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 Events
Academy of Finland and University of Sydney
Thu 8 Sep 2011, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
WMAX 110 (PIMS) (Schedule and location subject to change)
The Aharonov-Bohm effect and the Calderon problem for connection Laplacians
WMAX 110 (PIMS) (Schedule and location subject to change)
Thu 8 Sep 2011, 3:30pm-4:30pm

Abstract

The Aharonov-Bohm effect is a quantum mechanical phenomenon where electrons passing through a region of vanishing magnetic field gets scattered due to topological effects. It turns
out that this phenomenon is closely related to the cohomology of forms with integer coefficients. We study this relationship from the point of view of the Calder´n problem and see that it can be captured in how Cauchy data of the connection laplacian determines uniquely the holonomy representation of the connection.

The work was partially supported by Finnish Academy of Science and by NSF Grant No.DMS-0807502.
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UBC
Tue 13 Sep 2011, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
WMAX 110 (PIMS)
A Self-dual Polar Factorization for Vector Fields
WMAX 110 (PIMS)
Tue 13 Sep 2011, 3:30pm-4:30pm

Abstract

We show that any non-degenerate vector field u in L^{\infty}(\Omega, \R^N), where \Omega is a bounded domain in \R^N, can be written as {equation} \hbox{u(x)= \nabla_1 H(S(x), x) for a.e. x \in \Omega}, {equation} where S is a measure preserving point transformation on \Omega such that S^2=I a.e (an involution), and H: \R^N \times \R^N \to \R is a globally Lipschitz anti-symmetric convex-concave Hamiltonian. Moreover, u is a monotone map if and only if S can be taken to be the identity, which suggests that our result is a self-dual version of Brenier's polar decomposition for the vector field u as u(x)=\nabla \phi (S(x)), where \phi is convex and S is a measure preserving transformation. We also describe how our polar decomposition can be reformulated as a self-dual mass transport problem.
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Lu Li
UBC
Tue 20 Sep 2011, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
WMAX 110 (PIMS)
Backward uniqueness for the heat equation in cones
WMAX 110 (PIMS)
Tue 20 Sep 2011, 3:30pm-4:30pm

Abstract

I will talk about the backward uniqueness of the heat equation in unbounded domains. It is known that a bounded solution of the heat equation in a half-space which becomes zero at some time must be identically zero, even though no assumptions are made on the boundary values of the solutions. In a recent example, Luis Escauriaza showed that this statement fails if the half-space is replaced by cones with opening angle smaller than 90 degrees. In a joint work with Vladimir Sverak we show the result remains true for cones with opening angle larger than 110 degrees. Our proof covers heat equations having lower-order terms with bounded measurable coefficients.
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U. Paris-Sud, Orsay
Mon 26 Sep 2011, 4:00pm SPECIAL
Diff. Geom, Math. Phys., PDE Seminar / Probability Seminar
MATX 1100
The Langevin process and the trace formula
MATX 1100
Mon 26 Sep 2011, 4:00pm-5:00pm

Abstract

I will explain the probabilistic interpretation of the hypoelliptic Laplacian L_b . To L_b, one can associate the diffusion on the manifold X that is a solution of the differential equation b^2 x'' = −x' + w'. For b = 0, we get x' = w', the equation of Brownian motion, and for b = +∞, we obtain the equation of geodesics x'' = 0. I will explain the rigorous results one can derive on the corresponding heat kernels via the Malliavin calculus. These will include uniform Gaussian decay of the hypoelliptic heat kernel over a symmetric space.
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U. Paris-Sud, Orsay
Tue 27 Sep 2011, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
WMAX 110 (PIMS)
Orbital integrals and the hypoelliptic Laplacian
WMAX 110 (PIMS)
Tue 27 Sep 2011, 3:30pm-5:00pm

Abstract

Third talk in the series. If G is a reductive Lie group with Lie algebra g, orbital integrals are key ingredient in Selberg’s trace formula. I will explain how one can think of the evaluation of orbital integrals as the computation of a Lefschetz trace. Using in particular the Dirac operator of Kostant, the standard Casimir operator of X = G/K is deformed to a hypoelliptic operator L_b acting on the total space of a canonically flat vector bundle on X, that contains TX as a subbundle. The symbol of this hypoelliptic operator is exactly the one described in the previous talks. When descending the situation to a locally symmetric space, the spectrum of the original Casimir remains rigidly embedded in the spectrum of the hypoelliptic deformation. Making b → +∞ gives an explicit evaluation of semisimple orbital integrals.
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