Complex Analysis, Operator Theory, and Applications to Mathematical Physics

Followup Fall 2006

Organizers: Friedrich Haslinger (University of Vienna)

Emil Straube (Texas A&M University)

Harald Upmeier (University of Marburg)

In this program, the interplay between multivariable complex analysis, operator theory and quantization is explored in two related but different situations.

Part I is devoted to regularity properties of the d-bar-Neumann operator on (weakly) pseudoconvex domains, with special emphasis on compactness. Compactness has long been known to have implications for the operator theory of certain operators naturally associated with a domain, such as Toeplitz operators. Recently, interesting connections of these regularity properties to paramagnetism and diamagnetism in certain Schrödinger operators have been discovered. These questions involve a fascinating mixture of potential theory, complex analysis, partial differential equations, and operator theory.

In part II, the class of domains studied is characterized by a (holomorphic) group action. Since the group involved is non-commutative, one obtains "non-commutative'' Hardy and Bergman spaces of holomorphic functions and, accordingly, the algebraic behavior of Toeplitz operators on these Hilbert spaces is more complicated (commutators are not compact). Nevertheless, this "Toeplitz-Berezin'' quantization, e.g. on bounded symmetric domains, has already found applications to the quantization problem, even in the infinite dimensional setting of quantum field theory.

The interaction between the two themes is expected to be very fruitful. We propose to bring together at the Schrödinger Institute, for various lengths of time, experts from several complex variables, partial differential equations, Schrödinger operators, and operator theory.

The following scientists have already confirmed that they plan to participate in the program:

P. Charpentier (Universite´ Bordeaux) Nov. 8, 15.15, Boltzmann Lecture Hall Extremal basis and local estimates for the Szegö projection	Nov.5 - Nov.11
K. Diederich (Universität Wuppertal) Nov. 29, 16.15, Math. Koll., Dep.of Math. Real and complex analyticity	Nov.27-Dec.16
R. Dwilewicz (University of Missouri, Rolla) Nov.15, 15.15, Boltzmann Lecture Hall Global holomorphic approximations of Cauchy-Riemann functions	Nov.12-Nov.17
D. Ehsani (Penn State) Nov. 9, 11.15, Boltzmann Lecture Hall The d-bar Neumann operator and cancellation of singularities on the bi-disk	Nov.8 - Nov.12
F. Haslinger (Universität Wien, ESI) Nov. 14, 15.15, Boltzmann Lecture Hall Compactness of the solution operator to d-bar in weighted L^2 - spaces	Oct.25 -Nov.17
B. Helffer (Orsay) Nov. 15, 11.15, Boltzmann Lecture Hall Subelliptic estimates for some systems of complex vector fields: quasihomogeneous case	Nov.10-Nov.15
M. Kolar (University of Brno) Nov. 14, 17.15, Boltzmann Lecture Hall Local equivalence of Levi degenerate hypersurfaces.	Nov.10-Nov.15
S. Kolodziej (University of Krakow), Nov. 8, 16.15, Boltzmann Lecture Hall Hölder continuity of solutions to the complex Monge-Amp`ere equation with the right hand side in Lp	Nov.7- Nov. 12
B. Lamel (Universität Wien) Nov. 14, 16.15, Boltzmann Lecture Hall Jet determination of finite mappings	Oct.25- Nov.17
D. Popovici (Universite´ Toulouse) Nov. 9, 14.15, Boltzmann Lecture Hall Singular Morse Inequalities	Nov.5 - Nov.12
A. Raich (Texas A&M) Nov. 15, 14.15, Boltzmann Lecture Hall Pointwise heat kernel estimates with applications to complex analysis	Nov.12-Nov.17
S. Sahutoglu (University of Michigan, Ann Arbor) Nov. 15, 16.15, Boltzmann L.H. On a Sufficient Condition for the Existence of Stein Neighborhood Bases	Nov.12-Nov.18
G. Schneider (Universität Wien) Nov.15, 10.15, Boltzmann Lecture Hall Hankel operators with anti-holomorphic symbols on generalized Fock-spaces	Nov.12-Nov.18
A. Strasburger (Warsaw University), Nov. 9, 15.00, Boltzmann Lecture Hall Some properties of differential operators of gradient type associated with spherical harmonics	Nov.5 - Nov.11