CAREER: Symplectic 4-Manifolds and Singular Symplectic Surfaces

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This project studies new frontiers in the geometry of 4-dimensional spaces. These spaces play important roles in the mathematical areas of topology and geometry, as well as in models of important physical systems. Because humans cannot visualize four spatial dimensions, it is necessary to develop mathematical and diagrammatic tools to study these spaces. Historically, 4-dimensional spaces have displayed unusual phenomena, distinct from spaces of either lower or higher dimensions. A persistent theme in the discovery of such phenomena is the role of symplectic geometry. Although symplectic geometry was initially formulated to encode the equations of Hamiltonian physics, it has grown into an important abstract mathematical topic that is particularly powerful for studying 4-dimensional spaces. This project aims to use cutting-edge tools to make novel advances in 4-dimensional symplectic geometry. Because this is a rapidly growing field, training a well-prepared and diverse next generation is particularly important. To that end, the PI will create a new undergraduate course, run speaking workshops for graduate students, develop a paired graduate-undergraduate reading program, and organize workshops, summer schools, and conferences.The project will study symplectic 4-manifolds and singular symplectic surfaces. The first primary goal is to create diagrammatic methods to study the subtleties of diffeomorphism classes of symplectic 4-manifolds. The second goal is to produce new symplectic isotopy classifications for smooth and singular symplectic surfaces. The third goal is to perform new calculations of existing symplectic invariants and to define new invariants for symplectic 4-manifolds and their submanifolds. The project will employ four mathematical methods to study symplectic 4-manifolds and surfaces: branched coverings of braided surfaces, trisections, (de)singularization, and Weinstein handlebodies. The fourth goal is to create a pipeline of new mathematicians well trained in the study of manifolds and skilled in scientific communication and research. The project includes training of students at the graduate and undergraduate levels in geometry, topology, and 4-dimensional spaces, as well as in clear communication and mentoring skills.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.