Invited Talks: Titles and Abstracts
Nabile Boussaid (Université Franche-Comté — Besançon)
Virtual levels and virtual states of linear operators in Banach spaces. Applications to Schrödinger operators (slides)
Abstract: In this talk, I will present a joint work with Andrew Comech (Texas A&M). Our primary interest is on the limiting absorption principle. Such a tool is useful in spectral analysis. It is also used to obtain dispersive estimates and then to analyze long time evolution of associated nonlinear problems.
In dimension 1 and 2, in contrast to higher dimensions, the free linear Schrödinger operator has no limiting absorption principle near the threshold. This makes all this classical perturbative approach much more involved.
The absence of limiting absorption principle (LAP) in the vicinity of some point is equivalent to the presence of a virtual level at this point. But it is also known that virtual levels are unstable by perturbations leading to bifurcation of eigenvalues.
Our work is an attempt to understand the different characterizations of virtual levels and to provide limiting absorption principle for small perturbations of the free Schrödinger operators in dimension 1 and 2.
Eugenia Malinnikova (Stanford University)
Some inequalities for solutions of elliptic PDEs and Laplace eigenfunctions
Abstract: Almgren's frequency function is a useful tool to study quantitative behavior of solutions to elliptic PDEs. We will apply the frequency function to obtain generalizations of the Remez and Bernstein inequalities to solutions of elliptic equations. We also discuss related inequalities for the eigenfunctions of the Laplace–Beltrami operator. The talk is based on joint works with Alexander Logunov and with Stefano Decio.
Andrea Nahmod (University of Massachusetts – Amherst)
Probabilistic scaling, propagation of randomness and invariant Gibbs measures
Abstract: In this talk, we will start by describing how classical tools from probability offer a robust framework to understand the dynamics of waves via appropriate ensembles on phase space rather than particular microscopic dynamical trajectories. We will continue by explaining the fundamental shift in paradigm that arises from the "correct" scaling in this context and how it opened the door to unveil the random structures of nonlinear waves that live on high frequencies and fine scales as they propagate. We will then discuss how these ideas broke the logjam in the study of the Gibbs measures associated to nonlinear Schrödinger equations in the context of equilibrium statistical mechanics and of the hyperbolic Φ43 model in the context of constructive quantum field theory. We will end with some open challenges about the long-time propagation of randomness and out-of-equilibrium dynamics.
Christopher Judge (Indiana University)
Recent progress on the hot spots conjecture
Abstract: Physically, the hot spots conjecture of Jeff Rauch is the assertion that the local maxima of a temperature distribution on a perfectly insulated homogeneous material will migrate towards the boundary as time tends to infinity provided the initial distribution is generic. Mathematically, it is the assertion that the second eigenfunction of the Neumann Laplacian on a bounded convex domain has no (interior) local extrema. I will survey relatively recent results and the methods used to obtain these results. Part of the survey will concern my joint work with Sugata Mondal.
Iosif Polterovich (Université de Montréal)
Pólya's eigenvalue conjecture: some recent advances
Abstract: The celebrated Pólya's conjecture (1954) in spectral geometry states that the eigenvalue counting functions of the Dirichlet and Neumann Laplacian on a bounded Euclidean domain can be estimated from above and below, respectively, by the leading term of Weyl's asymptotics. The conjecture is known to be true for domains which tile the Euclidean space, however it remains largely open in full generality. In the talk we will explain the motivation behind this conjecture and discuss some recent advances, such as the proof of Pólya's conjecture for the disk, and its extension to the eigenvalues of a magnetic Schrödinger operator with an Aharonov–Bohm potential. The latter answers a question posed by Frank and Hansson (2008). The talk is based on joint works with Nikolay Filonov, Michael Levitin and David Sher.
Jacob Shapiro (Princeton University)
Classification of disordered insulators in 1D (slides)
Abstract: In this talk I will describe some of the mathematical aspects of disordered topological insulators. These are novel materials which insulate in their bulk but (may) conduct along their edge; the quintessential example is that of the integer quantum Hall effect. What characterizes these materials is the existence of a topological index, experimentally measurable and macroscopically quantized. Mathematically this is explained by applying algebraic topology to the space of appropriate quantum mechanical Hamiltonians; I will survey some recent results mainly concentrating on the classification problem in one dimension, where the problem reduces to studying spaces of unitaries (resp. orthogonal projections) which essentially-commute with a fixed projection.
András Vasy (Stanford University)
The black hole stability problem (slides)
Abstract: I will discuss analytic and geometric tools that lead to the understanding of black hole stability with a positive cosmological constant (Kerr-de Sitter spacetimes) and their extensions which play a role in the vanishing cosmological constant case (Kerr). This is based on joint works with Dietrich Haefner, Peter Hintz and Oliver Petersen.
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