CAREER: Higher Algebra and Symplectic Geometry

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Symplectic geometry is a geometry that efficiently encodes laws of motion dictating classical problems in physics. The last few decades have seen a burst of activity in the field thanks to the emergence of powerful algebraic tools called Fukaya categories. At the same time, the development of a new language for “spectral algebra”—an algebra that mixes traditional notions of adding and multiplying with more contemporary tools for studying shapes of arbitrarily high dimensions—has allowed us to organize sophisticated structures using algebraic intuitions. These two storylines have been highly fruitful, but have yet to cross-pollinate. This project aims to construct a long-sought-after bridge, to not only produce spectral methods for studying symplectic geometry, but to establish symplectic tools for studying spectral algebra. The project will also support numerous educational initiatives aimed at enriching and diversifying the mathematics community. These include the creation of a math podcast for and by students, a workshop for students to learn contemporary mathematical techniques of interest, and various co-curricular activities aimed at fostering communities of emerging mathematicians. The technical and collaborative heart of the project is the formalization of factorizable structures on moduli stacks of broken holomorphic objects. By showing that the usual moduli of holomorphic objects appearing in the Floer theory of Liouville sectors give rise to deformation problems encoded in factorizable sheaves on the moduli stacks, the project aims to construct spectral wrapped Fukaya categories for Liouville sectors as solutions to these deformation problems.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.