Hancock Building, Chicago, IL
Photo by Chris Rycroft adapted with help from Chris Kottke
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Schedule
| Saturday | |
|---|---|
| 9:15 | Coffee |
| 10:00 | Dean Baskin, Texas A&M University The Feynman propagator for the Klein–Gordon equation |
| 11:00 | Karen Butt, University of Chicago Marked Poincaré rigidity |
| 12:00 | Lunch |
| 2:00 | Sandro Coriasco, Università di Torino Microlocal analysis of a class of time-fractional partial differential equations |
| 3:00 | Coffee |
| 3:30 | Joey Zou, Oakland University A Gutzwiller trace formula for singular potentials |
| 4:30 | Chris Kottke, Reed College Analysis on quasi-fibered-boundary (QFB) manifolds |
| Sunday | |
| 9:15 | Coffee |
| 10:00 | Yiran Wang, Emory University Reconstruction of the observable Universe from the Sachs-Wolfe effects |
| 11:00 | Jonathan Luk, Stanford University High-frequency limits in general relativity |
Abstracts
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Dean Baskin
The Feynman propagator for the Klein–Gordon equation
The Feynman propagator for a wave equation is a solution operator distinguished by always propagating in the direction of the Hamilton vector field. In this talk, I will describe a construction of the Feynman propagator for the Klein–Gordon equation on Minkowski space with an asymptotically static potential. We realize the Feynman propagator as the inverse of a mapping between Sobolev spaces with additional regularity near the asymptotic sources of the Hamiltonian flow; the estimates establishing this inverse are based on propagation of regularity estimates in Vasy's 3-body scattering calculus. This work is joint with Moritz Doll and Jesse Gell-Redman.
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Karen Butt
Marked Poincaré rigidity
For a closed negatively curved manifold, in analogy to the marked length spectrum, we consider marked dynamical data, called the marked Poincaré determinant, which measures the unstable volume expansion of the geodesic flow around closed geodesics. Using geometric and microlocal techniques (generalized X-ray transform), we show that the marked Poincaré determinant determines the metric up to homothety among metrics in a neighborhood of a hyperbolic metric dimension 3. This is joint work with Humbert, Erchenko, Lefeuvre, and Wilkinson.
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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.
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Chris Kottke
Analysis on quasi-fibered-boundary (QFB) manifolds
As introduced by Mazzeo and Melrose, a fibered boundary manifold is a complete Riemannian manifold with an asymptotic fibration structure with respect to which the fiber maintains bounded length while the base expands in an asymptotically conic manner. Such structures, which can be suitably compactified to manifolds with boundary, are common to many examples of non-compact Calabi-Yau and hyperKähler manifolds in real dimension 4. However, the natural extensions of these examples to higher dimensions exhibit a more complicated 'quasi-fibered boundary' (QFB) geometry, in which the asymptotic fibration structure becomes stratified and may be resolved by compactification to a manifold with corners. I will describe joint work with F. Rochon on the development of a calculus of pseudodifferential operators adapted to this QFB geometry, with applications to the L2 Hodge theory of certain higher dimensional hyperKähler manifolds, including a proof of the Vafa-Witten conjecture for Hilbert schemes of points on ℂ2 and a new case of Sen's conjecture for the moduli space of SU(2) magnetic monpoles on ℝ3.
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Jonathan Luk
High-frequency limits in general relativity
It is known that a high-frequency limit of solutions to the Einstein vacuum equation need not be vacuum, and that "effective matter" may arise in the limiting spacetime. I will discuss progress in characterizing such limits, and explain its relation with the propagation of microlocal defect measures in nonlinear settings. I will also present a recent construction such that the limiting effective matter is given by a massless Vlasov field. The talk is based on joint works with Cécile Huneau.
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Yiran Wang
Reconstruction of the observable Universe from the Sachs-Wolfe effects
The integrated Sachs-Wolfe (ISW) effect is a property of the Cosmic Microwave Background (CMB), in which photons are gravitationally redshifted, causing the anisotropies in CMB. An outstanding question is what information of the gravitational perturbations can be inferred from the ISW effect. In this talk, we explore the possibility of a tomography approach, similar to the X-ray CT in medical imaging. In particular, we consider the X-ray transform for null-geodesics in Lorentzian geometry (called the light ray transform). Together with the PDE model for CMB, we show the microlocal inversion of the transform and address the partial data problem with observations only near the Earth. Also, we discuss recent advances in numerical simulation and related geometric inverse problems.
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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.
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.
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