New Bridges to Gromov-Witten Theory

In this project, the PI will work on Gromov--Witten theory, an area in mathematics lying in the intersection of algebraic geometry and symplectic topology, with implications to string theory in physics. Gromov--Witten theory is concerned with counting curves of a given type which intersect in special configurations. Such counts, in good situations, give rise to enumerative invariants of the space, called Gromov--Witten invariants. These invariants play a key role in modern enumerative algebraic geometry, as well as in theoretical physics, in the context of string theory. In this project, the PI will apply techniques to calculate Gromov-Witten invariants to several other questions coming from different areas of mathematics, to obtain major advances in mirror symmetry, in representation theory of quivers, and in the study of compactifications of moduli spaces of algebraic varieties. This award will also support graduate students working with the PI in these areas. The project is built around five main strands. In the first strand the project intends to develop new degeneration tools for calculating Gromov--Witten invariants with arbitrary insertions of a large class of spaces. The second strand is concerned with constructions of explicit examples of mirrors to log Calabi--Yau varieties and orbifolds using tropical geometric tools. In the third strand, the project intends to provide concrete descriptions of functorial compactifications of the moduli space of log Calabi--Yau pairs using mirror symmetry. The fourth strand of the project will provide a correspondence between Donaldson--Thomas theory of quivers and Gromov--Witten theory for cluster varieties. Finally, the project intends to implement tropical algebro-geometric tools to study the topology of near symplectic manifolds with a toric structure. The key new ingredient used throughout these strands is the recent advance in our understanding of Gromov--Witten theory using logarithmic geometry. This is a modern variant of algebraic geometry, developed in particular to understand solutions to degeneration problems, which naturally appear in the context of enumerative and tropical algebraic geometry. 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.