Logarithmic Moduli Spaces for Symplectic Geometry: Construction, Applications, and Beyond

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Symplectic manifolds are geometric objects that generalize the concept of phase-space in classical mechanics. During the past four decades, the field of symplectic geometry has evolved rapidly, leading to new connections with other significant areas of research, such as algebraic geometry, low dimensional topology, and high energy physics. This award supports research on fundamental objects known as holomorphic curves and their corresponding invariants and algebraic structures. In particular, the investigator will address foundational questions, such as the construction of well-behaved families of holomorphic curves in the presence of objects known as divisors. This research contains specific projects that can be carried out by graduate students and postdocs. The investigator will organize annual mini-symposia for introducing undergraduate students to research opportunities in geometry and topology, and their applications to other fields. He will also initiate a math club at the public library aimed at high school students.The main objective of this proposal is to construct moduli spaces of holomorphic curves for arbitrary pairs of symplectic manifolds and normal crossing symplectic divisors, satisfying particular properties. Construction of such moduli spaces requires a compactification, an analytical framework for deformation theory, addressing the transversality issue, and proving a gluing theorem. These moduli spaces have immediate applications in enumerative geometry, Mirror Symmetry, construction of Fukaya categories, and other active areas of research in symplectic geometry, algebraic geometry, and string theory. In collaboration with M. McLean and A. Zinger, the PI introduced topological notions of normal crossing symplectic divisor and variety. They have constructed tools such as regularizations and logarithmic tangent bundle for working with such objects. Recently, the PI developed a novel compactification and a deformation theory based on the logarithmic tangent bundle. He will use this setup to work on the remaining steps of the construction. The main project is to define Gromov-Witten invariants relative to an arbitrary normal crossing divisor. Other projects include proving a degeneration formula for Gromov-Witten invariants, finding the relations with the algebraic approach, and exploiting the applications to Mirror Symmetry. This project is jointly funded by Geometric Analysis and the Established Program to Stimulate Competitive Research (EPSCoR).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.