake Hall
We
generally go to lunch beforehand.
Please join us at 457 Lake Hall at 12:30 pm.
This seminar features talks in the fields of
partial differential equations, functional analysis, differential geometry and
topology, and mathematical physics.
The organizers are: Maxim Braverman,
Bob McOwen, Mikhail Shubin, Peter
Topalov
Talks:
·
February 17, 2012
Speaker: Alexander Shnirelman (
Title: Shooting problem for the ideal incompressible fluid
Abstract :
Consider the motion of ideal incompressible fluid in a bounded
2-dimensional domain (or compact surface) manifold. The fluid configuration
is defined by positions of all fluid particles, i.e. by an area preserving
diffeomorphism. The configuration space of the
fluid is the group SDiff(M) of all such diffeomorphisms. Any flow (in the absence of external
forces) is a geodesic on
SDiff(M) in the L2 metric. Geodesic is
defined by the initial position (which may be assumed to be the identity
map) and the initial velocity, which is a divergence-free
vector field on M. The (geodesic)
exponential map exp assigns to
every velocity v the map exp(v).
Theorem: For any diffeomorphism
f in SDiff(M) there exists v such that exp(v).
This theorem looks superficially like the classical Hopf-Rinow
theorem, but in fact they have nothing in common. The proof is based on some
new ideas of microlocal and global analysis.
·
January 20, 2012
Speaker: Andres Hubach (
Title: Exotic Characteristic Classes
Abstract : Exotic characteristic classes can be obtained by applying a version of the Chern-Weil construction to certain infinite rank bundles that occur in geometry and mathematical physics, such as the tangent bundle to loop spaces. These classes can be used to show the nontriviality of the K -theories of these bundles.
·
January 6, 2012
Speaker: Feride Tiglay (Fields Institute)
Title:
Fredholm determinants and breakdown for an integrable evolution
equation
Abstract :
For some integrable equations it is possible to find
explicit formulas for solutions. I will present such a construction for a
recently introduced integrable equation and how it
can be used to prove a breakdown theorem for the Cauchy problem.
·
November 18, 2011
Speaker: Zhan Dapeng (Michigan State University)
Title: Conformally invariant random shapes and Schramm-Loewner evolution
Abstract : Many two-dimensional lattice models were observed by statistical physicists to satisfy conformal invariance when their meshes are very small. These conjectures were solved using the newly developed Schramm-Loewner evolution (SLE) introduced by Oded Schramm in 1999. The SLE process generates a random fractal curve growing in a plane domain, and its behavior depends on a positive parameter. The SLE with different parameters are proved to be the scaling limits of different lattice models. In this review talk I will briefly talk about the definition of SLE, several lattice models which converge to SLE, and my research on SLE.
·
October 7, 2011
Speaker: Hamid Hezari (MIT)
Title: Volume of Nodal Sets of Eigenfunctions
Abstract : Yau's conjecture
states that the volume of the nodal set of
·
September 23,
2011
Speaker: Claudio Procesi (
Title: A normal form of the non-linear Schrodinger equation
Abstract : We discuss a class of normal forms of the
completely resonant nonlinear Schrodinger equation on a torus (in any
dimension). We stress the geometric and combinatorial constructions
arising from this study. This normal form has many remarkable properties and can
be used to construct quasi-periodic solutions for the NLS
by applying a suitable KAM algorithm.