Location: 380F. (Will be 381T in winter 2015!)
Time: Monday, 4pm.
Abstract: One of the most beautiful chapters of theoretical physics is the analysis of excitation spectrum of the Bose gas due to Bogoliubov. Clearly, this analysis is nonrigorous. However it has been qualitatively confirmed by experiments and underlies our understanding of one of the most remarkable physical phenomena -- the superfluidity.
I will describe a rigorous result saying that, appropriately interpreted, Bogoliubov's picture is correct in a certain limit that involves a large number of particles and large volume. According to this result, in this limit the joint spectrum of the many-body Schrödinger Hamiltonian and the momentum has a special shape predicted by Bogoliubov.
I do not expect that everybody in the audience is familiar with the physics jargon. The results and tools that I am going to describe can be understood and appreciated purely in the context of spectral theory of operators on Hilbert spaces.
Abstract: Sharp estimates for the solutions to elliptic PDEs in $L^\infty$ in terms of the corresponding norm of the boundary data follow directly from the maximum principle. It holds on arbitrary domains for all (real) second order divergence form elliptic operators $- \mathrm{div} A \nabla$. The well-posedness of boundary problems in $L^p$, $p<\infty$, is a far more intricate and challenging question, even in a half-space. In particular, it is known that some smoothness of $A$ in $t$, the transversal direction to the boundary, is needed.
In the present work we establish the well-posedness in $L^p$ of the Dirichlet problem for all divergence form elliptic equations with real (possibly non-symmetric) coefficients independent on the transversal direction to the boundary. Equivalently, we show that for all such operators the $L$-harmonic measure is quantifiably absolutely continuous with respect to the Lebesgue measure. The lack of smoothness and lack of symmetry in the coefficients defy most of the previously known methods. We introduce a new strategy and use the celebrated Kato problem estimate, adapted Hodge decomposition, square function/non-tangential maximal function bounds, epsilon-approximability and $A^\infty$ criteria for harmonic measure, among other tools.
This is joint work with S. Hofmann, C. Kenig, and J. Pipher.
Abstract: We will introduce the analytic torsion on locally symmetric spaces and describe some of its asymptotic properties. The talk will focus on the analytic aspects of the theory. In particular, we will describe the torsion in the non-compact, finite-volume setting as well as a gluing formula for the torsion in that case.
Abstract: Spectral theory (even microlocal analysis) has been very useful for studying the long time behavior of recurrent Markov chains. The quantitative theory of absorbing Markov chains is in its infancy. Here there is a `quasi-stationary distribution' (the first Dirichlet eigenfunction) and one may ask `How close are we to quasi-stationarity if the process has not been absorbed up to time n?' One can also ask about time to absorption (the top Dirichlet eigenvalue) and the `shape' of the quasi-stationary distribution (McKeen-Vlasov equation). In joint work with Laurent Miclo, we show how the Doob transform produces a recurrent chain with upper and lower bounds for rates of convergence.
Abstract: The quantum ergodicity theorem of Shnirelman, Colin de Verdiere and Zelditch gives that for an orthonormal sequence of eigenfunctions of the laplacian on a compact manifold with ergodic geodesic, outside a density one subsequence, the eigenfunctions equidistribute. The geodesic flow on the sphere is very much not ergodic, and indeed quantum ergodicity (QE) fails on the sphere for the standard sequence of spherical harmonics. On the other hand Zelditch has shown QE holds for a random orthonormal basis in this case. We prove QE for joint eigenfunctions of laplacian and an averaging operator over a finite collection of rotations (with some restrictions). We also give a new approach to a QE theorem on graphs by Anantharaman and Le Masson. Joint work with Shimon Brooks and Etienne Le Masson.