Analysis and PDE Seminar Homepage, Autumn-Winter-Spring 2018/2019

Location: 384I.

Time: Monday, 4pm.

• Wednesday, October 10, BAMAS at Stanford, Room 383N

  • 2:30-3:30pm

    Peter Hintz (MIT)

    ``Trapping in perturbations of Kerr spacetimes''

    Abstract: We study the trapped set of spacetimes whose metric decays to a stationary Kerr metric at an inverse polynomial rate. In the first part of the talk, I will focus on the dynamical aspects of this problem and show that the trapped set is a smooth submanifold which converges to that of the stationary metric at the same rate. In the second part, I will explain how to use this to prove microlocal estimates at the trapped set for solutions of wave equations on such spacetimes.
  and

  • 4-5pm

    Semyon Dyatlov (Berkeley/MIT)

    ``Long time propagation and fractal uncertainty principle''

    Abstract: I will show a frequency-independent lower bound on mass of eigenfunctions on surfaces of variable negative curvature. This was previously done in the case of constant curvature in joint work with Jin, relying on the fractal uncertainty principle proved in joint work with Bourgain. I will focus on the new components needed to handle the case of variable curvature, in particular propagation of quantum observables up to local Ehrenfest time. Joint work in progress with Long Jin and Stéphane Nonnenmacher.

• Wednesday, November 14, BAMAS at Berkeley, Evans Hall

  • 2:40-3:30pm, 740 Evans

    Maciej Zworski (Berkeley)

    ``Internal waves for (linearized) fluids and 0th order pseudodifferental operators''

    Abstract: Colin de Verdière and Saint-Raymond have recently found a fascinating connection between modeling of internal waves in stratified fluids and spectral theory of 0th order pseudodifferential operators on compact manifolds. The purpose of this talk is to show how a version of their results follows from the now standard radial estimates for pseudodifferential operators and some results about Lagrangian surfaces in classical and wave (quantum) settings. Some numerical simulations and comments about the case of positive viscosity will also be provided. Joint work with S. Dyatlov. (For the brave souls who attended the UCB Harmonic Analysis and Differential Equations Student Seminar on the same topic this talk will provide, after re-introduction of the problem, some technical details avoided then.)
  and

  • 4:10-5pm, 736 Evans

    Yiran Wang (Stanford)

    ``Determination of space-time structures from gravitational perturbations''

    Abstract: We consider inverse problems for the Einstein equations with source fields. The problem we are interested in is to determine space-time structures e.g. topological, differentiable structures of the manifold and the Lorentzian metric, by generating small gravitational perturbations and measuring the responses near a freely falling observer. We discuss some unique determination results for Einstein equations with scalar fields and electromagnetic fields under a microlocal linearization stability condition. A key component of our approach is to analyze the new waves generated from the nonlinear interaction of multiple gravitational waves using microlocal techniques. The talk is based on joint works with M. Lassas and G. Uhlmann.

• Monday, February 25

  Theo Johnson-Freyd (Perimter Institute)

  ``Bott periodicity from quantum Hamiltonian reduction''

  Abstract: The "quantization dictionary" posits that constructions in noncommutative algebra often parallel constructions in symplectic geometry. I will explain an example of this dictionary: I will produce the 8-fold periodicity of Clifford algebras as an example of quantum Hamiltonian reduction of a free fermion quantum mechanical system. No knowledge of the words "quantization", "Clifford algebra", "free fermion", or "Hamiltonian reduction" will be assumed. The exceptional Lie group G2 will make a cameo appearance.

• Wednesday, March 6, 3:15pm, Room 383N (geometry seminar slot)

  Walter Schachermayer (Vienna)

  ``Pathwise Otto calculus''

  Abstract: We revisit the [JKO98] variational characterization of diffusion as entropic gradient flux, and provide for it a probabilistic interpretation based on stochastic calculus. It was shown by Jordan, Kinderlehrer, and Otto in [JKO98] that, for diffusions of Langevin-Smoluchowski type, the Fokker-Planck probability density flow minimizes the rate of relative entropy dissipation, as measured by the distance traveled in terms of the quadratic Wasserstein metric. We obtain novel, stochastic-process versions of these features, valid along almost every trajectory of the diffusive motion in both the forward and, most transparently, the backward, directions of time, using a very direct perturbation analysis; the original results follow then simply by taking expectations. As a bonus, we derive the Cordero-Erausquin version of the so-called HWI inequality relating relative entropy, Fisher information and Wasserstein distance.

• Wednesday, March 13, BAMAS at Stanford, Room 384H

  • 2-3pm

    Rafe Mazzeo (Stanford)

    ``Analytic aspects of Kapustin-Witten theory''

    Abstract: Kapustin and Witten introduced a new set of gauge-theoretic equations, which were later proposed by Gaiotto and Witten as a tool to access some old and new manifold invariants. I will describe recent progress on the analytic foundations of this subject. Joint work with Witten and with S. He.
  and

  • 3:20-4:20pm

    Gabriel Paternain (Cambridge)

    ``Nonlinear detection of connections''

    Abstract: I will discuss the geometric inverse problem of recovering a connection from the parallel transport along geodesics of a compact Riemannian manifold with strictly convex boundary or along light rays in Minkowski space. This problem is motivated by other geometric inverse problems and is tackled with a range of techniques including energy estimates, regularity results for the transport equation associated with the geodesic flow and microlocal analysis.

• Friday, April 12, BAMAS at Berkeley, in 740 Evans

  • 2:10-3pm

    Jared Wunsch (Northwestern)

    ``Schrödinger equations with conormal potentials''

    Abstract: Consider the semiclassical Schrödinger equation $(h^2\Delta+V-E)u=0,$ where, instead of being smooth, $V$ is allowed to be singular across a hypersurface. The singularity in the potential turns out to have very interesting consequences for the structure of solutions $u;$ in effect, WKB solutions include not just contributions from classical propagation across the interface but also reflected singularities, in what amounts to a quantum diffraction effect (meaning one that is not visible at the level of classical Hamiltonian dynamics). I will discuss the propagation and reflection of semiclassical singularities in this setting, and also its consequences for the existence of quantum resonances in systems where trajectories escape to infinity under classical flow but not under the branched flow where we allow reflections. This is joint work with Oran Gannot.

  and

  • 4:10-5pm

    Hamid Hezari (UC Irvine)

    ``The inverse spectral problem for strictly convex domains''

    Abstract: We discuss the recent developments in the inverse length spectral theory of smooth strictly convex domains, including the works of Avila-De Simoi-Kaloshin and Kaloshin-Sorrentino on the Birkhoff conjecture, and De Simoi-Kaloshin-Wei on the length spectral rigidity of nearly circular domains with a reflectional symmetry. In a joint work with Zelditch we explore the inverse laplace spectral problem for nearly circular ellipses, among all smooth domains without any symmetry or convexity assumption.

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