Photo by Andrew E. Larsen
Spherical Gaussian field by Dmitry Belyaev
Photo collage by Nora Wunsch
Schedule
Talks will take place in Lunt 105.
| Saturday | |
|---|---|
| 9:30 | Coffee |
| 10:00 | Guillaume Bal, University of Chicago Asymmetric Transport in Topological Insulators |
| 11:00 | Amir Vig, University of Michigan Cancellations in the Wave Trace |
| 12:00 | Lunch |
| 2:00 | Katrina Morgan, Northwestern University Wave propagation on rotating cosmic string spacetimes |
| 3:00 | Coffee |
| 3:30 | Pierre Albin, University of Illinois Urbana–Champaign Dirac-type operators on stratified spaces |
| 4:30 | Perry Kleinhenz, Michigan State University Energy decay for the damped wave equation |
| Sunday | |
| 9:30 | Coffee |
| 10:00 |
Allen Fang, Princeton University A new proof for the nonlinear stability of slowly-rotating Kerr–de Sitter |
| 11:00 |
Chris Sogge, Johns Hopkins University Curvature and harmonic analysis on compact manifolds |
Registration
The conference will be open to the mathematical public. Participants are asked to register here.
Abstracts
-
Pierre Albin
Dirac-type operators on stratified spaces
Stratified spaces arise naturally even when studying smooth objects, e.g., as algebraic varieties, orbit spaces of smooth group actions, and many moduli spaces. There has recently been a lot of activity developing analysis on these spaces and studying topological invariants such as the signature. I will report on work with Jesse Gell-Redman on the index of Dirac-type operators satisfying a 'Witt' condition and hence having a natural choice of Fredholm domain and then describe work in progress with Gell-Redman and Paolo Piazza that goes beyond the Witt condition.
-
Guillaume Bal
Asymmetric Transport in Topological Insulators
Asymmetric transport along interfaces separating insulating bulks has been observed in many (acoustic, electromagnetic, electronic, mechanical) settings and shown to be robust to perturbations. This surprising robustness to defects, which may be seen as an obstruction of Anderson localization, has a topological origin. The talk proposes a classification of partial differential operators modeling such systems by means of confining domains walls. A first associated topological invariant is the index of a Fredholm operator computed explicitly by a Fedosov–Hormander formula, which implements in Euclidean space an Atiyah–Singer index theorem. We also introduce a physical observable, a form of edge conductivity, to characterize asymmetric transport. The observable is itself associated to a second topological invariant whose calculation is less direct. We present a bulk-edge correspondence stating that the two invariants in fact agree. The theory is illustrated with generalizations of systems of Dirac equations and applied to e.g. single-layer and twisted bilayer graphene-based topological insulators. -
Allen Fang
A new proof for the nonlinear stability of slowly-rotating Kerr–de Sitter
Black hole stability has seen numerous important developments in the past decade. In my talk, I will focus on the nonlinear stability of the slowly-rotating Kerr–de Sitter family. Nonlinear stability of the slowly-rotating Kerr–de Sitter family was first proven by Hintz and Vasy in 2016 using microlocal techniques. In my talk, I will present a new proof of the nonlinear stability of slowly-rotating Kerr–de Sitter spacetimes that utilizes the vectorfield method to uncover the high-frequency spectral gap in Kerr–de Sitter, connecting the physical-space and frequency-space approaches. The proof also uses a new scheme to prove nonlinear stability, avoiding Nash–Moser. -
Perry Kleinhenz
Energy decay for the damped wave equation
The damped wave equation describes the motion of a vibrating system exposed to a damping force. For the standard damped wave equation, exponential energy decay is equivalent to the Geometric Control Condition (GCC). The GCC requires every geodesic to meet the positive set of the damping coefficient in finite time. A natural generalization is to allow the damping coefficient to depend on time, as well as position. I will give an overview of the classical results and discuss how a time dependent generalization of the GCC implies exponential energy decay. I will also mention some results for unbounded damping when the GCC is not satisfied. -
Katrina Morgan
Wave propagation on rotating cosmic string spacetimes
A rotating cosmic string spacetime has a singularity along a timelike curve corresponding to a one-dimensional source of angular momentum. Such spacetimes are not globally hyperbolic: they admit closed timelike curves near the so-called "string". This presents challenges to studying the existence of solutions to the wave equation via conventional energy methods. In this work, we show that forward solutions to the wave equation (in an appropriate microlocal sense) do exist. Our techniques involve proving a statement on propagation of singularities and using the resulting estimates to show existence of solutions. This is joint work with Jared Wunsch. -
Chris Sogge
Curvature and harmonic analysis on compact manifolds
We shall explore the role that curvature plays in harmonic analysis on compact manifolds. We shall focus on spectral projection estimates and Strichartz estimates for solutions of the Schrödinger equation. We focus on gains that arise when the sectional curvatures of the manifold are negative. -
Amir Vig
Cancellations in the Wave Trace
The wave trace contains information on the asymptotic distribution of eigenvalues for the Laplacian on a Riemannian manifold. It is well known that its singular support is contained in the length spectrum, which allows one to infer geometric information only under a length spectral simplicity or other nonresonance type condition. We construct large families of domains for which there are multiple geodesics of a given length having different Maslov indices, leading to a cancellation of arbitrarily many orders in the wave trace at that length. This shows that there are potential limitations in using the wave trace for inverse spectral problems and more fundamentally, that the Laplace spectrum and length spectrum are inherently different objects, at least as far the wave trace is concerned.
Organizers
Kiril Datchev (Purdue University),
Antônio Sá Barreto (Purdue University),
David Sher (DePaul University), and Jared Wunsch (Northwestern University).
David Sher (DePaul University), and Jared Wunsch (Northwestern University).
Funding
This meeting is supported by the NSF and Northwestern University.
Some travel funding may be available from Northwestern's RTG for students and postdocs who are US citizens or permanent residents. If interested, email MMMM2023@math.northwestern.edu, attaching a CV and a paragraph explaining your interest in the meeting.
Some travel funding may be available from Northwestern's RTG for students and postdocs who are US citizens or permanent residents. If interested, email MMMM2023@math.northwestern.edu, attaching a CV and a paragraph explaining your interest in the meeting.
