Schedule for: 16w5066 - Geometric and Spectral Methods in Partial Differential Equations

A welcome drink will be served at the hotel.

It is now well known that a lower bound on the Ricci curvature yields geometric and analytic estimates on the volume of balls, heat kernel, Poincaré inequality... We will discuss on the possibility that some of these properties are induced by spectral properties of Schrödinger operator of the type \Delta-\lambda ricci_- (ricci__ being the lowest eigenvalue of the Ricci tensor)

The essential spectrum of the Laplacian on functions over a noncompact Riemannian manifold has been extensively studied. It is known that on hyperbolic space a spectral gap appears, whereas is has been conjectured that on manifolds with uniformly subexponential volume growth and Ricci curvature bounded below the essential spectrum is the nonnegative real line. Much less is known for the spectrum of the Laplacian on differential forms. In our work we prove a generalization of Weyl's criterion for the essential spectrum of a self-adjoint and nonnegative operator on a Hilbert space. We use this criterion to study the spectrum of the Laplacian on k-forms over an open manifold. We first show that the spectrum of the Laplacian on 1-forms always contains the spectrum of the Laplacian on functions. We also study the spectrum of the Laplacian on k-forms under a continuous deformation of the metric. The results that we obtain allow us to compute the spectrum of the Laplacian on k-forms over asymptotically flat manifolds. This is joint work with Zhiqin Lu.

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 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 Kahler QAC-metrics having Ricci potential decaying sufficiently fast at infinity. We can then obtain QAC Calabi-Yau metrics in the Kahler classes of these metrics by solving a corresponding complex Monge-Ampere equation. This is a joint work with Ronan Conlon and Anda Degeratu.

The central topic of the talk will be a description of the nature of the maximal and minimal domains of the operators of a first order elliptic complex of cone operators on a compact manifold with boundary. This is, for both the minimal and maximal domains, a more subtle problem than that of a single elliptic cone operator. This is joint work with Thomas Krainer (arXiv:1611.06526).

In this talk we will consider sequences of compact oriented Riemannian manifolds with smooth boundary and study their convergence with respect to intrinsic flat distance. This distance, defined by Sormani and Wenger using work of Ambrosio and Kirchheim, generalizes the flat distance used by Federer and Fleming.

This talk is based on joint work with Nelia Charalambous, in which we consider the spectra of drifting (aka weighted or Bakry-Émery) Laplace operators on Riemannian manifolds. We shall discuss eigenvalue estimates and Weyl's law in this setting. The proof of Weyl's law is via the short time asymptotic expansion of the heat trace, and so we will discuss this expansion. In this work, we assume only finite regularity of the weight function, and we shall see that the behavior of the short time asymptotics of the heat trace determines, and conversely is determined by the regularity of the weight function.

Let $X$ be an orientable smooth manifold without boundary. The surgery sequence associated to $X$, due to Browder, Novikov, Sullivan and Wall, is a fundamental object in differential topology. Browder and Quinn also developed a version of this sequence for smoothly stratified spaces. The goal of this talk is to explain how it is possible to use the signature operator in order to map the surgery sequence in topology to a sequence of K-theory groups for C^*-algebras, called the analytic surgery sequence. The original result is due in the smooth case to Higson and Roe but I will instead explain an alternative approach developed by Schick and myself. I will also explain how, building on joint work with Albin, Leichtnam, Mazzeo it is possible to map the Browder-Quinn sequence associated to a Cheeger space to the analytic surgery sequence. This talk is based on joint work with Thomas Schick and ongoing work, still in progress, with Pierre Albin.

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.

In this talk I will explain recent joint work with Rafe Mazzeo, Hartmut Weiss and Frederik Witt on the asymptotics of the natural $L^2$ metric on the Hitchin moduli space of rank-$2$ Higgs bundles. It will be shown that on the regular part of the Hitchin fibration this metric is well-approximated by the so-called semiflat metric coming from the algebraic completely integrable system moduli space is endowed with. This result confirms some aspects of a more detailed conjectural picture made by Gaiotto, Moore and Neitzke.

We will review K-theoretical constructions related to twisted K-homology and relate them to the Baum-Connes conjecture

The index theorem of Atiyah-Patodi-Singer describes the index of a twisted Dirac operator on an even dimensional Riemannian manifold with totally geodesic boundary as the interior integral of the usual index density and an additional contribution coming the boundary known as the eta invariant. In my talk I will describe the calculation of the eta invariants for the Berger metrics on odd dimensional spheres using a fairly explicit formula for the transgression form arising from boundaries, which are not totally geodesic.

There are two kinds of solutions of the Cauchy problem of first order, the viscosity solution and the more geometric minimax solution and in general they are different. We show how they are related: iterating the minimax procedure during shorter and shorter time intervals one approaches the viscosity solution. This can be considered as an extension to the contact framework of the result of Q. Wei in the symplectic case.

A famous theorem of Patterson and Sullivan shows that, on hyperbolic manifolds, the bottom of the spectrum of the Laplacian is related to the dynamics of the fundamental group via its critical exponent. In this talk, we will give an alternative simple proof of Patterson-Sullivan theorem, and we will explain which new insights it gives to the relationship between dynamics and bottom of the spectrum in negative curvature.

Moduli spaces arising in gauge theory are often non-compact, and the issue of suitable compactification is an important general question. We shall describe the compactification of the (euclidean) monopole moduli spaces and discuss as an application the Sen conjecture, which concerns L^2 harmonic forms on these moduli spaces.

In this talk, based on joint work with Plamen Stefanov and Gunther Uhlmann, I discuss the geodesic X-ray transform on a Riemannian manifold with boundary. The geodesic X-ray transform on functions associates to a function its integral along geodesic curves, so for instance in domains in Euclidean space along straight lines. The X-ray transform on symmetric tensors is similar, but one integrates the tensor contracted with the tangent vector of the geodesics. I will explain how, under a convexity assumption on the boundary, one can invert the local geodesic X-ray transform on functions, i.e. determine the function from its X-ray transform, in a stable manner. I will also explain how the analogous result can be achieved on one forms and 2-tensors up to the natural obstacle, namely potential tensors (forms which are differentials of functions, respectively tensors which are symmetric gradients of one-forms). Here the local transform means that one would like to recover a function (or tensor) in a suitable neighborhood of a point on the boundary of the manifold given its integral along geodesic segments that stay in this neighborhood (i.e. with both endpoints on the boundary of the manifold). Our method relies on microlocal analysis, in a form that was introduced by Melrose. I will then also explain how, under the assumption of the existence of a strictly convex family of hypersurfaces foliating the manifold, this gives immediately the solution of the global inverse problem by a stable `layer stripping' type construction. Finally, I will discuss the relationship with, and implications for, the boundary rigidity problem, i.e. determining a Riemannian metric from the restriction of its distance function to the boundary.

There are several questions about the behavior of Laplace eigenfunctions that have proved to be extremely hard to deal with and remain unsolved. Among these are the study of their number of critical points, the study of the size of their zero set, the study of the number of connected components of their zero set, and the study of the topology of such components. A natural approach is to randomize the problem and ask the same questions for the zero sets of random linear combinations of eigenfunctions. In this talk I will present several results in this direction. This talk is based on joint works with Boris Hanin and Peter Sarnak.

The Cheeger-Muller theorem, first conjectured by Ray and Singer in 1973, gives equality between the analytic torsion and the Reidemeister torsion on a Riemannian manifold equipped with a flat Euclidean vector bundle. We prove a version of the Cheeger-Muller theorem on manifolds with cusp singularities. The proof uses geometric microlocal analysis techniques pioneered by Melrose, in particular the method of analytic surgery: we consider a family of smooth manifolds which degenerate to a manifold with cusps and study the behavior of the torsion under this degeneration. This is joint work with P. Albin (UIUC) and F. Rochon (UQAM).

The Weil-Petersson metrics on the Riemann moduli spaces of complex structures for an $n$-fold punctured oriented surface of genus $g,$ in the stable range $g+2n>2,$ are shown to have complete asymptotic expansions in terms of Fenchel-Nielsen coordinates at the exceptional divisors of the Knudsen-Deligne-Mumford compactification. The results are then applied to obtain the asymptotic expansions of the Ricci curvature and the sectional curvature of the moduli spaces. This is joint work with Richard Melrose.

Complex line bundles are classified up to isomorphism by integer cohomology in degree two, and it is of interest to look for similarly geometric objects which are associated to higher degree cohomology. Gerbes (of which there are various versions, due respectively to Giraud, Brylinski, Hitchin and Chattergee and Murray) are such objects associated to H^3, and various notions of "higher gerbes" have been likewise defined. However, these often to run into technical difficulties and annoyances typically associated with higher categories. We propose a natural geometric notion of higher gerbes as "multi simplicial line bundles", which avoids many of the difficulties. Moreover, every cohomology class is represented by one of these objects in the guise of a line bundle on the iterated free loop space with a so-called "fusion product" for each loop factor. This is joint work with Richard Melrose.

Manifolds with infinite cylindrical ends have continuous spectrum of increasing multiplicity as energy grows, and in general embedded resonances and eigenvalues can accumulate at infinity. However, we prove that if geodesic trapping is sufficiently mild, then such an accumulation is ruled out, and moreover the cutoff resolvent is uniformly bounded at high energies. This talk is based on joint work with Tanya Christiansen.

We show that high energy resolvent estimates for a Schrödinger operator on a manifold with infinite cylindrical ends imply the existence of resonance-free regions and consequences for the asymptotic expansion of solutions of the wave equation. While this is familiar for several classes of manifolds, this case is made more delicate by the presence of infinitely many thresholds in the continuous spectrum. This talk is based on joint work with Kiril Datchev.

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.

I will talk about two results that are motivated by the recent results of De Simoi-Kaloshin-Wei and Avila-De Simoi-Kaloshin. The first result concerns inverse spectral problems for strictly convex domains with one reflectional symmetry and the second concerns length spectral rigidity of ellipses. The two key ingredients are variational methods and asymptotic of periodic billiard orbits of rotation numbers 1/q.

I will report on recent joint work with Luc Hillairet which clarifies the distribution of resonances created by diffractive trapping associated to propagation of singularities among cone points. We find that there is a logarithmic curve of resonances below the real axis, and then a further logarithmic gap beneath. We also derive a quantization condition on the real parts of the resonances on this first curve.

We will study the semiclassical eigenvalue distribution in clusters and sub-clusters of the Hamiltonian of the hydrogen atom in a magnetic field. The result involves averages of the magnetic field terms in H along the classical orbits of the Kepler problem with fixed energy and fixed projection of the classical angular momentum in the direction of the magnetic field.

In joint work with R.\ Wolf, we prove compactness of a restricted set of real-valued, compactly supported potentials $V$ for which the corresponding Schr\"odinger operators $H_V$ have the same resonances, including multiplicities. More specifically, let $B_R(0)$ be the ball of radius $R > 0$ about the origin in $R^d$, for $d=1$ or $d=3$. Let $\mathcal{I}_R (V_0)$ be the set of real-valued potentials in $C_0^\infty( B_R(0))$ so that the corresponding Schr\"odinger operators have the same resonances, including multiplicities, as $H_{V_0}$. We prove that the iso-resonant set $\mathcal{I}_R (V_0)$ is a compact subset of $C_0^\infty (B_R(0))$ in the $C^\infty$-topology.

Let M_t, t\neq 0 be a family of compact hyperbolic surfaces which as t\to 0 degenerates to a surface M_0 with two cusps, via pinching a neck. We show a quantization condition for eigenvalues of the Laplacian on M_t in compact subsets of (1/4, \infty), with the subprincipal term determined from the scattering matrix of M_0. We use the Lefschetz fibration model for the degeneration and its metric resolution due to Melrose-Zhu. This is work in progress joint with Richard Melrose.

Poisson manifolds are rich geometric objects generalizing symplectic manifolds. Unfortunately it can be quite challenging to say anything meaningful about these manifolds in general. In this talk we will discuss how rescaled tangent bundles, a familiar tool of geometric microlocal analysis, allow us to gain traction and understand nice classes of Poisson manifold.

We will review the non-trivial Ricci soliton on $\mathbb{CP}^2\#\overline{\mathbb{CP}^2}$ constructed by Koiso and Cao. In particular we present a different point view which allows to have a better understanding of its Ricci and scalar curvatures. We will present it as a manifold where uniqueness to Yamabe equation is obtained.

Given a compact Riemannian manifold (M, g) without bound- ary of dimension m ≥ 3 and under some symmetry assumptions, we estab- lish existence and multiplicity of positive and sign changing solutions to the following Yamabe type equation −divg(a∇u) + bu = c|u|2∗−2u on M where divg denotes the divergence operator on (M, g), a, b and c are smooth functions with a and c positive, and 2∗ = 2m denotes the critical Sobolev m−2 exponent. In particular, if Rg denotes the scalar curvature, we give some examples where the Yamabe equation −4(m − 1)∆gu + Rgu = κu2∗−2 on M. m−2 admits an infinite number of sign changing solutions. We also study the lack of compactness of these problems in a symmetric setting and how the symmetries restore it at some energy levels. This allows us to use a suitable variational principle to show the existence and multiplicity of such solutions. This is joint work with M ́onica Clapp.