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