Analysis and PDE Seminar Homepage, Autumn-Winter-Spring 2016/2017

Location: 384I.

Time: Monday, 4pm.

• Wednesday, August 24, 11am, Room 383N (unusual day/time/room):

  Jan Derezinski (Warsaw)

  ``Almost homogeneous Schrödinger operators''

  Abstract: First I will describe a certain natural holomorphic family of closed operators with interesting spectral properties. These operators can be fully analyzed using just trigonometric functions. I try to market them under the name "a toy model of renormalization group". Then I will discuss 1-dimensional Schroedinger operators with a 1/x^2 potential with general boundary conditions, which I studied recently with S.Richard. Even though their description involves Bessel and Gamma functions, they turn out to be equivalent to the previous family.

• Wednesday, August 31, 2pm, Room 383N (unusual day/time/room):

  Frédéric Rochon (UQAM)

  ``QAC Calabi-Yau manifolds''

  Abstract: We will explain how to construct new examples of quasi-asymptotically conical (QAC) Calabi-Yau manifolds that are not quasi-asymptotically locally Euclidean (QALE). Our strategy consists first in introducing a natural compactification of QAC-spaces by manifolds with fibred corners and to give a definition of QAC-metrics in terms of a natural Lie algebra of vector fields on this compactification. Using this and the Fredholm theory of Degeratu-Mazzeo for elliptic operators associated to QAC-metrics, we can in many instances obtain Kähler QAC-metrics having Ricci potential decaying sufficiently fast at infinity. We can then obtain QAC Calabi-Yau metrics in the Kähler classes of these metrics by solving a corresponding complex Monge-Ampère equation. This is a joint work with Ronan Conlon and Anda Degeratu.

• Monday, October 3:

  Jonathan Luk (Stanford)

  ``Stable shock formation for solutions to the multidimensional compressible Euler equations in the presence of non-zero vorticity''

  Abstract: It is well-known since the foundational work of Riemann that plane symmetric solutions to the compressible Euler equations may form shocks in finite time. For a class of simple plane symmetric solutions, we prove that the phenomenon of shock-formation is stable under perturbations of the initial data that break the plane symmetry with potentially non-vanishing vorticity. In particular, this is the first constructive shock-formation result for which the vorticity is allowed to be non-vanishing at the shock. We show in particular that the vorticity remains bounded all the way up to the shock, and that the dynamics are well-described by the irrotational compressible Euler equations. This is a joint work with J. Speck (MIT), which is partly an extension of an earlier joint work with J. Speck (MIT), G. Holzegel (Imperial) and W. Wong (Michigan State).

• Wednesday, October 12, 3:15pm, Room 383N (unusual day/time/room, geometry seminar slot):

  Georgi Raikov (Pontificia Universidad Catolica de Chile)

  `` Resonances and SSF singularities for a magnetic Schroedinger operator''

  Abstract: I will consider the 3D Schroedinger operator with constant magnetic field, perturbed by a rapidly decaying electric potential.

  First, I will discuss the asymptotic behavior of the corresponding Krein spectral shift function (SSF) near the Landau levels which play the role of spectral thresholds. I will show that the SSF has singularities near these thresholds, which admit a fairly explicit description in terms of appropriate Berezin-Toeplitz operators.

  Further, I will introduce the resonances of the perturbed Schroedinger operator, and will discuss their asymptotic distribution near the spectral thresholds. I will show that under suitable assumptions on the pertubing potential, there are infinitely many resonances near every fixed Landau level, and the main asymptotic term of the corresponding resonance counting function is again related to the Berezin-Toeplitz operators arising in the description of the SSF singularities.

  If time allows, some extensions to Pauli and Dirac operators with non-constant magnetic fields could be briefly outlined.

• Friday, October 28, BAMAS at Berkeley, Room 736, Evans Hall:

  • 2:30-3:30: Peter Hintz (Berkeley)

    ``Non-linear stability of Kerr-de Sitter black holes''

    Abstract: In joint work with András Vasy, we prove the stability of the Kerr-de Sitter family of black holes in the context of the initial value problem for the Einstein vacuum equations with positive cosmological constant, for small angular momenta but without any symmetry assumptions on the initial data. I will describe the general framework which enables us to deal systematically with the diffeomorphism invariance of Einstein's equations, and thus how our solution scheme finds a suitable (wave map type) gauge within a carefully chosen finite-dimensional family of gauges. In particular, I will explain our microlocal proof of a key ingredient of this framework, called `constraint damping,' a device first introduced in numerical relativity.
  
  and

  • 4-5pm: Laurent Michel (Nice/Stanford)

    ``Metastability for semiclassical random walk''

    Abstract: We study the return to equilibrium for a semiclassical random walk associated to a multiple well transition density. We describe accurately the small eigenvalues of the associated operator. The proof is based on a supersymmetric approach. As a preliminary we prove a general factorization result on pseudo differential operators. Joint work with J.-F. Bony and F. Hérau

• Monday, October 31 (double header), both talks in 384I:

  • 2:30-3:30pm

    Jussi Behrndt (TU Graz)

    ``Spectral shift functions and Dirichlet-to-Neumann maps''

    Abstract: In this talk we discuss a representation formula for the spectral shift function of a pair of self-adjoint operators via an abstract operator valued Titchmarsh-Weyl m-function. The general result is applied to different self-adjoint realizations of second-order elliptic partial differential operators on smooth domains with compact boundaries, Schrödinger operators with compactly supported potentials, and finally, Schrödinger operators with singular potentials supported on hypersurfaces. In these applications the spectral shift function is determined in an explicit form with the help of (energy parameter dependent) Dirichlet-to-Neumann maps. The talk is based on joint work with Fritz Gesztesy and Shu Nakamura.
    

  • 4-5pm

    András Vasy (Stanford)

    ``The stability of Kerr-de Sitter black holes (Part II: The analysis)''

    Abstract: In this lecture, based on joint work with Peter Hintz, I will discuss analytic aspects of our proof of the stability of slowly rotating Kerr-de Sitter black holes, following the October 26 geometry seminar in which geometric aspects were discussed.

    Kerr-de Sitter black holes are rotating black holes in a universe with a positive cosmological constant, given by an explicit formula, due to Kerr and Carter. They are parameterized by their mass and angular momentum, much like Kerr black holes in a space-time with vanishing cosmological constant.

    I will discuss analytic (PDE) aspects of the stability question for these black holes in the initial value formulation. Namely, appropriately interpreted (fixing a gauge), Einstein's equations can be thought of as quasilinear wave equations, and then the question is if perturbations of the initial data along a Cauchy hypersurface produce solutions which are close to, and indeed asymptotic to, a Kerr-de Sitter black hole, typically with a different mass and angular momentum. The key aspects are arranging appropriate gauge conditions and the analysis of linear wave equations on asymptotically Kerr-de Sitter backgrounds, including with coefficient possessing limited regularity.

• Monday, November 14:

  Laurent Michel (Nice/Stanford)

  ``On the small eigenvalues of the Witten Laplacian''

  Abstract: The Witten Laplacian was introduced (by E. Witten) in the early 80’s to give an analytic proof of the Morse inequalities. The first optimal description of the small eigenvalues of this operator was obtained in 2004 (Bovier-Gayrard-Klein, Helffer-Klein-Nier) under some generic assumptions on the landscape potential f. In this talk, we present the approach of Helffer-Klein-Nier and show some recent progress to get rid of the generic assumption.

• Monday, December 5 (joint with symplectic seminar, 383N):

  Sergei Kuksin (Paris 7)

  ``Small-amplitude solutions for space-multidimensional hamiltonian PDEs under periodic boundary conditions''

  Abstract: I will discuss the problem of studying the long-time behaviour of small solutions for nonlinear Hamiltonian PDEs on T^d, and will explain that in certain sense the behaviour for solutions of space-multidimensional equations (d>1) significantly differs from that for the 1d systems. The talk is based on my recent joint work with H.Eliasson and B.Grebert, to appear in GAFA, arXiv 1604.01657.

• Friday, December 9, BAMAS at Stanford, Room 384H:

  • 1:30-2:30pm

    Semyon Dyatlov (MIT)

    ``Resonances for open quantum maps''

    Abstract: Quantum maps are a popular model in physics: symplectic relations on tori are quantized to produce families of N x N matrices and the high energy limit corresponds to the large N limit. They share a lot of features with more complicated quantum systems but are easier to study numerically. We consider open quantum baker's maps, whose underlying classical systems have a hole allowing energy escape. The eigenvalues of the resulting matrices lie inside the unit disk and are a model for scattering resonances of more general chaotic quantum systems. However in the setting of quantum maps we obtain results which are far beyond what is known in scattering theory.

    We establish a spectral gap (that is, the spectral radius of the matrix is separated from 1 as N tends to infinity for all the systems considered. The proof relies on the notion of fractal uncertainty principle and uses the fine structure of the trapped sets, which in our case are given by Cantor sets, together with simple tools from harmonic analysis, algebra, combinatorics, and number theory. We also obtain a fractal Weyl upper bound for the number of eigenvalues in annuli. These results are illustrated by numerical experiments which also suggest some conjectures. This talk is based on joint work with Long Jin.

  and

  • 2:45-3:45pm

    Maciej Zworski (Berkeley)

    `` Ruelle zeta function at zero for surfaces of variable curvature''

    Abstract: For surfaces of constant negative curvature the Selberg trace formula shows that the order of vanishing of the Ruelle zeta function at 0 is given by the absolute value of the Euler characteristic of the surface. Using simple microlocal arguments we prove that this remains true for any negatively curved sufficiently smooth surface. Joint work with S. Dyatlov.

• Monday, January 9, double header, both talks in 384I:

  • Volker Schlue (Paris) 2:30-3:30

    ``On the stability of expanding black hole cosmologies''

    Abstract: In general relativity, the Kerr de Sitter spacetime represents a model of a black hole in an expanding universe. Recently Hintz and Vasy established the non-linear stability of the black hole exterior (up to the "cosmological horizon") as solutions to the Einstein equations with positive cosmological constant. I will present recent progress on my understanding of the dynamics beyond the cosmological horizon, and show in particular that the Weyl curvature decays at far distances from the black hole. A significant difficulty of the stability problem in this "cosmological region" is that the asymptotic geometry is a priori unknown, and in particular cannot be expected to be represented globally by a member of the Kerr de Sitter family.

  • Dean Baskin (Texas A&M) 4-5:

    ``Asymptotics of the radiation field on asymptotically Minkowski spaces''

    Abstract: I will report on joint work with Andras Vasy and Jared Wunsch which determines the global asymptotic behavior of solutions of the wave equation on asymptotically Minkowski spaces. The rate of decay (and, indeed, all exponents in the asymptotic expansion at time-like infinity) is determined by the locations of the resonances of an associated operator on an asympotically hyperbolic space. If time permits, I will describe joint work with Jeremy Marzuola showing that a similar theorem holds for spacetimes given by the product of the real line and a cone.

• Monday, February 6, BAMAS at Berkeley, Room 740 Evans:

  • 2:40-3:30pm

    Jonathan Luk (Stanford)

    ``Strong cosmic censorship in spherical symmetry for two-ended asymptotically flat data''

    Abstract: I will present a recent work (joint with Sung-Jin Oh) on the strong cosmic censorship conjecture for the Einstein-Maxwell-(real)-scalar-field system in spherical symmetry for two-ended asymptotically flat data. For this model, it was previously proved (by M. Dafermos and I. Rodnianski) that a certain formulation of the strong cosmic censorship conjecture is false, namely, the maximal globally hyperbolic development of a data set in this class is extendible as a Lorentzian manifold with a $C^0$ metric. Our main result is that, nevertheless, a weaker formulation of the conjecture is true for this model, i.e., for a generic (possibly large) data set in this class, the maximal globally hyperbolic development is inextendible as a Lorentzian manifold with a $C^2$ metric.
  and

  • 4:10-5pm

    Jeff Galkowski (Stanford/McGill)

    ``Pointwise Bounds for Steklov Eigenfunctions''

    Abstract: Let (\Omega,g) be a compact, real-analytic Riemannian manifold with real-analytic boundary \partial \Omega. The harmonic extensions of the boundary Dirchlet-to-Neumann eigenfunctions are called Steklov eigenfunctions. We show that the Steklov eigenfuntions decay exponentially into the interior in terms of the Dirichlet-to-Neumann eigenvalues and give a sharp rate of decay to first order at the boundary and proving a conjecture of Hislop and Lutzer. The estimates follow from sharp estimates on the concentration of the FBI transforms of solutions to analytic pseudodifferential equations Pu=0 near the characteristic set \{\sigma(P)=0\}. This talk is based on joint work with John Toth.

• Monday, February 27:

  Francisco Martin (Granada)

  ``Translating solitons of the mean curvature flow''

  Abstract: A translator is a surface in $\R^3$ that (up to a tangential diffeomorphism) moves with velocity $v=(0,0,-1)$ by MCF. Equivalently, the mean curvature at each point is $H= (0,0,-1)^{\perp}.$ Besides vertical planes, one of the simplest examples of complete translators is the grim reaper cylinder. In this talk we will describe several existence and uniqueness results for complete translators which are graphs over planar domains. This is a joint work with D. Hoffman, T. Ilmanen and B. White.

• Thursday, March 2 at noon, in 383N (special analysis/PDE/geometry seminar):

  Michael Singer (UCL)

  ``Multiscale clustering and smooth monopole families''

  Abstract: The moduli space of euclidean monopoles of charge k is a complete non-compact smooth riemannian manifold M_k of dimension 4k. The `infinity’ of the manifold has different regions: for each (proper) partition of k = k_1 + … + k_n, there is a region approximated (locally) by a product C_n x M_{k_1} x … x M_{k_n}, where C_n is a space of n widely separated points in R^3. In trying to figure out how these different regions fit together in the moduli space, it is essential to have a systematic understanding of multiscale clustering or `clusters-within-clusters’. I shall describe work in progress which will provide such a systematic understanding, and, if time permits, explain the applications to compactification of monopole moduli spaces.

• Monday, March 20, BAMAS at Stanford, Room 384I:

  • 2:30-3:30pm

    Jared Wunsch (Northwestern)

    ``Resonances generated by conic diffraction''

    Abstract: I will report on recent work, joint with Luc Hillairet, that refines our understanding of the strings of resonances along logarithmic curves generated by multiply diffracted trapped rays.
  
  and

  • 4-5pm

    Gunther Uhlmann (U Washington/HKUST)

    ``Travel Time Tomography, Boundary Rigidity and Lens Rigidity''

    Abstract: We will consider the inverse problem of determining the sound speed or index of refraction of a medium by measuring the travel times of waves going through the medium. This problem arises in global seismology in an attempt to determine the inner structure of the Earth by measuring travel times of earthquakes. It has also several applications in optics and medical imaging among others. The problem can be recast as a geometric problem: Can one determine a Riemannian metric of a Riemannian manifold with boundary by measuring the distance function between boundary points? This is the boundary rigidity problem. We will also consider the problem of determining the metric from the scattering relation, the so-called lens rigidity problem. The linearization of these problems involve the integration of a tensor along geodesics, similar to the X-ray transform. We consider also the partial data case, where you are making measurements on a subset of the boundary. This is joint work with Plamen Stefanov and Andras Vasy

• Monday, April 3:

  Heinrich Freistuhler (Universität Konstanz)

  ``A Causal Five-Field Theory of Dissipative Relativistic Fluid Dynamics''

  Abstract

• Wednesday, April 5, 4:15pm, Room 384H (applied math seminar slot):

  Michael Taylor (University of North Carolina)

  ``Hardy spaces, Cauchy integrals, and Toeplitz operators on uniformly rectifiable domains''

• Monday, April 17:

  Apoorva Khare (Stanford)

  ``The Schoenberg-Rudin theorem, Rayleigh quotients, and Schur polynomials''

  Abstract: Which functions preserve positive semidefiniteness (psd) when applied entrywise to the entries of psd matrices? This question has a long history beginning with Schur, Schoenberg, and Rudin, who classified the positivity preservers of matrices of all dimensions. The study of positivity preservers in fixed dimension is harder, and a complete characterization remains elusive to date. In fact it was not known if there exists any analytic preserver with negative coefficients. We prove such an existence result, and in fact a characterization, for classes of polynomials. Central to the proof are novel determinantal identities involving Schur polynomials. Our result can be reformulated as a variational problem; this leads to novel connections between Rayleigh quotients and Schur polynomials. (Joint with A. Belton, D. Guillot, and M. Putinar.)

• Monday, May 15, BAMAS at Stanford, Room 384H:

  • 2:15-3:15pm

    Peter Hintz (Berkeley)

    ``Resonances for obstacles in hyperbolic space''

    Abstract: We consider scattering by star-shaped obstacles in hyperbolic space and show that resonances satisfy a universal bound $\Im\lambda\leq -1/2$; in odd dimensions and for small obstacles with diameter $\rho$, we improve this to $\Im\lambda < -C/\rho$ for a universal constant $C$. Our proofs largely rely on the classical vector field approach of Morawetz. We also explain how to relate resonances for small obstacles to scattering resonances in Euclidean space. This talk is based on joint work with Maciej Zworski.
  
  and

  • 4-5pm

    Francois Monard (UCSC)

    ``X-ray transforms and tensor tomography on surfaces''

    Abstract: On (M,g) a non-trapping Riemannian surface with boundary, the tensor tomography problem consists of inferring (i) what is reconstructible of a symmetric tensor field from knowledge of its integrals along geodesics through that surface, and (ii) how to reconstruct it. In the Euclidean case and zero-th order tensors (i.e., functions), this is the well-known X-Ray (or Radon) transform and it serves as the theoretical backbone of Computerized Tomography.
    In a geometric setting, the answer to questions (i) and (ii) depends on the order of tensors considered, the underlying geometry, and what functional setting for the X-ray transform is chosen. In this talk I will review recent results on these aspects, and will discuss reconstruction approaches for functions and tensor fields, some valid in rather general settings, others requiring more Euclidean explicitness.

• Monday, May 22:

  Georgios Moschidis (Princeton)

  ``A proof of the instability of AdS spacetime for the Einstein--null dust system''

  Abstract: The AdS instability conjecture is a conjecture related to the initial-boundary value problem for the vacuum Einstein equations with a negative cosmological constant in general relativity. It states that generic, arbitrarily small perturbations to the initial data of the AdS spacetime, under evolution by the vacuum Einstein equations with reflecting boundary conditions on conformal infinity, lead to the formation of black holes. Formulated in 2006 by Dafermos and Holzegel, this conjecture has attracted a vast amount of numerical and heuristic works by several authors, focused mainly on the simpler setting of the spherically symmetric Einstein--scalar field system. In this talk, we will provide the first rigorous proof of the AdS instability conjecture in the simplest possible setting, namely for the spherically symmetric Einstein--massless Vlasov system, in the case when the Vlasov field is moreover supported only on radial geodesics. This system is equivalent to the Einstein--null dust system, allowing for both ingoing and outgoing dust. In order to overcome the "trivial" break down occuring once the null dust reaches the centre $r=0$, we will study the evolution of the system in the exterior of an inner mirror with positive radius $r_{0}$ and prove the conjecture in this setting. After presenting our proof, we will briefly explain how the main ideas can be extended to more general matter fields.

• Thursday, June 1, 1:30-2:30pm, Room 383N (unusual day/time/room, joint with geometry):

  Hart Smith (University of Washington)

  ``Dispersive estimates for the wave equation on manifolds with bounded curvature tensor''

  Abstract: TBA

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