Geometric Function Theory in Several Complex Variables

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The main part of the proposal by the P.I.is to continue his work (in collaboration with Prof. S. Ivashkovich) on almost complex manifolds. Almost complex manifolds are not an un-motivated generalization of complex manifolds. After the work of Gromov, their relevance became unquestionable, with deep applications to geometry and in particular to symplectic geometry. On one hand they allow a remarkable flexibility that complex manifolds lack, on the other hand they sometimes provide a much clearer approach to problems in several complex variables. Recent work in the area, in particular by Ivashkovich and the P.I., and by Nefton Pali, shows that truly basic questions still need investigation. The P.I. intends to work on analytic questions, e.g. the Kobayashi-Royden metric or the construction of pluri-subharmonic functions by the method of Poletsky. The geometric role of pluri-subharmonic functions on almost complex manifolds has been well illustrated recently in the works of Gaussier-Sukhov and of Ivashkovich and the P.I.Related to the above there are several questions that the P.I. intends to work on, especially the Cartan Lemma for holomorphic matrices with uniform bounds. The problem was so far solved only recently and only in complex dimension 1 by the P.I. and Prof B. Berndtsson, using a refined maximum principle. This proposal is on the Complex Analysis side but at the confluence of several active areas of Mathematics. Some geometry is involved, differential inequalities play an important role. The field in which the P.I. works is mainly the theory of several complex variables. Many ``real problems'' receive their best explanation by using complex methods, even if the initial problem does not seem to call for using complex numbers. It goes back to the origin, for solving algebraic equations, but it continued for example in the theory of Fourier integrals (that are taught in engineering classes), and it continues in the most recent developments. Complex analysis has deep and rich connections to the theory of ordinary or partial differential equations, which allows the study of physical phenomena (at the macroscopic level). It applies to the study of dynamical systems. Complex Analysis has also various links to Geometry which is more directed to a qualitative (rather than quantitative) description of natural phenomena. It is in this meaningful context that the P.I. proposes to work.