Location: Room 383N.
Time: Wednesdays, 3-4pm, or 4-5pm if the geometry seminar does not meet.
Abstract: The Fourier transform maps $ L^2(S^2)$ to $ L^4(R^3)$. We show that there exist functions which extremize the associated inequality, and that any extremizing sequence of nonnegative functions has a convergent subsequence. This was previously known for paraboloids, where all extremizers are Gaussians and vice versa. Complex extremizers and extremizing sequences are related to nonnegative ones in a simple way. All critical points of the associated nonlinear functional are real analytic. Constant functions are local extremizers, but we do not know whether they are global extremizers, nor whether extremizers are unique modulo symmetries of the problem. The proofs involve concentration compactness ideas, inequalities for convolutions, facts about Fourier integral operators, symmetrization, a characterization of approximate characters, a perhaps nonstandard regularity theorem, an idea from additive combinatorics, facts about spherical harmonics and Gegenbauer polynomials, and several explicit computations.
andAbstract: We consider the semclassical Metropolis operator on a bounded domain. We obtain a precise description of its spectrum that give useful bounds on rates of convergence for the Metropolis algorithm. As an example, we treat the random placement of N hard discs in the unit square, the original application of the Metropolis algorithm.
We will discuss the uniqueness of expanding self-similar solutions to the network flow in a fixed topological class. We prove the result via the a parabolic Allen-Cahn approximation. Moreover, we prove that any regular evolution of connected tree-like network (with an initial condition that might be not regular) is unique in a given a topological class.
Abstract: Understanding the geodesics on a singular space is of interest both from the point of view of studying the inner geometry of such spaces and also from a PDE point of view, since the geodesics are the expected trajectories of singularities of solutions of the wave equation. In the case of conical singularities, it was proved by Melrose and Wunsch that the geodesics hitting the conical point foliate a neighborhood of that point smoothly. In other words, there is a smooth exponential map based at the singular point. We consider a different class of singularities and prove that while the exponential map based at the singularity is not smooth, its precise asymptotic behavior (to any order) near the singularity can be described completely in terms of certain blow-ups of the space and of its cotangent bundle. Joint work with V. Grandjean.
Abstract: The radiation field introduced by Friendlander provides a direct approach to the asymptotic expansion of solutions to the wave equation at null infinity. We use this concept to study the asymptotic behavior of solutions to Einstein Vacuum equations, which are close to Minkowski space, at null infinity. By imposing harmonic gauge, the Einstein Vacuum equations reduce to a system of quasilinear wave equations on R_{t,x}^{1+n}. We show that if the space dimension n>=5 the Moller wave operator is an isomorphism from Cauchy data satisfying the constraint equations to the radiation fields satisfying an analog of constraint equations on small neighborhoods of suitable weighted b-type Sobolev spaces.
Abstract: For the Toeplitz quantization of complex-valued functions on a 2n-dimensional torus we prove that the expected number of eigenvalues of small random perturbations of a quantized observable satisfies a natural Weyl law. In numerical experiments the same Weyl law also holds for ``false'' eigenvalues created by pseudospectral effects. The talk is based on joint work with TJ Christiansen.
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Abstract: We present a pseudodifferential calculus generalizing the 'fibred cusp calculus' introduced by Mazzeo and Melrose. The generalization is in two directions: First, in a direction similar to previous work by Vaillant, the calculus allows to construct parametrices which satisfy a weaker ellipticity requirement than 'full ellipticity', namely we do not require invertibility of the normal operator at the boundary. This is important since many operators of interest, for example the Hodge Laplacian, tend to have non-invertibe normal operator. Second, we allow multiple fibrations of the boundary. Such operators arise for example as Hodge Laplace operator on locally symmetric spaces of Q-rank one. In the case of two fibrations these are locally of the form P(x,y,z,w; x^3 \partial_x, x^2 \partial_y, x\partial_z,\partial_w). This is joint work with E. Hunsicker.
Abstract: Mathematically, quantum decay rates appear as imaginary parts of poles of the meromorphic continuation of Green's functions. As energy grows, decay rates are related to properties of geodesic flow and to the structure at infinity. For a cusp, infinity is `small', which typically slows decay. However, I will present a class of examples for which decay rates go to infinity with energy even in the presence of a cusp. This is part of a more general investigation of resonances on manifolds with hyperbolic ends.