Problems in Complex Analysis and Complex Geometry

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This project addresses some fundamental problems that lie at the interface of complex geometry, partial differential equations, and mathematical physics. The unifying theme is complex structures and moduli defined by solutions of partial differential equations, their geometric obstructions, and their singularities. The methods build on the theory of partial differential equations, but the central component will be the development of new techniques and ideas from complex analysis.Complex structures and moduli first appeared in pure mathematics, but they have since revealed themselves to be at the center of a wide range of phenomena, from applied mathematics to both statistical and high-energy physics. In fact, some of the problems in the present project arise directly from questions about critical phenomena and supersymmetry. Progress in any part of the project should have wide repercussions, both in mathematics and in related fields. The principal investigator has trained a steady stream of undergraduates, graduate students, and postdoctoral researchers in the past, and the project will be crucial in allowing him to continue his efforts in this direction.