Manifolds are shapes that locally resemble Euclidean space. This project focuses on manifolds that are closed in the sense that they have no boundary edges and do not extend to infinity. A closed one-dimensional manifold is equivalent to the circle, while a closed (orientable) two-dimensional manifold is equivalent to the sphere, the surface of a donut, or the surface of a “donut” with two or more holes. Closed three-dimensional manifolds cannot be so easily visualized, while closed four-dimensional manifolds can have very complicated structures and are not well-understood. Four-dimensional manifolds, with three spatial directions and one temporal direction, are used in general relativity as models for the universe. Four-dimensional manifolds also play a central role in gauge theories developed to unify three of the four known fundamental forces (the electromagnetic, weak, and strong interactions). The first goal of the project is to complete a mathematical proof of a prediction from supersymmetric quantum field theory, one that relates two different gauge theories used to help understand four-dimensional manifolds. The second goal of the project is to advance understanding of the possible structures of four-dimensional manifolds, a source of fascination and inspiration for mathematicians and physicists for nearly a century. The classification of possible structures of three-dimensional manifolds advanced tremendously in recent decades, but four-dimensional manifolds remain mysterious, despite intense effort by mathematicians to analyze them. The third goal of the project is to develop methods to relate different approaches to understanding the structure of three-dimensional manifolds. The project involves graduate students in the research. To help train the next generation of mathematicians, the principals also will continue their tradition of organizing seminars and conferences, contributing expository articles to help engage a broader audience interested in learning about careers and research in mathematics, mentoring undergraduate and graduate students and postdoctoral researchers, and encouraging the interest of high-school students in mathematics through summer programs and outreach activities at the National Museum of Mathematics.
The first goal of the project is to complete a proof of Witten's formula relating the Donaldson and Seiberg-Witten invariants of a closed, oriented, smooth four-dimensional manifold with admissible topology and simple type, employing a mathematically rigorous method based on moduli spaces of non-Abelian monopoles. The work will apply a new approach to gluing solutions to non-linear partial differential equations that arise in geometric analysis to establish a proof of an expected gluing theorem for non-Abelian monopoles. The second goal of their project is complete a proof of the conjectured Bogomolov-Miyaoka-Yau inequality for simply connected four-dimensional manifolds of Seiberg-Witten simple type and having non-zero Seiberg-Witten invariants. The approach uses a new version of Morse theory for singular analytic spaces applied to the singular moduli space of non-Abelian monopoles to prove existence of solutions to another non-linear partial differential equation – the anti-self-dual Yang-Mills equation on a rank-two Hermitian vector bundle with prescribed topology over a four-dimensional manifold. The third goal of the project is to derive relations between the instanton and Seiberg-Witten Floer homologies of closed three-dimensional manifolds, potentially relating fundamental groups and contact structures.
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.