Geometric Methods in Complex Analysis

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The proposal deals with geometric methods in complex analysis -- the study of functions of complex variables. Development of this field is one of the great contributions of modern mathematics. In fact, many different directions of research in modern mathematics stem from problems in complex analysis. Conversely, to study complex analysis one needs many tools from other areas - such as differential equations, topology, functional analysis, etc. In addition, complex analysis has numerous applications to other sciences. It is this inherent interconnection with other branches of mathematics, physics, and computer science that has made complex analysis one of the leading research areas in mathematics for many decades. More specifically, the proposed research is concerned with the study of the properties of real submanifolds in complex Euclidean spaces, or, more generally, complex manifolds. Their significance comes from the fact that such manifolds are models for many objects and processes that appear in applications in other areas of mathematics, such as symplectic topology and contact geometry, but also in other sciences such as theoretical physics and electrical engineering. For example, approximation of smooth complex-valued functions by simple computable functions, such as polynomials, can be understood through the study of various convexity problems of their graphs - natural example of real submanifolds. On the other hand, real submanifolds often capture valuable geometric properties of the ambient manifolds, and thus they can play an important role in understanding the geometry and topology of complex spaces. The proposal consists of several concrete problems that lie at the heart of this topic. The author has published a number of papers dedicated to the subject. The new directions for further research outlined in the proposal are based on the original ideas and innovative techniques that the author has developed in these papers. In general terms, successful realization of the proposal will lead to new insights and will play a role in the future development of the subject.