Speaker: David Barrett

Title: Leray transforms and duality for domains in projective space

Abstract: This talk will explore the interplay between the following three topics: (A) the Mobius-invariant geometry of real hypersurfaces in complex projective space; (B) duality results relating functions on a domain in projective space to functions on the dual complement; (C) Leray's integral transform for linearly convex domains.

Speaker: Antonio Bove

Title: Sums of squares of complex vector fields: variations on Kohn's operator

Abstract: Recent work with D. S. Tartakoff is presented. It concerns a model of sum of squares of complex vector fields and a sharp Gevrey hypoellipticity threshold is obtained. The discussion is related to previous work with Derridj, Kohn and Tartakoff.

Speaker: Makhlouf Derridj

Title: On the subellipticity of some hypoelliptic quasihomogeneous systems of complex vector fields (joint work with B. Helffer)

Speaker: Peter Ebenfelt

Title: Superrigidity of CR mappings into nondegenerate quadrics

Speaker: Miroslav Englis

Title: Generalized Toeplitz operators and weighted Bergman kernels

Abstract: We describe the boundary singularity of weighted Bergman kernels on smoothly bounded strictly pseudoconvex domains with respect to weights which are fractional powers of a defining function, and, more generally, of the reproducing kernels of Sobolev spaces of holomorphic functions of any real order. This generalizes the classical result of Fefferman for the unweighted Bergman kernel. An extension to weights with logarithmic singularities at the boundary, and some other related results, will also be outlined.

Speaker: Charles Fefferman

Title: Interpolation of functions

Abstract: The talk explains how to compute a function F in C^m(R^n) taking given values at N given points, with C^m norm within a factor of C of least possible, where C depends only on m and n. The computation is efficient -- it takes O(N log N) operations -- but the constant C is huge. Some ideas to reduce C to 2 are discussed. (Joint work with Bo'az Klartag.)

Speaker: Gerald Folland

Title: Compact Heisenberg manifolds

Abstract: Let M be the quotient of the Heisenberg group by a discrete co-compact subgroup. We study the following two problems: 1. What are the eigenvalues and eigenforms of the Kohn Laplacian on M? 2. How can M be realized as the boundary of a bounded strongly pseudoconvex domain? The answer to the second question reveals an intimate connection with the theory of Abelian varieties.

Speaker: Franc Forstneric

Title: Bordered Riemann surfaces in C^2

Abstract: (Joint work with Erlend Fornass Wold) We prove that the interior of any compact complex curve with smooth boundary in C^2 admits a proper holomorphic embedding into C^2. This implies new results on embedding bordered Riemann surfaces properly into C^2.

Speaker: Peter Greiner

Title: On subelliptic PDE's and their subRiemannian geometry

Abstract: I shall discuss nonholonomic geodesics,real and complex, in Carnot-Caratheodory geometry which play a significant role in the construction of inverse kernels, heat kernels, funda- mental solutions, etc. for subelliptic Laplacians. The examples that I study via a Hamiltonian formalism may be reduced to questions concerning the behaviour of abelian integrals.

Speaker: Philip Harrington

Title: Bounded Plurisubharmonic Exhaustion Functions on Non-smooth Pseudoconvex Domains

Absract: Bounded plurisubharmonic exhaustion functions have proven to be useful tools in the study of the $\bar\partial$-Neumann operator on non-smooth pseudoconvex domains. We will show that in Stein manifolds, H\"older continuous plurisubharmonic exhaustion functions always exist for bounded pseudoconvex domains with Lipschitz boundaries, which implies that the $\bar\partial$-Neumann operator is bounded in $W^s$ for some $s>0$. If the boundary is $C^1$, a weaker condition can be used to show that for every $0 < s <\frac{1}{2}$ there is a weighted $\bar\partial$-Neumann operator which is bounded in $W^s$, thus implying $W^s$ solvability for the Cauchy-Riemann operator.

Speaker: Xiaojun Huang

Title: On the equivalence problem for Bishop surfaces with a vanishing Bishop invariant

Abstract: A hypersurface in $(z,w)\in {\Bbb C}^2$, which is locally defined by an equation of the form: $Im(w)=|z|^2+o(|z|^2)$, is called a strongly pseudoconvex hypersurface. A real surface in $(z,w)\in {\Bbb C}^2$, which is locally defined by an equation of the form: $w=|z|^2+o(|z|^2)$, is called a Bishop surface with a vanishing Bishop invariant. In this talk, I will discuss a recent joint paper with Wanke Yin. In this work, we give a solution to the equivalence problem for Bishop surfaces with a vanishing Bishop invariant. As a consequence, we answer, in the negative, a problem that Moser asked in 1985 after his work with Webster. This will be done in two main steps: We first derive the formal normal form for such surfaces. We then show that two real analytic Bishop surfaces with a vanishing Bishop invariant are holomorphically equivalent if and only if they have the same formal normal form (up to a trivial rotation). Our normal form is constructed by an induction procedure through a completely new weighting system from what is used in the literature. Our convergence proof is done through a new hyperbolic geometry associated with the surface.

Speaker: Martin Kolar

Title: Chern-Moser operators and the Catlin multitype

Abstract: We use a generalization of the Chern - Moser operator to study local equivalence and symmetries of Levi degenerate hypersurfaces. In dimension two, such operators are well defined for hypersurfaces of finite type. In higher dimensions, this role is taken by finite Catlin multitype. Some basic properties of the generalized Chern-Moser operators will be derived.

Speaker: Svatopluk Krysl

Title: Symplectic spinors and symplectic Dirac operators

Abstract: Symplectic spinors are proper symplectic analogues of the orthogonal spinors. In the symplectic case, the role of the spin group plays the so called metaplectic group. While in the case of the orthogonal group, the spinor representation is finite dimensional, in the symplectic case, the symplectic spinors form an infinite dimensional Hilbert space. The symplectic Dirac operator as well as its higher spin analogues will be introduced. For it, we introduce a distinguished class of infinite dimensional irreducible representations of the metaplectic group. We shall derive a theorem on an explicit relation of the spectrum of the symplectic Dirac operator to the spectrum of the symplectic Rarita-Schwinger operator. Here, a technical notion of symplectic Killing spinor appears. We shall derive some consequences of the existence of a non-zero symplectic Killing spinor for the underlying symplectic manifold. (In the orthogonal case, Killing spinors are related to super-symmetries of the studied theory and its existence automatically implies that the underlying space is Einstein.

Speaker: Bernard Lamel

Title: Equivalence in the infinite type setting

Speaker: Jiri Lebl

Title: Singular Levi-flat hypersurfaces in projective space

Abstract: It is well known that there are no nonsingular Levi flat hypersurfaces in projective space of high enough dimension. There are however plenty of singular real analytic Levi-flat hypersurfaces. As Levi-flat hypersurfaces share many properties with complex varieties, it is natural to ask if an analogue of Chow's theorem holds, i.e. are all Levi-flat hypersurfaces in projective space algebraic. While this is not true in general, we can give a positive answer if the Levi-flat hypersurface is locally defined using a meromorphic function. I will give several examples demonstrating some subtleties involved and will also give some results relating degeneracy (in the Segre variety sense) to the size of the singularity in the algebraic case.

Speaker: Laszlo Lempert

Title: Holomorphic Banach bundles over compact complex manifolds

Abstract: Motivated by work of I. Gohberg and J. Leiterer starting in the 1960s, I consider holomorphic Banach bundles over (finite dimensional) compact complex manifolds, and introduce the notion of two such bundles being compact perturbations of one another. The main result is a finiteness theorem; an easily stated special case says that if the cohomology groups of one bundle are finite dimensional, then the same holds for its compact perturbations.

Speaker: Jeff McNeal

Title: Non-holomorphic projections and biholomorphic maps.

Abstract: On a smoothly bounded domain in C^n, D, we can consider the space H(D; T) of "T-twisted" holomorphic functions on D as a subspace of the weighted square-integrable functions on D, L^2(D:W). I'll discuss how biholomorphic mappings of D are related to the operator projecting L^2(D;W) onto H(D;T), for certain choices of T and W. The connections discussed are inspired by the work of Bell and Ligocka from the late 70s. Extension of biholomorphic mappings of D follow from regularity results about the twisted d-bar Neumann problem associated to T and W.

Speaker: Andreea Nicoara

Title: The Non-Noetherianity of the Denjoy-Carleman Quasianalytic Rings

Abstract: The Denjoy-Carleman quasianalytic classes are subrings of the ring of smooth functions consisting of functions with Taylor expansions that do not necessarily converge but on which the Taylor morphism is injective, i.e. a function with zero derivatives up to infinite order is the zero function. It was proven by Childress in 1976 that these subrings do not have the Weierstrass Division Property, so the standard argument used for holomorphic functions and real-analytic functions to show these two are Noetherian rings cannot be used here. It is thus an open problem as to whether the Denjoy-Carleman quasianalytic classes are Noetherian rings or not. I will discuss very recent work on this problem using methods from model theory (a branch of logic) that I have done jointly with Liat Kessler (MIT).

Speaker: Louis Nirenberg

Title: Remarks on fully nonlinear elliptic partial differential equatons.

Abstract: Some results are presented in connection with establishing properties of solutions.The talk will be expository.

Speaker: Raphael Ponge

Title: A new hypoelliptic operator on almost CR manifolfds

Abstract: The aim of the talk is to present the construction, out of the Kohn-Rossi complex, of a new hypoelliptic operator on almost CR manifolds equipped with a real structure. The operator acts on all $(p,q)$-forms, but when restricted to $(p,0)$-forms and $(p,n)$-forms it is a sum of squares up to sign factor and lower order terms. Therefore, only the Hormander's bracket condition is needed to have hypoellipticity on those forms.

Speaker: Gerd Schmalz

Title: Hypersurfaces in C^2 with a 2-parametric family of automorphisms

Abstract: A combination of M. Kolar's recent results on automorphisms of hypersurfaces of finite type and S. Lie's classical method is used to classify the hypersurfaces of finite type in $\mathbb C^2$ with a two-parametric family automorphisms. This is joint work with V. Ezhov and M. Kolar.

Speaker: Yum-Tong Siu

Title: Impact of Kohn's work on recent developments in algebraic geometry

Abstract: J.J. Kohn introduced in his 1979 Acta paper the notion of multiplier ideal sheaves. This talk will survey the developments in algebraic geometry from applications of multiplier ideal sheaves.

Speaker: Jan Slovak

Title: Inclusions of parabolic geometries

Abstract: (Joint work with Boris Doubrov) All Fefferman type constructions for parabolic geometries where the underlying manifolds do not change are studied. Recently, two such constructions appeared. Indeed, conformal structures naturally associated with non-degenerate rank 2 vector distributions on 5-dimensional manifolds were studied by Nurowski and those associated with non-degenerate rank 3 distributions on 6-dimensional manifolds by Bryant. In both cases there are natural parabolic geometries associated with these distributions, which serve as an intermediate structure between the distribution and the conformal geometry. Using classical results by A. Onischchik, we classify all possibilities of such inclusions of parabolic geometries. Apart of known examples, a new series of embeddings of 2--graded $B_\ell$ geometries into 1--graded $D_{\ell+1}$ geometries has been detected. These geometries correspond to the generic $\ell$--dimensional distributions of codimension $\frac12\ell(\ell-1)$, and the Bryant's example fits into this series with $\ell=3$.

Speaker: Petr Somberg

Title: Invariant prolongation and commutative modification of Cartan connection for CR geometry

Abstract: The Cartan connection is playing an important role in parabolic geometries (the most typical examples being the conformal and CR geometries). The normal Cartan connection on the corresponding principal fiber bundle induces the associated (normal) covariant derivative on the corresponding tractor bundles. A strong motivation for an alternative normalization condition for covariant derivatives on tractor bundles is coming from a study of the prolongation of a certain class of overdetermined systems of PDE's. In the lecture, a new normalization condition is introduced for a general parabolic geometries and, in particular, special cases (including the CR geometry) are discussed.

Speaker: Vladimir Soucek

Title: Continuous families of invariant differential operators in conformal and CR geometries.

Abstract: A construction of invariant differential operators in conformal and CR geometries (or more generaly, in parabolic geometries) are quite involved in some cases. In particular, a construction of invariant differential operators P(n) having the n-th power of the laplacian (resp. of the sublaplacian) as its symbol were discussed recently from several points of view and their relations to the Q-curvature Q(n) was explained. In conformal geometry, A. Juhl has introduced recently a family of invariant differential operators mapping sections of appropriate line bundles on a Riemannian manifold X to sections of line bundles on its boundary M depending on a continuous(!) parameter and he, together with R. Graham, has shown the role of the family in relation to the operators P(n) and the curvature Q(n). In the lecture, we shall show how to construct an analogue of the Juhl family of invariant differential operators in CR geometry. The main tool will be homomorphisms of (generalized) Verma modules and their branching rules (which are of independent interest).

Speaker: Emil Straube

Title: Some questions concerning regularity properties of the d-bar-Neumann operator.

Abstract: I will discuss/motivate the following three questions. (1) Is there a proof that subelliptic estimates imply the existence of bounded plurisubharmonic functions whose Hessians blow up like a power of the reciprocal of the boundary distance that does not proceed via finite type of the boundary? (2) On a domain where the Levi form has at most one degenerate eigenvalue, compactness of the d-bar-Neumann operator implies hypoellipticity of d-bar in the space of locally square integrable functions. This implication should not hinge on the Levi form having only one degenerate eigenvalue. (3) What are the implications of regularity properties of the d-bar-Neumann operator for the existence of Stein neighborhood bases for the closure of the domain?

Speaker: Brian Street

Title: The $\square_b$ heat equation on pseudoconvex manifolds of finite type via the wave equation

Abstract: In 2001, Nagel and Stein studied the $\square_b$ heat operator, $e^{-t\square_b}$, on pseudoconvex domains of finite type in $\mathbb{C}^2$. They demonstrated that the Schwartz kernel of $e^{-t\square_b}$ satisfies estimates analogous to the off diagonal estimates of the classical heat operator, while $e^{-t\square_b}-\pi$ satisfies on diagonal estimates (where $\pi$ denotes the Szeg\"o projection). In this talk, we show that standard methods of studying the classical heat equation using the finite propagation speed of the wave equation can be adapted to this situation, providing a simple proof of Nagel and Stein's results. Because of the simplicity of this proof, it is quite easy to adapt it to other, similar, situations. In addition, we mention that the same methods can be used to study multipliers $m(\square_b)$.

Speaker: Dror Varolin

Title: Interpolation and sampling from singular subvarieties in C^n

Abstract: This is joint work with Stanislav Ostrovsky. A complex subvariety W of C^n is said to be interpolating if any weighted L^2 holomorphic function on W can be extended to a weight L^2 holomorphic function on C^n, i.e., the restriction to W is surjective. W is said to be sampling if restriction to W is injective. In joint work with Ortega-Cerda and Schuster, I considered the case where W is a smooth hypersurface. In this talk I will discuss the case where W is a possibly singular hypersurface.

Speaker: Stephen Yau

Title: Explicit Construction of Moduli Space for Complete Reinhardt Domains via Bergman Functions

Abstract: We introduce higher order Bergman functions for complete Reinhardt domains in a variety with isolated singularities. These Bergman functions are invariant under biholomorphic maps. We use Bergman functions to determine all the biholomorphic maps between two such domains. We construct an infinite family of numerical invariants from the Bergman functions for such domains in A_n-variety {(x, y, z) ^ C^3 : xy = z^(n+1)}. These infinite family of numerical invariants are actually a complete set of invariants for either the set of all strictly pseudoconvex domains or the set of all pseudoconvex domains with real analytic boundaries in A_n-variety. In particular the moduli space of these domains in A_n variety is constructed explicitly as the image of this complete family of numerical invariants. Recall that A_n-variety is the quotient of cyclic group of order n + 1 on C^2. We prove that the moduli space of complete Reinhardt domains in A_n variety coincides with the moduli space of the corresponding complete Reinhardt domains in C^2. Since our complete family of numerical invariants are explicitly computable, we have solved the biholomorphically equivalent problem for large family of domains in C^n.

Speaker: Vojtech Zadnik

Title: Remarks on chains in CR geometry

Abstract: We remind the notion of chains for CR structures of hypersurface type. Both the CR structure and the system of chains can be described as Cartan geometries and it turns out they can be related in a very direct way, i.e. without prolongation. We outline the principles of that construction and enjoy some applications. In particular, it follows that the CR structure is actually determined by the family of chains. Further, in the model case, there is a rich additional structure on the space of all chains, which we also plan to investigate during the lecture.

Speaker: Dimitri Zaitsev

Title: Chern-Moser type normal forms for almost CR structures

Abstract: We propose constructions extending the Chern-Moser normal form to non-integrable Levi-nondegenerate (hypersurface type) almost CR structures. One of them translates the Chern-Moser normalization into pure intrinsic setting, whereas the other directly extends the (extrinsic) Chern-Moser normal form by allowing non-CR embeddings that are in some sense "maximally CR". One of the main differences with the classical integrable case is the presence of the non-integrability tensor at the same order as the Levi form, making impossible a good quadric approximation - a key tool in the Chern-Moser theory.