Higher Algebraic Structures in Symplectic Geometry and Applications

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Symplectic geometry is a branch of modern mathematics lying at the crossroads of many other subfields including differential topology, algebraic geometry, dynamical systems, and theoretical physics. Its equations describe many natural phenomena, including the motions of classical mechanics, yet its mathematical structures are surprisingly rich and subtle, necessitating a wide array of new tools. This project seeks to construct new mathematical objects called "symplectic invariants", and apply them to several geometric problems to help understand when symplectic dynamics can distort one shape into another. This project will also contribute to dissemination and education by organizing large-scale seminars and workshops, mentoring graduate students, and crafting new ways to visualize abstract geometry via computer visualizations and 3D printing.More specifically, this project will focus on exploiting higher algebraic structures in Floer theory and symplectic field theory. These structures have been known to experts for some time, but only recently have been shown to play a powerful role in embedding obstructions and other geometric problems. The PI will continue their investigation of "higher symplectic capacities", developing new algorithms to facilitate computations and exploring the resulting enumerative combinatorics. This project will also expand the foundations and purview of these higher algebraic structures, including their role in Weinstein geometry and connections with Fukaya categories and homological mirror symmetry. Furthermore, the PI will continue to develop a framework for studying numerical simulations in symplectic geometry, with an emphasis on both theoretical ideas and practical computer software.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.