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Schedule/Abstracts

The talks will take place at the Garrett Theological Seminary, Room 205, 2121 Sheridan Road, Evanston, IL 60201

Tentative schedule of talks:


Titles and Abstracts

Analytic Toeplitz and pseudo-differential operators; analytic wave-front
Louis Boutet de Monvel
Abstract: I will review the definition of analytic pseudo-differential operators or Toeplitz operators; also various 'geometric' ways of seeing the analytic wavefront set.

Cartan-Remez type inequalities for analytic and plurisubharmonic functions
Alex Brudnyi
Abstract: We start by proving the famous Cartan lemma and the Remez inequality for analytic and plurisubharmonic functions. Then we deduce from these results some results about behavior of compact families of analytic and plurisubharmonic functions over convex subsets of a Euclidean space and present several applications of these results in analysis.

Holomorphic Extension and the Complex Monge-Ampère Equation
Daniel Burns
Abstract: Some time ago, Boutet de Monvel proved a Paley-Wiener type theorem for compact real analytic manifolds, relating the exponential rate of decay of the Fourier coefficients of a real analytic function with respect to an analytic elliptic operator with the radius of holomorphic extension of said function into the complex domain, i.e., into a small neighborhood of $M$ in its complexification. We discuss what happens when one wants to study this extension globally. This leads to holomorphic differential geometric properties of global complexifications, and especially exhaustions by solutions of the homogeneous complex Monge-Ampère equation. We also discuss another approach to these questions using the Toeplitz operators of Boutet de Monvel and Guillemin. Time permitting, we would also discuss the relation of this geometry and the Monge- Ampère solution to the Ricci flow in two dimensions and the algebraicization of complex manifolds with special exhaustions. This is joint work in various parts with Raul Aguilar, Zhou Zhang and Victor Guillemin.

Optimal off-diagonal upper bounds for Bergman-Szego kernels associated to high powers of positive line bundles with smooth non-analytic metrics
Michael Christ
Abstract: Given a positive line bundle on a compact complex manifold, one can form its positive powers. A smooth Hermitian metric on the bundle gives rise to metrics on those powers. Associated to these metrics are Bergman-Szego orthogonal projections from all square integrable sections, onto the holomorphic sections. Assuming that the metric is smooth but not necessarily analytic, we formulate and prove optimal asymptotic upper bounds for the Bergman-Szego kernels away from the diagonal, as the power tends to infinity.

Partial rigidity of degenerate CR embeddings into spheres
Peter Ebenfelt
Abstract: We shall consider degenerate CR embeddings $f$ of a strictly pseudoconvex hypersurface $M\subset \mathbb{C}^{n+1}$ into a sphere ${\mathbb S}$ in a higher dimensional complex space $\mathbb{C}^{N+1}$. The degeneracy of the mapping $f$ will be characterized in terms of the ranks of the CR second fundamental form and its covariant derivatives. In 2004, the speaker, together with X. Huang and D. Zaitsev, established a rigidity result for CR embeddings $f$ into spheres in low codimensions. A key step in the proof of this result was to show that degenerate mappings are necessarily contained in a complex plane section of the target sphere (partial rigidity). In the 2004 paper, it was shown that if the total rank $d$ of the second fundamental form and all of its covariant derivatives is less than $n$ (here, $n$ is the CR dimension of $M$), then $f(M)$ is contained in a complex plane of dimension $n+d+1$. The converse of this statement is also true, as is easy to see. When the total rank $d$ exceeds $n$, it is no longer true, in general, that $f(M)$ is contained in a complex plane of dimension $n+d+1$, as can be seen by examples. In this talk, we shall show that (well, explain how) when the ranks of the second fundamental form and its covariant derivatives exceed the CR dimension $n$, then partial rigidity may still persist, but there is a "defect" $k$ that arises from the ranks exceeding $n$ such that $f(M)$ is only contained in a complex plane of dimension $n+d+k+1$. Moreover, this defect occurs in general, as is illustrated by examples.

Introduction to metaplectic FBI transforms and applications
Michael Hitrik
Abstract: The metaplectic Fourier-Bros-Iagolnitzer (FBI) transform allows one to pass from the standard Hilbert space $L^2(\mathbb{R}^n)$ to an exponentially weighted space of holomorphic functions on $\mathbb{C}^n$. Pseudodifferential operators can be transported to the FBI transform side, and in this way, one obtains some flexible and powerful techniques for their analysis, particularly in the analytic case. In my mini-course, which is intended to be elementary, I hope to be able to cover the following topics:

  • Complex symplectic geometry. Positive Lagrangian planes in the complexified phase space.

  • FBI-Bargmann transforms and Bergman kernels.

  • Pseudodifferential operators on the FBI transform side. Relation to Toeplitz operators.

  • Pseudodifferential operators with holomorphic symbols and the quantization-multiplication formula.

  • Bohr-Sommerfeld quantization conditions for non-selfadjoint differential operators with analytic coefficients in dimension one.

FBI transform and the complex Poisson kernel on a compact analytic Riemannian manifold
Gilles Lebeau (Slides)
Abstract: The aim of these lectures is to give a detailed proof of a theorem of L. Boutet de Monvel formulated in 1978 in the short article Convergence dans le domaine complexe des séries de fonctions propres (C.R.A.S. Paris, t.287, série A, (1978) pp 855--856), and also to explain how the Poisson kernel used by L. Boutet de Monvel is related to the FBI transform introduced by J. Sjöstrand in his book published in Astérisque, 95, 1982, Singularités analytiques microlocales.

Quantizing a Riemannian manifold
László Lempert
Abstract: Typically, the first step in the quantization of a physical system is finding a Hilbert space whose vectors represent the quantum states of the system. Assuming we understand the classical configuration space, a Riemannian manifold $M$, geometric quantization provides a way to construct this Hilbert space. The Kähler version of geometric quantization constructs the quantum Hilbert space as the space of square integrable holomorphic sections of a certain line bundle over the tangent bundle $TM$, which is often the same thing as holomorphic $L^2$ functions on $TM$. For this to be meaningful, one needs to choose a complex structure on $TM$ and a weight function (because $L^2$ refers to a weighted $L^2$ space). The talk will discuss my joint results with Szöke on how one can make these choices and whether the quantum Hilbert spaces corresponding to different choices are canonically isomorphic.

Holomorphic extension of fundamental solutions of elliptic linear partial differential operators of higher order with analytic coefficients
Serge Lukasiewicz
Abstract: We prove that every fundamental solution of an elliptic linear partial differential operator with analytic coefficients and simple complex characteristics in an open set $\Omega \subset \mathbb{R}^n$ can be continued at least locally as a multi-valued analytic function in $\mathbb{C}^n$ up to the complex bicharacteristic conoid. This holomorphic extension is ramified around the bicharacteristic conoid and belongs to the Nilsson class. We already proved it for operators of the second order, so the proof will be explained for operators of degree bigger than 4. This is a simplified model to study the singularities of the Bergman Kernel for strictly pseudo-convex domains with analytic boundary. We'll give also some applications in physics.

Propagation of analytic singularities for the Schrödinger equation
André Martinez
Abstract: We consider the Schrödinger equation associated perturbations of the flat Euclidian metric. For short range perturbations, we characterize the analytic wave front set of the solution to the Schrödinger equation in terms of that of the free solution in the forward nontrapping region for negative values time, and in the backward nontrapping region for positive values time. The same kind of results also holds in the long-range case (where potentials growing subquadratically at infinity are allowed), but the free quantum evolutions has to be modified in order to take into account the effects of the perturbation at infinity. This talk is based on joint works with S. Nakamura and V. Sordoni, published in Advances in Mathematics 222 (2009) 1277--1307 and in Comm. Part. Diff. Eq., Vol. 35 (2010) 2279-2309.

Ultra-analytic smoothing properties of the radially symmetric spatially homogeneous Boltzmann equation without angular cutoff
Karel Pravda-Starov
Abstract: We discuss the phase space properties of the Boltzmann collision operator and prove that the Cauchy problem associated to the radially symmetric spatially homogeneous non-cutoff Boltzmann equation with Maxwellian molecules enjoys the same Gelfand-Shilov regularizing effect as the Cauchy problem defined by the evolution equation associated to a fractional harmonic oscillator. This is a joint work with Yoshinori Morimoto (Kyoto University), Nicolas Lerner (Université Paris 6) and Chao-Jiang Xu (Université de Rouen and Wuhan University).

Spectral analysis on interior transmission eigenvalues
Luc Robbiano
Abstract: The problem of interior eigenvalues is related to a scattering problem. There are eigenvalues for a system of two elliptic problems coupled by the boundary conditions. In this presentation we prove that there exist an infinity of interior transmission eigenvalues and that the eigenfunctions associated span a dense space in $L^2$ space. The main difficulties are that the problem is not self-adjoint and it is not an elliptic problem in the sense that the solutions do not gain two more derivative than the data.

Can Toeplitz quantization solve a fully nonlinear PDE?
Yanir Rubinstein
Abstract: Several years ago Zelditch and the speaker conjectured that Toeplitz quantization can be used to produce a solution to the initial value problem for the homogeneous Monge-Ampere equation. We give some background to this problem and report on some partial results in this direction.

Quantization on Lie groups
Michael Ruzhansky
Abstract: In this talk we will present recent results on the global analysis of pseudo-differential operators on compact (and on nilpotent) Lie groups, with several applications to harmonic analysis and the theory of pseudo-differential operators.

Analytic microlocal analysis using holomorphic functions with exponential growth
Johannes Sjöstrand
Abstract: The purpose of this course is to explain some basic elements and ideas in the approach, assembled in my book Singularités analytiques microlocales (Astérisque no 95, 1982) and that have been used in later works. We plan to treat the following subjects:

  • Formal analytic symbols and pseudodifferential operators.

  • Exponentially weighted spaces of holomorphic functions ( $H_{\phi}$ spaces).

  • Stationary phase (steepest descent).

  • Pseudodifferential operators and Fourier integral operators acting on $H_{\phi}$ spaces.

  • Gaussian integral transforms and $WF_a$.

  • Analytic regularity and propagation of singularities for differential operators with analytic coefficients.

  • The WKB method in the analytic case.

  • Quasimodes and non-self-adjoint operators.

The Poisson transform on a compact real analytic manifold
Matt Stenzel
Abstract: We study the Poisson transform defined by mapping a function $f$ on a compact, real analytic manifold to the analytic continuation of $e^{-t \sqrt{\Delta}}f$ to a Grauert tube complexification of $X$. We show that the unitary part of this map has many properties in common with the Segal-Bargmann transform on a compact Lie group defined by analytically continuing the heat kernel transform, $e^{-t\Delta}f$, to the complexified Lie group. When $X$ is a homogeneous space we show that the inverse of the Poisson transform is the unitary part of the map defined by restricting a holomorphic function to the real points, $X$.

Eigenfunction nodal oscillation bounds on Riemann surfaces
John Toth
Abstract: Let $(M, g)$ be a real analytic compact Riemannian surface. Denote by $\varphi_{\lambda}$ the eigenfunctions of the Laplace operator $\Delta_g$ with $-\Delta_g \varphi_{\lambda} = \lambda^2 \varphi_{\lambda}$. Let $H$ be a real analytic curve on $M$. Under certain "goodness" assumptions on $H$, we will describe some recent results (joint with Y. Canzani) on asymptotic upper bounds for the intersection number $\sharp \{\varphi_{\lambda}^{-1}(0)\cap H\}$ as $\lambda \rightarrow \infty$.

Oscillatory modules
Boris Tsygan
Abstract: It is well known that the asymptotics of the product of pseudo-differential operators can be described by an associative algebra called a deformation quantization of the algebra of functions on the cotangent bundle. Similarly, asymptotics of Fourier integral operators and Lagrangian distributions can be described in terms of modules over deformation quantization algebras. Deformation quantization algebras can be defined for any symplectic manifold. Oscillatory modules are modules over these algebras that are endowed with an additional structure. On the one hand, they reflect more information about asymptotics; on the other hand, their category, in examples, is related to the Fukaya category of the symplectic manifold and to the microlocal category defined by Tamarkin.

Grauert tubes and nodal sets of eigenfunctions
Steve Zelditch
Abstract: Nodal (zero) sets of eigenfunctions of the Laplacian of eigenvalue $\lambda^2$ are analogues of real algebraic varieties of degree lambda, but very little is known about them. In the case of real analytic Riemannian manifolds $(M, g)$ they are know to have hypersurface volume $\simeq \lambda$. My talk is about analytic continuations of eigenfunctions to Grauert tubes and the volume and distribution of complex nodal sets. In particular, there are explicit limit distributions in the ergodic and completely integrable cases.

Use of the FBI transform in various problems in analytic microlocal analysis
Claude Zuily
Abstract: We shall rewiew in this lecture some results obtained with L. Robbiano several years ago. These results concern various problems in analytic microlocal analysis such as the uniqueness in the Cauchy problem, the smoothing effect for Schrodinger equation and the Strichartz estimate. They have in common the fact that they use the Sjöstrand theory of the FBI transform. This lecture will not contain recent results so it should be considered more as a training lecture.

Pollicott-Ruelle resonances from the microlocal point of view: sharp upper bounds and decay of correlations
Maciej Zworski
Abstract: Faure and Sjöstrand have recently shown how the ideas of Helffer and Sjöstrand developed for the study of scattering resonances of operators with analytic coefficients apply in the investigation of Pollicott-Ruelle resonances. These resonances appear as power spectra of correlations in Anosov dynamical systems. I will show how microlocal weights can be used to give sharp upper bounds on the resonance counting function (work with Datchev and Dyatlov) and expansions of correlations (work with Nonnenmacher, following earlier developments by Liverani and Tsujii).