Analysis and PDE Seminar Homepage, Autumn-Winter-Spring 2015/2016

Location: 384I.

Time: Monday, 4pm.

• Monday, October 12:

  Peter Perry (University of Kentucky)

  IST and PDE

  Abstract: This talk reports in part on joint work with Russell Brown (Kentucky), Katherine Ott (Bates College), Michael Music (Kentucky), Jiaqi Liu (Kentucky), and Catherine Sulem (Toronto).

  In recent years, inverse scattering theory (IST) has been developed into a mathematically rigorous and effective tool for studying global well-posedness and long-time behavior of completely integrable dispersive PDE's in one and two dimensions. I will discuss Deift and Zhou's pioneering work on the cubic NLS in one dimension (2003) and more recent work:
  (1) (Liu, Perry, Sulem) the derivative nonlinear Schrodinger equation, a 1+1 dimensional equation that models the dynamics of Alfven waves propagating along an ambient magnetic field in a long-wave, weakly nonlinear scaling regime;
  (2) (Perry) the Davey-Stewartson equation, a dispersive equation in two dimensions that describes modulated nonlinear surface gravity waves propagating over a horizontal sea bed, and
  (3) (Music, Perry) the Novikov-Veselov equation, a two-space dimensional analogue of the celebrated Korteweg-de Vries equation that includes the celebrated Kadomtsev-Petviashvili equations as scaling limits.
  I will also discuss prospects for further progress on long-term behavior for integrable dispersive PDE's and small perturbations from integrability.

  References:
  (1) P. Deift, X. Zhou. Long-time asymptotics for solutions of the NLS equation with initial data in a weighted Sobolev space. Dedicated to the memory of Jürgen K. Moser. Comm. Pure Appl. Math. 56 (2003), no. 8, 1029-1077
  (2) P. Perry, Global Well-Posedness and Long-time Asymptotics for the Defocussing Davey-Stewartson II Equation in $H^{1,1}(R^2)$ arXiv:1110.5589v3 (Accepted pending revisions for J. Spectral Theory).
  (3) P.A. Perry, K.A. Ott, R.M. Brown. Action of a scattering map on weighted Sobolev spaces in the plane arXiv:1501.04669 (submitted to J. Funct. Anal.)
  (4) Mi. Music, P. A. Perry. Global solutions for the zero-energy Novikov-Veselov equation by inverse scattering. arXiv:1502.02632 (submitted to Comm. Math. Phys.)
  (5) J. Liu, P. A. Perry, C. Sulem. Global existence for the Derivative Nonlinear Schrodinger Equation by the method of inverse scattering. To be posted on arXiv.

• Friday, October 16, Bay Area Microlocal Analysis Seminar at Berkeley, 736 Evans Hall:

  • 2:10-3pm: Nicolas Burq (Paris XI, Orsay)

    From Strichartz estimates to uniform Lp resolvent estimates (joint with with D.Dos Santos and K. Krupchyk)

    Abstract: We prove uniform Lp resolvent estimates for the stationary damped wave operator. The uniform Lp resolvent estimates for the Laplace operator on a compact smooth Riemannian manifold without boundary were first established by Dos Santos Ferreira-Kenig-Salo and advanced further by Bourgain-Shao-Sogge-Yao. Here we provide an alternative proof relying on the techniques of semi- classical Strichartz estimates. This approach allows us also to handle non-self-adjoint perturbations of the Laplacian and embeds very naturally in the semi-classical spectral analysis framework.

  • 3:40-4:30pm: Franz Luef (Norwegian University of Science and Technology, Trondheim)

    Modulation spaces and applications to pseudodifferential operators

    Abstract: Modulation spaces are a class of Banach spaces that have turned out to be of great use in harmonic analysis, time-frequency analysis and quanitzation. Many classical function spaces, such as generalized Sobolev spaces or the Sjoestrand class, are modulation spaces. In this talk I present the basic theory of modulation spaces and describe the link to pseudodifferential operators, including the Gabor wave front set introduced by Rodino and Wahlberg. The latter turns out to coincide with the global wave front set of Hoermander.

• Monday, October 19:

  Xuwen Zhu (Stanford)

  Eleven dimension supergravity equations on edge manifolds

  Abstract: We study the eleven dimensional supergravity equations on $\mathbb{B}^7 \times \mathbb{S}^4$ considered as an edge manifold. We compute the indicial roots of the linearized system using the Hodge decomposition, and using the edge calculus and scattering theory we prove that the moduli space of solutions, near the Freund--Rubin states, is parameterized by three pairs of data on the bounding 6-sphere.

• Monday, October 26:

  Chris Kottke (Northeastern University)

  Partial compactification and metric asymptotics of monopoles

  Abstract: I will describe a partial compactification of the moduli space, M_k, of SU(2) magnetic monopoles on R^3, wherein monopoles of charge k decompose into widely separated `monopole clusters' of lower charge going off to infinity at comparable rates. The hyperkahler metric on M_k has a complete asymptotic expansion, the leading terms of which generalize the asymptotic metric discovered by Bielawski, Gibbons and Manton in the case that the monopoles are all widely separated. This is joint work with M. Singer, and is part of a larger work in progress with R. Melrose and K. Fritzsch to fully compactify the M_k as manifolds with corners and compute their L^2 cohomology.

• Monday, November 16:

  Aynur Bulut (Michigan/MSRI)

  Global well-posedness results for nonlinear wave and Schrödinger equations in the ill-posed regime via probabilistic techniques

  Abstract: In this talk, we give an overview of several results on global well-posedness of the nonlinear wave and Schrödinger equations with an emphasis on techniques which allow for treatment of initial data supercritical with respect to the scaling of the nonlinearity. For this class of data, the initial value problems are ill-posed — uniform continuity of the solution map cannot hold — and we are therefore required to go beyond the usual class of deterministic constructions based on fixed-point methods. In particular we consider the problems in a probabilistic setting, with randomized initial data, where the global results are asked to hold for data occurring almost surely in the randomization — that is, excluding a measure zero set of initial data. We have pursued this philosophy in several recent works for the nonlinear wave and Schrödinger equations on compact manifolds, including the two-dimensional and three-dimensional unit balls B_2 and B_3 in Euclidean space, with radial data and vanishing Dirichlet boundary conditions, as well as results on the product domain B_2 \times T. In this probabilistic low regularity framework, where the energy is often not directly available, the most delicate cases are handled using uniform-in-time control provided by finite-dimensional truncations of the equation and derivation of uniform-in-time a priori bounds from invariance properties of the associated Gibbs measures. Careful choices of function spaces are required in order to complete the relevant estimates.

• Monday, November 30, Bay Area Microlocal Analysis Seminar at Stanford, Room 380-384H:

  • Patrick Gérard (Université Paris-Sud, Orsay, and MSRI), 2:30-3:30pm

    Geometry of the phase space and wave turbulence for the cubic Szegö equation

    Abstract : I will review the main properties of the cubic Szegö equation, a toy model for nonlinear wave interaction in one space dimension enjoying some Lax pair structure. Emphasis will be put on the structure of the action-angle variables in the phase space and how they are connected to long time strong transition to high frequencies. This is a joint work with Sandrine Grellier.

  • Alexis Drouot (Berkeley), 4-5pm

    Scattering resonances for highly oscillatory potentials

    Abstract: We study resonances of compactly supported potentials $V_\varepsilon(x) = W(x,x/\varepsilon)$ where $W : \R^d \times \R^d/(2\pi \Z)^d \to \C$, $d$ odd. That means that $V_\varepsilon(x)$ is a sum of a slowly varying potential, $W_0(x)$, and one oscillating at frequency $1/\varepsilon$. For $W_0 = 0$ we prove that $V_\varepsilon$ has no resonances above the line $\Im \lambda = -A \ln(1/\varepsilon)$ -- except a simple resonance of modulus $\sim \varepsilon^2$ when $d=1$. In the case $W_0 \neq 0$ we prove that resonances in fixed strips admit an expansion in powers of $\varepsilon$. We use this result to produce an effective potential converging uniformly to $W_0$ as $\varepsilon \rightarrow 0$ and whose resonances approach resonances of $V_\varepsilon$ modulo $O(\varepsilon^4)$. This work proves a conjecture of Duchene-Vukicevik-Weinstein.

• Monday, January 11:

  Heiko Gimperlein (Heriot-Watt University, Edinburgh)

  "The wave equation and analytic factorization of Lie group representations"

  Abstract: In this talk, we outline how simple properties of the wave equation provide detailed information about functions of the Laplacian on a real Lie group G. While the basic idea goes back to Cheeger, Gromov and Taylor in the 1980's, its application to questions in representation theory seems to be new. We are going to use it to prove a factorization theorem of Dixmier-Malliavin type for the space of analytic vectors E^{\omega} for representations of G on, for example, a Banach space: There exists a natural algebra of superexponentially decreasing analytic functions A(G), such that E^{\omega} = A(G) * E^{\omega}. Applications to analytic representation theory will be mentioned. In particular, we obtain a new approach to the Kashiwara-Schmid theorem on unique analytic globalizations of Harish-Chandra modules. (joint with B. Kroetz, C. Lienau and H. Schlichtkrull)

• Friday, January 29, Bay Area Microlocal Analysis Seminar at Berkeley, Room 740 Evans:

  • Vadim Kaloshin (University of Maryland), 2:40-3:30pm

    ``On deformational spectral rigidity of convex symmetric planar domains''

    Abstract: One can associate to a planar convex domain $\Omega \subset \R^2$ two types of spectra: the Laplace spectrum consisting of eigenvalues of a Dirichlet problem and the length spectrum consisting of perimeters of all periodic orbits of a billiard problem inside $\Omega$. The Laplace and length spectra are closely related, generically the first determines the second. M. Kac asked if the Laplace spectrum determines a domain $\Omega$. There are counterexamples. During the talk we show that a planar axis symmetric domain close to the circle can't be smoothly deformed preserving the length spectrum unless the deformation is a rigid motion. This gives a partial answer to a question of P. Sarnak. This is a joint work with J. De Simoi and Q. Wei.

  • Andras Vasy (Stanford), 4:10-5pm

    ``The Feynman propagator and its positivity properties''

    Abstract: In this talk, partially on joint work with Jesse Gell-Redman, Nick Haber and Michal Wrochna, I will explain the properties of the Feynman propagator, i.e. the inverse of the wave operator on `Feynman function spaces', in various settings. I will also explain its positivity properties, and the connection to spectral and scattering theory in Riemannian settings, as well as to the classical parametrix construction of Duistermaat and Hormander.

• Monday, February 8:

  Sa'ar Hersonsky (University of Georgia)

  ``Electrical Networks and Stephenson's Conjecture''

  Abstract: The Riemann Mapping Theorem asserts that any simply connected planar domain which is not the whole of it, can be mapped by a conformal homeomorphism onto the open unit disk. After normalization, this map is unique and is called the Riemann mapping.
  In the 90's, Ken Stephenson, motivated by a circle packing approximation scheme suggested by Thurston (and first proved to converge by Rodin-Sullivan), predicted that the Riemann Mapping may be approximated by a different scheme, i.e., by a sequence of finite networks endowed with particular choices of conductance constants. These networks are naturally defined in terms of the contact graph of any circle packing.
  We will affirm Stephenson's Conjecture in a greater generality.

• Monday, February 22, Bay Area Microlocal Analysis Seminar at Stanford, Room 384I:

  • Gilles Lebeau (Université de Nice), 2:30-3:30pm

    ``On the holomorphic extension of the Poisson Kernel''

    Abstract: Let $\Omega$ be an open subset of $\mathbb R^d$ with analytic boundary. The Poisson kernel K(x,y), with $x\in \Omega, y\in \partial\Omega$, is the solution of the following elliptic boundary value problem, where $\triangle$ denotes the usual Laplace operator $$ \triangle_x K(x,y)=0 \quad \text{in} \quad \Omega,\quad \quad K(x,y)\vert_{\partial\Omega}= \delta_{x=y}.$$ In this lecture, we are interested in the holomorphic extension in $x\in \mathbb C^d$ of $K(x,y)$ near a given point $y\in \partial\Omega$. Very little is known about this problem. We will first recall old and classical results on the regularity of the Dirichlet problem. Then we will state "conjectures" on the location of the singularities of the holomorphic extension of $K$ and will describe some particular cases where it holds true. Finally, we will explain how this problem is related to propagation of singularities and complex billiard dynamics.

  • Jeff Galkowski (Stanford), 4-5pm

    ``A Quantum Sabine Law for Resonances in Transmission Problems''

    Abstract: We prove a quantum Sabine law for the location of resonances in transmission problems. In this talk, our main applications are to scattering by strictly convex, smooth, transparent obstacles and highly frequency dependent delta potentials. In each case, we give a sharp characterization of the resonance free regions in terms of dynamical quantities. In particular, we relate the imaginary part of resonances to the chord lengths and reflectivity coefficients for the ray dynamics and hence give a quantum version of the Sabine law from acoustics.

• Monday, March 7, 2:30-3:30pm, Room 380F:

  Dan Freed (UT Austin)

  ``Reflection positivity and invertible topological phases''

  Abstract: This talk surveys an application of homotopy theory to condensed matter physics and quantum field theory. In joint work with Mike Hopkins we implement an extended version of reflection positivity (Wick-rotated unitarity) for invertible topological quantum field theories and compute the abelian group of deformation classes of such theories in terms of bordism spectra. An immediate application is to the classification of special "short-range entangled" topological phases of matter.

• Monday, March 14:

  Michael Singer (UCL)

  ``Monopole moduli spaces and Sen’s conjecture''

  Abstract: Sen’s conjecture is a statement about the space of L^2 harmonic forms on the moduli space of euclidean monopoles. I shall describe the conjecture and recent work aimed at proving it.

• Monday, April 11:

  Junyong Zhang (Stanford/Beijing Institute of Technology)

  ``Global-in-time Strichartz estimates for Schrödinger and wave equations on non-trapping scattering manifolds''

• Monday, April 25:

  Pablo R. Stinga (Iowa State)

  ``Fractional nonlocal space-time equations''

  Abstract: We report on recent advances on the regularity theory of fractional nonlocal space-time equations like the master equation considered by L. Caffarelli and L. Silvestre. As a byproduct of our ideas we show novel mixed-norm estimates for parabolic equations. The latter results not only recover previous works by N. V. Krylov but also extend them to their fullest extent.

• Monday, May 2:

  Jan Derezinski (Warsaw)

  ``The Klein-Gordon equation on curved spacetimes and its propagators''

  Abstract: The Klein-Gordon equation has several natural Green's functions, often called propagators. The so-called Feynman propagator, used in quantum field theory, has a clear definition on static spacetimes. I will discuss, partly on a heuristic level, its possible generalizations to the non-static case. I will also describe a curious open problem about the self-adjointness of the Klein-Gordon operator.

• Monday, TBA:

  Andrew Waldron (UC Davis)

  ``TBA''

• Monday, May 23 (joint with symplectic geometry):

  Yanir Rubinstein (Maryland)

  ``The degenerate special Lagrangian equation''

  Abstract: In joint work with J. Solomon, we introduce the degenerate special Lagrangian equation and develop the basic analytic tools to construct and study its solutions. This equation governs geodesics in the space of positive Lagrangians. It has intriguing relations to uniqueness, existence and stability notions related to special Lagrangians, Lagrangian mean curvature flow, the topology of Hamiltonian isotopy classes, and the (strong) Arnold conjecture.

• Tuesday, May 24, 3pm, Room 383N (unusual time/room):

  Yakov Shlapentokh-Rothman (Princeton)

  ``Scattering theory and blue-shift instabilities on Kerr exterior and interior spacetimes''

  Abstract: In the first part of the talk, which concerns joint work with Dafermos and Rodnianski, we will present results corresponding to the "existence and uniqueness of scattering states" and "asymptotic completeness" for the wave equation on Kerr exterior spacetimes. These results are stated with respect to function spaces which allow for the solutions to exhibit certain singular behaviors along the event horizon. In the second part of the talk, which concerns joint work with Dafermos, we will show that it is in fact necessary to allow such singular behavior, and furthermore, exploiting essentially the same mechanism, we will also show that solutions to the wave equation generically have a singular behavior along the Cauchy-Horizon in Kerr black hole interiors.

• Monday, December 5:

  Semyon Dyatlov (Stanford)

  ``TBA''

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