Location: Room 383N.
Time: Wednesdays, 3-4pm, or 4-5pm if the geometry seminar does not meet.
Abstract: Resonances are complex numbers analogous to eigenvalues for a class of operators on noncompact domains. Physically, they may correspond to decaying waves, with the real part giving the frequency and the imaginary part the rate of decay. After an introduction to resonances, we will concentrate on the problem of understanding the growth of the resonance counting function for Schrödinger operators. While in the one-dimensional case its asymptotics are known, its behavior in higher dimensions appears quite subtle.
Abstract. In this talk I describe some recent result as well as the work in progress on the neck pinching of surfaces under under mean curvature flow.
Abstract. The Calderón problem in the anisotropic case consists in determining a matrix of coefficients in an elliptic equation from boundary measurements of solutions. This inverse problem is the mathematical model for Electrical Impedance Tomography (EIT), which has been proposed as a diagnostic method in medical imaging and nondestructive testing. In geometric terms, the problem is to determine a Riemannian metric from the Cauchy data of harmonic functions on a manifold. Our approach is based on Carleman estimates. We characterize those Riemannian manifolds which admit a special limiting Carleman weight. By using these weights we construct special harmonic functions of complex geometrical optics type, and prove uniqueness results in inverse problems for a class of Riemannian manifolds. This is a joint work with D. Dos Santos Ferreira (Paris 13), C. Kenig (Chicago), and G. Uhlmann (Washington).
Abstract: I will present some results about the Weyl asymptotics for the damped wave equation on a negatively curved manifold. I will give a fractional Weyl upper bound for the number of eigenvalues with given imaginary part. It is notoriously difficult to prove a lower bound - in fact, it is already difficult to prove existence of infinitely many eigenvalues in a given horizontal strip. I will show a very particular model (twisted Laplacian on an arithmetic surface) where it is possible.
andAbstract: We would like to present some recent work together with Johannes Sjöstrand, dealing with the spectral analysis of non-selfadjoint perturbations of selfadjoint semiclassical operators in dimension 2. Specifically, assuming that the classical flow of the unperturbed part is completely integrable, we analyze spectral contributions coming from both Diophantine and rational invariant Lagrangian tori. Estimating the tunnel effect between the two types of tori, we obtain an accurate description of the spectrum in a suitable complex window, provided that the strength of the non-selfadjoint perturbation is not too large. We also hope to talk about the ongoing work, where we study the global distribution of the imaginary parts of the eigenvalues in the entire spectral band.