Quantitative Symplectic Geometry in Higher Dimensions

University Of Southern California, Los Angeles CA

Investigators

Abstract

Abstract: Symplectic geometry is a branch of modern mathematics which lies at the interface of many areas in modern geometry and theoretical physics. Among other things, it can be used to place subtle and surprising restrictions on the dynamics of a broad class of systems (think motions of celestial bodies or the molecules in a gas). Understanding the scope of these restrictions and how to effectively utilize them in naturally arising physical systems is an exciting active area of research. In this project, the PI will develop new tools and paradigms to study the quantitative side of symplectic geometry. The PI will also initiate the study of "digital symplectic geometry", which aims to link deep geometric tools with numerical simulations and methods from machine learning. In addition, the PI will also be actively involved in various community-building and mentorship activities, including organizing intensive learning retreats for the wider Southern California geometry and topology community and mentoring undergraduates in research projects at the interface of pure and applied mathematics. In more detail, a central theme is to develop filtered refinements and higher extensions of powerful emerging tools from pseudoholomorphic curve theory (symplectic field theory, Floer homology, Gromov–Witten invariants, Fukaya categories, ...) in order to systematically investigate rigidity and flexibility properties of Hamiltonian flows. Along the way, this project will explore new types of geometric enumerative problems (e.g. multidirectional tangency constraints) whose solutions uncover intriguing connections with algebra and combinatorics (scattering diagrams, Q-Gorenstein smoothings, cluster algebras, quiver representations, ...). Three major components of this project are: (1) undertaking a novel strategy for tackling higher dimensional symplectic ellipsoid embeddings by exploiting links between symplectic field theory and singular algebraic curves and importing ideas from Gross–Siebert mirror symmetry, (2) exploring new frontier research avenues for real-valued invariants called symplectic capacities, notably by pursuing a quantitative theory of Fukaya categories and by pushing higher symplectic capacities into the world of contact geometry, and (3) building a new computational framework called digital symplectic geometry which aims to analyze Hamiltonian dynamics and nonsqueezing phenomena by performing numerical simulations and incorporating methods from computer optimization and machine learning. The projects in this proposal will develop deep and surprising connections between quantitative symplectic geometry and various other areas, including algebraic curve theory, birational geometry, mirror symmetry, and more. For instance, the PI will construct new interesting families of singular algebraic plane curves which also shed light on scattering diagrams arising from cluster algebras and quiver representations. The PI will also develop and exploit connections between symplectic singular Lagrangian fibrations and algebraic deformations of Fano varieties, and to bridge recent work on Weinstein manifolds and their qualitative Fukaya categories with the quantitative theory of symplectic capacities. Furthermore, the proposed projects in digital symplectic geometry aim to create new mechanisms for generating and testing conjectures in Hamiltonian dynamics. 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.