Lefschetz Fibrations, Their Noncommutative Counterparts, and Formal Groups

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The goal of this project is to integrate recent insights, originating from theoretical physics, into classical areas of geometry. One can consider these purely mathematical questions as a "theoretical laboratory", which allows one to quickly explore the structure of cutting-edge ideas, abstracting away the full complexity of their physical origin. Besides the expected scientific benefit, the project contains specific parts designed for graduate and undergraduate research. Undergraduate research is an increasingly important part of the training of next-generation scientists and mathematicians. A particular effort has been made in this project to find issues of current relevance which allow students to take charge, under suitable mentorship.In Kontsevich's formulation of the string theory notion of mirror symmetry, this becomes a relation between symplectic geometry and algebraic geometry, formulated in a common algebraic language of noncommutative geometry. The project intends to further develop noncommutative geometry thinking in symplectic geometry. This is useful as an organizing principle for the information arising from pseudo-holomorphic curve methods. In the framework of the project, it will lead to new methods for understanding and computing that information. One key question under consideration is the dependence of categorical structures on the Novikov parameter. Mirror symmetry also has an arithmetic aspect. That motivates another part of the project, which is to bring structures common in number theory, such as formal groups, to bear on symplectic geometry.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.