Head of program P1-0288: prof. dr. Oleksiy Kostenko

Duration: 01/01/2022 - 12/31/2027

Abstract: We shall investigate several interrelated groups of problems in complex analysis, complex geometry, complex dynamics, Fourier analysis, partial differential equations, and analysis on infinite metric graphs. In the field of complex analysis and geometry we shall study questions related to the class of Oka manifolds, their position in complex geometry with respect to other standard classes of manifolds, and we shall continue developing Oka-theoretic methods for applications in the theory of minimal surfaces, in complex contact geometry, and elsewhere. We shall look into the possibility of providing a geometric characterization of the class of Oka domains with compact complements in Cn, the union problem for Oka manifolds, Oka properties of elliptic surfaces, the connection between Oka manifolds and Campana special manifolds, their relationship with metric positivity of compact projective and Kaehler manifolds, and the study of degenerations of Euclidean fibres in holomorphic Stein fibrations. We shall investigate the Calabi-Yau problem for minimal surfaces in nonstandard geometries. We will find bounds on derivatives of conformal minimal surfaces in model domains and develop the hyperbolicity theory of domains in Rn by way of minimal surfaces. We shall develop the theory of minimal hulls of compact sets in Rn. We will look for new geometric invariants of holomorphic contact structures on complex Euclidean spaces. We will try to solve the existence problem for algebraic Kobayashi hyperbolic contact structures on C3. We will adapt the methods of Oka theory to obtain new construction techniques for holomorphic Legendrian curves. We shall investigate the possibilities offered by the Penrose theory of twistor spaces in the interplay between superminimal surfaces in self-dual Einstein 4-manifolds and complex contact 3-manifolds. We will study Cauchy-Riemann singularities of real submanifolds in complex manifolds and their normal forms in codimension 2. We will study multidimensional complex dynamics with emphasis on the analysis and classification of wandering Fatou components of holomorphic automorphisms of Euclidean spaces, endomorphisms of complex projective spaces, and local dynamics of holomorphic maps tangent to the identity. We shall develop new discretization methods for computing the inverse nonlinear Fourier transform by way of superposition of elementary model solutions. We shall investigate the role played by our new p-ellipticity condition which we have introduced in the study partial differential equations. In particular, we shall focus on the role of p-ellipticity in the development of dimensionless estimates for bilinear and trilinear embeddings. We shall develop new analytic methods in the study of infinite metric graphs - the quantum graphs - with emphasis on non-locally finite graphs.

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