Mathematicians are developing a new way to count two-dimensional shapes inside six-dimensional spaces, using equations borrowed from particle physics. The project sits at the intersection of two powerful but separate approaches to understanding manifolds—high-dimensional geometric shapes that appear throughout physics, data science, and machine learning. One approach uses gauge fields (like electromagnetic fields) governed by equations similar to Maxwell’s; the other uses pseudo-holomorphic curves, which probe the geometry of symplectic manifolds. The researcher aims to connect these fields by studying generalized Seiberg–Witten equations, which govern a poorly understood class of gauge fields. A key goal is to prove that counting pseudo-holomorphic curves and solutions to these equations yields a new invariant of six-dimensional symplectic manifolds—a mathematical fingerprint that remains unchanged when the space is deformed. This is fundamental mathematics with no immediate practical application. However, similar work on gauge theory and pseudo-holomorphic curves has already reshaped how mathematicians distinguish three- and four-dimensional spaces and classify how curves sit inside them. A deeper understanding of these equations could eventually inform the geometry behind machine learning, dynamical systems, or theories of gravity—fields where manifolds are central but the mathematics remains incomplete.
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This project is in differential geometry, a branch of mathematics that uses calculus to answer fundamental questions about manifolds: geometric shapes that are high-dimensional analogues of curves and surfaces. Manifolds appear naturally in many contexts across mathematics, computer science, and physics—such as polynomial and differential equations, dynamical systems, data analysis, machine learning, and the theory of gravity—making their study a central goal of modern geometry. Over the last forty years, two groundbreaking approaches have emerged in the study of manifolds, both through the use of solutions to differential equations. The first approach uses gauge fields, such as the electromagnetic field and other fields considered in particle physics. These fields are governed by equations similar to Maxwell’s equations of electromagnetism. The second approach uses pseudo-holomorphic curves, a special class of two-dimensional shapes that can be used to probe the geometry of higher-dimensional shapes known as symplectic manifolds. Research on gauge fields and pseudo-holomorphic curves has led to significant advances in our understanding of manifolds and is still rapidly evolving. The proposed project lies at the intersection of these two research areas and explores new, surprising connections between them. In the study of gauge fields, it aims to answer key questions about generalized Seiberg–Witten equations, which govern a little-understood class of gauge fields. These equations are conjectured to have important applications to the problem of distinguishing manifolds of dimension three and four, as well as classifying the many complicated ways in which curves and surfaces can exist within such manifolds. While many such applications have been proposed in physics and geometry, developing them requires a deeper understanding of the solutions to these equations. This project will investigate what happens to solutions of generalized Seiberg–Witten equations when the underlying manifold is deformed. This is generally a challenging but important problem, as mathematicians seek properties of manifolds that are topological—meaning they remain unchanged under deformations. The first objective is to solve this problem for manifolds of dimension two, i.e., surfaces, which would already be a significant step forward while being achievable within the grant’s timeframe. In symplectic geometry, this project aims to combine pseudo-holomorphic curves with gauge theory in a novel way to develop new tools for studying symplectic manifolds. Specifically, building on the first objective, the second objective is to prove that counting pseudo-holomorphic curves and solutions to generalized Seiberg–Witten equations yields a new invariant of symplectic manifolds of dimension six. This invariant would parallel those developed previously using algebraic methods, addressing the long-standing open problem of finding their counterpart in differential geometry. This project aligns with recent developments in the study of gauge fields and pseudo-holomorphic curves while exploring the surprising connections between these two seemingly unrelated areas of mathematics. It also incorporates the latest developments from other fields, such as geometric measure theory and nonlinear elliptic differential equations.
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