Curve counting beyond rational numbers

Massachusetts Institute Of Technology, Cambridge MA

Investigators

Abstract

This project focuses on symplectic manifolds, which are crucial objects in understanding the mathematics behind many physical phenomena, such as the movement of planets and the behavior of particles. The main goal is to find different ways to identify and count surfaces within these spaces. Understanding these two-dimensional objects can help us comprehend the more abstract, high-dimensional spaces they exist in. By analyzing this detailed geometric information, the project aims to tackle theoretical mathematical problems inspired by physics, such as those seen in Hamiltonian mechanics and string theory. Solving these theoretical problems can enhance our understanding of complex systems, potentially resulting in advancements in technology, healthcare, and general knowledge of nature. In addition, this project will provide research training opportunities for students. The counts of pseudo-holomorphic maps into symplectic manifolds are usually rational-valued due to the presence of nontrivial automorphisms. The project aims to answer questions in Hamiltonian dynamics and mathematical physics by developing curve counts with coefficients beyond rational numbers, including integers, complex K-theory, and complex cobordism. New curve-counting invariants inspired by cohomological operations and homotopy-theoretic enhancement of Floer theory will be developed along the way. The research topics include global Kuranishi charts for operations in the integral Hamiltonian Floer theory, Adams operations in enumerative geometry, Floer homotopy types over complex cobordism, and the study of periodic points of Hamiltonian diffeomorphisms. 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.