Complex analysis and convexity

Research Project, 2013 – 2016

The importance of convexity in complex analysis has been realized for a very long time. This project concerns, to a large part, the exploitation of one aspect of convexity whose relation to complex analysis does not seem to have been noted before: questions related to the Brunn-Minkowski inequality. It turns out that this inequality has very natural counterparts in complex analysis, that give useful information in a variety of areas. While the classical Brunn-Minkowski inequality deals with the volumes of (convex) sets, its complex counterpart deals with L^2-norms of holomorphic functions. It has applications in seemingly unrelated areas like interpolation theory, moduli problems for classes of complex manifolds, extremal Kahler metrics and algebraic geometry. Another part of the project concerns the adaptation of a standard tool from complex analysis to convexity theory: the theory of closed positive forms and currents. These notions in the complex setting form a very useful tool to study complex varieties; it is hoped that their real counterparts will play a role in the study of tropical varieties, graphs and perhaps (quasi)crystals.

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