| Saturday | ||
|---|---|---|
| 9:30 | Breakfast | |
| 10:00 | Maciej Zworski, UC Berkeley Scattering resonances as viscosity limits | |
| 11:00 | Yaiza Canzani,
Harvard University and IAS Pointwise Weyl law and random waves | |
| 2:00 | Richard Melrose, MIT Nodal degeneration for the Laplacian and the Weil-Petersson metric | |
| 3:00 | Long Jin, UC Berkeley A local trace formula for Anosov flows | |
| 4:00 | Dean Baskin,
Texas A&M University High frequency estimates for the Helmholtz equation and applications to boundary integral equations | |
| 5:00 | Wine and cheese in Lunt common room | |
| Sunday | ||
| 9:30 | Breakfast | |
| 10:00 | Peter Hintz, Stanford University Quasilinear wave equations and microlocal analysis with rough coefficients | |
| 11:00 | Wilhelm Schlag, University of Chicago On long-term dynamics of nonlinear wave equations | |
| 1:00 | Lunch at Jared's | |
Abstracts
High frequency estimates for the Helmholtz equation and applications to boundary integral equations
I will discuss joint work with Euan Spence and Jared Wunsch on high frequency estimates for the Helmholtz equation in exterior domains as well as in interior domains subject to impedance boundary conditions. The motivation for these estimates comes from numerical analysis.
Pointwise Weyl law and random waves
There are several questions about Laplace eigenfunctions that have proved to be extremely hard to deal with and remain unsolved. Among these are the study of the asymptotic behavior of the Hausdorff measure of the zero set, the number of critical points and the number of nodal domains. A natural approach is to randomize the problem and ask all the previous questions for random linear combinations of eigenfunctions. It turns out that when doing so one needs to control with some accuracy the kernel of the spectral projection operator onto high frequency windows. In this talk I will explain how to get good off diagonal estimates for the kernel of the projection near non self-focal points.
Quasilinear wave equations and microlocal analysis with rough coefficients
We present a global approach to the study of nonlinear wave equations, based on microlocal analysis on compactified spaces with boundary. Specifically, we consider quasilinear waves, where the metric depends on the solution; analyzing the corresponding wave operator requires a robust framework for proving microlocal regularity results for operators with rough coefficients, in the spirit of work by Beals and Reed, and we will describe the key ingredients of this framework. Joint work with András Vasy.
A local trace formula for Anosov flows
In this talk, we present a local trace formula for Anosov flows on compact manifolds which relates Pollicott-Ruelle resonances to the periods of closed orbits. As an application, we show a weak lower bound for the counting function for resonances in a strip and thus the infinitude of the resonances.
Nodal degeneration for the Laplacian and the Weil-Petersson metric
I will describe joint work with Xuwen Zhu in which we analyze the
behaviour of the Laplacian on the fibres of a (multi-)Lefschetz fibration.
We use this to describe the behaviour of the constant curvature metric on a
Riemann surface undergoing nodal degeneration and applying this to the
universal case of moduli space deduce the asymptotics of the Weil-Petersson
metric, so extending recent work of Mazzeo and Swoboda.
On long-term dynamics of nonlinear wave equations
We will review some of the developments in the asymptotic
theory of semilinear wave equations of the focusing type. These equations
admit soliton solutions and we address the description of the dynamical
picture
after long time. Ideas from both PDE and dynamics will be used.
Scattering resonances as viscosity limits
In practically all situations resonances can be defined as limits of
L2 eigenvalues
of operators which regularize the Hamiltonian at infinity. For instance,
Pollicott--Ruelle resonances in the theory of dynamical systems are given by
viscosity limits: adding a Laplacian to the generator of an Anosov
flow gives an operator with a discrete spectrum; letting the coupling
constant go to zero
turns eigenvalues into the resonances (joint work with S Dyatlov).
This principle
seems to apply in all other settings where resonances can be defined and
I will explain it in the simplest case of scattering by compactly
supported potentials.
The method has also been numerically investigated in the chemistry
literature as
an alternative to complex scaling.
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