A. Stein manifolds and holomorphic mappings

We shall investigate open problems concerning the existence, approksimation and interpolation properties of holomorphic mappings of Stein manifolds into general complex manifolds. Main research topics will include:

(1) The Oka-Grauert principle and holomorphic fleksibility of complex manifolds

(2) Locally biholomorphic self-mappings and automorphisms of complex Euclidean spaces.

(3) Holomorphic immersions, submersions and foliations.

(4) Proper holomorphic mappings and embeddings.

B. Analysis and geometry of pseudoholomophic curves in almost complex manifolds. Research will focus on the following open problems:

(I) Let E be a compact totally real submanifold in an almost complex manifold (M,J). When does there exist a compact Riemann surfaces with boundary, holomorphically immersed in M and with boundary contained in E? This is one of the most basic and fundamental questions of complex analysis. Traditionally this question has been studied in connection to problems of polynomial and rational convexity; since the of M. Gromov (1986) it assumed an even more important role due to its relations to problems in symplectic topology and nonlinear analysis.

(II) Although the local theory of (small) pseudoholomorphic discs is fairly well understood, there are important open problems in the corresponding global theory. For example, does the set of all pseudoholomorphic discs (of certain smoothness class) in an almost complex manifold admit the structure of a Banach manifold? Is the same true for other pseudoholomorphic curves with smooth boundaries? In the integrable case an affirmative answer was given recently by Forstnerič (Manifolds of holomorphic mappings from strongly pseudoconvex domains, Asian J. Math., 11 (2007), 113-126), but the problem remains open in the nonintegrable case.

(III) Y. Eliashberg proved in 1990 that every almost complex structure J on a smooth manifold X of dimension 2n different from 4, which is a topological handlebody with handles of index at most n, is homotopic to an integrable Stein structure J' on X. The situation in dimension n=2 (for Stein surfaces) was clarified by R. Gompf (Ann. of Math. (2) 148 (1998), 619-693). Recently Forstnerič and Slapar showed that a Stein structure J' on X, homotopic to J, can be chosen such that there exists a J'-holomorphic map from X to another complex manifold Y in a given homotopy class of maps (Stein structures and holomorphic mappings, Math. Z. 256 (2007), 615-646). When the initial structure J on X is already integrable Stein, it would be natural to expect that the above conclusion holds by chosing a homotopy from J to J' within the class of integrable Stein structures on X, and without changing the underlying smooth structure on X when n=2. To answer this problem one needs to gain a better understanding of the change of the contact structure on the boundary of a strongly pseudoconvex domain when attaching to it a handle with totally real core along a legendrian submanifold. We shall try to clarify this important issue.

C. Selected problems in harmonic analysis (with Oliver Dragičević)

(1) Improvement of constants in estimates of norms of iterates of the Ahlfors-Beurling operator.

(2) Investigation of Lp-estimates for spectral multipliers, arising from the Hermite operator on spaces of arbitrary dimension, with emphasis on absolute constants independent both of the index p and of the dimension.

(3) The use of the Bellman function method for the operators from the Laguerreov semigroup.

(4) Investigation of elliptic differencial operators and especially of Schrödingerjevih operators.

(5) Investigations of powers of the Ahlfors-Beurling operator.

D. Hamiltonian systems with high degree of symmetry (with Pavle Saksida)

We shall investigate four areas of mathematical-physical problems concerning modified hamiltonian dynamical systems:

(1) Superposition rules for nonlinear integrable partial differencialnih equations.

(2) The influence of symmetry on transport phenomena in classical and quantum statistical mechanics. mehaniki.

(3) The theory of modified hamiltonian systems with high degree of symmetry.

(4) Algebraic geometry of singular loci of integrable systems.