Schedule/Abstracts
The talks will take place at the Garrett Theological Seminary, Room 205, 2121 Sheridan Road, Evanston, IL 60201
Tentative schedule of talks:
 Monday:
 Tuesday:
 Wednesday:
 Thursday:
 Friday:
Titles and Abstracts
Analytic Toeplitz and pseudodifferential operators; analytic wavefront
Louis Boutet de Monvel
Abstract: I will review the definition of analytic pseudodifferential operators or Toeplitz
operators; also various 'geometric' ways of seeing the analytic wavefront set.
CartanRemez 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 MongeAmpère Equation
Daniel Burns
Abstract: Some time ago, Boutet de Monvel proved a PaleyWiener 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 MongeAmpère 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 offdiagonal upper bounds for BergmanSzego kernels
associated to high powers of positive line bundles with smooth nonanalytic 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 BergmanSzego 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
BergmanSzego 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 FourierBrosIagolnitzer (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 minicourse,
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.

FBIBargmann transforms and Bergman kernels.

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

Pseudodifferential operators with holomorphic symbols and the quantizationmultiplication formula.

BohrSommerfeld quantization conditions for nonselfadjoint 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 855856),
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 multivalued
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 pseudoconvex 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 longrange 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) 12771307 and in Comm. Part. Diff. Eq., Vol. 35 (2010) 22792309.
Ultraanalytic smoothing properties of the radially symmetric spatially homogeneous Boltzmann equation without angular cutoff
Karel PravdaStarov
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 noncutoff Boltzmann equation with Maxwellian molecules enjoys the same GelfandShilov 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 ChaoJiang 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 selfadjoint 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 MongeAmpere 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 pseudodifferential operators on compact
(and on nilpotent) Lie groups, with several applications to
harmonic analysis and the theory of pseudodifferential
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 nonselfadjoint 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
SegalBargmann 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 pseudodifferential 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.
PollicottRuelle 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 PollicottRuelle 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).
