Analysis and PDE Seminar Homepage, Autumn 2012

Location: TBA

Time: Thursdays, 2:30-3:30pm, or 2:15-4:05pm if a double header.

• Thursday, October 4:

  Hiroshi Matano (University of Tokyo)

  ``Validity of formal asymptotic expansions in the Allen-Cahn equation and FitzHugh-Nagumo system''

• Tuesday October 30, 1:15pm, 383N (Note unusual day/time)

  Jeremy Marzuola (University of North Carolina, Chapel Hill)

  Quasilinear Schrödinger Equations

  Abstract: We discuss recent work with Jason Metcalfe and Daniel Tataru on local well posedness results for quasilinear Schrödinger equations. We will discuss both a natural functional framework, as well as the local smoothing, energy estimates and multilinear estimates required.

• POSTPONED, to Nov 26!

  Originally scheduled: Thursday, November 1:

  Eva Silverstein (Stanford)

  ``Inflation, de Sitter spacetime and the origin of structure''

  Abstract: Inflation is a (deceptively) simple idea in cosmology which ties directly to observational data on the one hand (real experiments), and to difficult problems in formulating quantum gravity on the other hand (thought experiments). These talks will start by explaining the theory of the origin of structure in the universe as being seeded by quantum fluctuations generated during a period of accelerated expansion. I will go on to describe the current observational and theoretical state of the subject and its connection to interesting mathematical structures such as de Sitter spacetime and string compactifications.

• Friday, November 2, Building 380, Room 380W, 2:30-3:30 and 4-5pm:

  Bay Area Microlocal Analysis Seminar at Stanford

  Mike Christ (Berkeley)

  ``Optimal off-diagonal bounds for Bergman/Szego kernels associated to positive line bundles with smooth metrics''

  and

  Maciej Zworski (Berkeley)

  ``Exponential decay of correlations in scattering and dynamical problems''

• Thursday, November 15 at 4pm in 383N:

  Jared Wunsch (Northwestern)

  ``Resolvent estimates and energy decay for the wave equation on conic manifolds''

  Abstract: I will discuss joint work with Dean Baskin on the rate of local energy decay for a solution of the wave equation on a manifold with conic singularities. In this situation, diffraction of waves by the cone points is a potential obstruction to local energy decay, but turns out, under suitable genericity assumptions, to be a very mild one. The proof proceeds via resolvent estimates, which also yield applications to the local smoothing estimate and Strichartz estimates (the latter joint work with Baskin and Marzuola) for the Schrödinger propagator.

• Monday, November 26, 4pm, Room 380-381U (special date/time/room):

  Eva Silverstein (Stanford)

  ``Inflation, de Sitter spacetime and the origin of structure''

  Abstract: Inflation is a (deceptively) simple idea in cosmology which ties directly to observational data on the one hand (real experiments), and to difficult problems in formulating quantum gravity on the other hand (thought experiments). These talks will start by explaining the theory of the origin of structure in the universe as being seeded by quantum fluctuations generated during a period of accelerated expansion. I will go on to describe the current observational and theoretical state of the subject and its connection to interesting mathematical structures such as de Sitter spacetime and string compactifications.

• Wednesday, November 28, 4pm, Room 383N, Geometry seminar:

  Semyon Dyatlov (UC Berkeley)

  ``Resonances for normally hyperbolic trapped sets''

  Abstract: Resonances are complex analogs of eigenvalues for Laplacians on noncompact manifolds, arising in long time resonance expansions of linear waves. We prove a Weyl type asymptotic formula for the number of resonances in a strip, provided that the set of trapped geodesics is r-normally hyperbolic for large r and satisfies a pinching condition. Our dynamical assumptions are stable under small smooth perturbations and motivated by applications to black holes. We also establish a high frequency analog of resonance expansions.

• Friday, November 30, 1pm, Room 380-383N (special date/time):

  Dan Knopf (UT Austin)

  Type-II singularities of Ricci flow

  Abstract: We will discuss recent progress (by the speaker and collaborators) in determining what rates of finite-time singularity formation  are possible for either compact or noncompact solutions of Ricci flow, and in constructing degenerate neckpinch solutions with prescribed  asymptotic behaviors.

• Friday, December 7

  Bay Area Microlocal Analysis Seminar at Berkeley, 2:30-5pm, Evans Hall, Room 891

  • Gunther Uhlmann (University of Washington and UC Irvine)

    ``Travel time tomography with partial data''

    Abstract: The travel time tomography problem consists in determining the anisotropic index of refraction or sound speed of a medium by making travel time measurements. We will survey what is known about this problem including some recent results on the partial data case. The latter are joint work with Andras Vasy.
  and

  • Austin Ford (Stanford)

    ``Examples of the structure and dispersion of waves on two-dimensional cones''

    Abstract: In recent years, there has been much effort to understand the dispersive properties of solutions to the wave and Schrödinger equations in various geometries. I will discuss in this talk the beginnings of extending this program to singular spaces, namely the settings of two-dimensional cones and related spaces. This will begin with the asymptotics of the Schrödinger group, and I will show how these lead to dispersive and Strichartz estimates for solutions to this equation on cones. I will also discuss joint work with Matt Blair, Sebastian Herr, and Jeremy Marzuola extending these estimates to solutions on polygonal domains and surfaces with (exact) conical singularities. The analogous results for solutions to the wave equation on these spaces will also be discussed (joint with Matt Blair and Jeremy Marzuola). Time permitting, I'll also mention current work with Andrew Hassell and Luc Hillairet with the goal of understanding the microlocal structure of these "classical waves" and the various implications knowing this structure would have.

• Tuesday, December 11, 4pm, Room 380W (special date/time):

  Plamen Stefanov (Purdue)

  ``Is a curved flight path in SAR better than a straight one?''

  Abstract: In the plane, we study the transform Rf of integrating a unknown function f over circles centered at a given curve \gamma. This is a simplified model of SAR, when the radar is not directed but has other applications, like thermoacoustic tomography, for example. We study the problem of recovering the wave front set WF(f). If the visible singularities of f hit \gamma once, we show that WF(f) cannot be recovered in, i.e., the artifacts cannot be resolved. If \gamma is the boundary of a strictly convex domain \Omega, we show that this is still true. On the other hand, in the latter case, if f is known a priori to have singularities in a compact set, then we show that one can recover WF(f), and moreover, this can be done in a simple explicit way, using backpropagation for the wave equation.

• Schedule for Winter/Spring 2012.
• Schedule for Autumn/Winter 2010-2011.
• Schedule for Winter-Spring 2010.
• Schedule for Autumn 2009.
• Schedule for Spring 2009.
• Schedule for Winter 2009.
• Schedule for Spring 2008.
• Schedule for Winter 2008.
• Schedule for Autumn 2007.