The conference will be open to the mathematical public, but registration is essential for building access.
Abstracts
Sandro Coriasco
Microlocal analysis of a class of time-fractional partial differential equations
We study regularity and decay properties of the solutions of the Cauchy problem for time-fractional partial differential equations, with tempered initial data, associated with a differential operator on space variables with polynomially bounded coefficients. We obtain a representation formula for the solution, modulo time-regular functions, smooth and rapidly decreasing with respect to the space variables. By means of the representation formula, the (decay and smoothness) singularities of the solution of the homogeneous Cauchy problem can be controlled, in terms of (global) wavefront sets of the initial data. This is joint work with Giovanni Girardi and Stevan Pilipović.
Joey Zou
A Gutzwiller trace formula for singular potentials
We discuss extending the Gutzwiller trace formula, which relates the regularized trace of a semiclassical Schrödinger propagator to dynamical data of the corresponding Hamiltonian dynamics, from the case of smooth potentials to that of conormal potentials with derivative discontinuities across some hypersurface. We show, in addition to a main term as expected from classical dynamics, that there is a subleading contribution from dynamics associated to “branching” Hamiltonian trajectories that are allowed to reflect from the hypersurface of potential singularity. We discuss the microlocal tools, using the b-calculus, needed to study the propagation of singularities, allowing us to focus on the dynamically relevant contributions to the trace, as well as the computations behind the trace formula, some variational calculus needed to make sense of the dynamical quantities appearing in the formula, and an explicit 1-dimensional example whose eigenvalues have explicit asymptotics that demonstrate the contributions from the branching trajectories. Joint work with Jared Wunsch and Mengxuan Yang.
This meeting is supported by the NSF and Northwestern University.
Some NSF support for participants is available. See the registration page for details. Please apply by March 7th for full consideration.