Research project is (co) funded by the Slovenian Research Agency.

**UL Member:** Faculty of Mathematics and Physics

**Code:** P1-0291

**Project:** Analysis and geometry

**Period:** 1. 1. 2015 - 31. 12. 2027

**Range per year:** 0,76 FTE **category:** C

**Head:** Franc Forstnerič

**Research activity**: Natural sciences and mathematics

**Citations for bibliographic records**

**Related projects:** J1-9104, J1-1690

**Project description:**

We are proposing research on contemporary topics in complex analysis and geometry, Oka-Grauert-Gromov theory, Cauchy-Riemann geometry, pluripotential theory, minimal surfaces, harmonic and Fourier analysis, partial differential equations and relations, theory of integrable systems, mathematical physics, invariants of Lie grupoids and algebroids.

The proposal is to a certain extent a continuation of our past and current research on topics where we gave several major contributions as is evident from our publications in leading mathematical journals during 2009-14. In addition, the proposal contains several new directions of investigation. We mention in particular new applications of the Oka-Grauert-Gromov theory, a subject that was brought in a coherent form during the last decade and is summarized in the monograph F.Forstnerič, Stein Manifolds and Holomorphic Mappings, Springer-Verlag (2011). It recently became clear that our methods are very useful in the classical theory of minimal surfaces, holomorphic null curves, and other objects represented by directed immersions of Riemann surfaces. This opens a new dimension in the theory of first order holomorphic partial differential relations. We expect major new applications of our method of gluing holomorphic sprays and of exposing boundary points. We shall investigate the connections between the theory of minimal surfaces and of null-plurisubharmonic functions in the general framework of Harvey-Lawson theory. Globevnik will continue his investigations concerning the extendibility of holomorphic functions, a topic on which he is a leading expert. We expect to obtain new results on boundary differential relations for holomorphic function on finitely connected planar domains. With the addition of U.Kuzman (PhD 2013) our research is expanding into almost complex geometry. O.Dragičević will focus on sharp estimates for spectral multipliers and sharp dimension-free bilinear estimates for operators in divergence form with complex coefficients. Slapar will work on the classification of normal forms of complex points in real submanifolds, an essential ingredient for understanding local polynomial hulls, and on the topological characterization of q-convex manifolds. P. Saksida will apply his results on nonlinear Fourier transform to the analysis of the sine-Gordon equation and to the nonlinear Schrödinger equation.