CAREER: A Symplectic Lens of Mirror, Arithmetic, and Homotopy

Symplectic topology grew out of geometric studies of motion and has become a central area connecting geometry, topology, number theory, and algebraic geometry. This project uses symplectic topology to understand hidden structures that relate the study of shapes, the counting of curves, and arithmetic phenomena. The project will advance fundamental mathematics by developing new links between different fields and by creating tools and perspectives that can be used in future research. The educational component is integrated with the research component. The investigator will involve undergraduate and graduate students in research, organize a regional symplectic topology seminar for graduate students, and continue to develop the Rutgers Symplectic Summer School for advanced graduate students and postdoctoral researchers. These activities will broaden access to current research, improve communication between junior and senior researchers, and help train the next generation of mathematicians. The research has two main directions. First, the investigator will study Frobenius structures on quantum D-modules and quantum connections from arithmetic, categorical, and homotopy-theoretic viewpoints. The project will investigate a p-adic version of the Gamma conjecture, use topological Hochschild homology of Fukaya categories and open-closed field theory to study Frobenius symmetries, and develop a Floer-homotopical approach using cyclotomic spectra and bordism theories of derived orbifolds. Second, the investigator will use orbifold Gromov-Witten theory to study symplectic cohomology of affine varieties and its relation to Gross-Siebert intrinsic mirror symmetry. Building on global Kuranishi charts and derived orbifolds, the project will study counts of pseudo-holomorphic curves with tangency conditions via root stacks and compare them with log Gromov-Witten theory. Expected outcomes include new connections between symplectic cohomology and Gross-Siebert mirror rings, progress on homological mirror symmetry for log Calabi-Yau varieties, and Weinstein-type results for contact boundaries of complex smooth affine varieties. 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. NSF Award ID: 2540393 | Program: 01002627DB NSF RESEARCH & RELATED ACTIVIT,01002930DB NSF RESEARCH & RELATED ACTIVIT,01003031DB NSF RESEARCH & RELATED ACTIVIT | Principal Investigator: Shaoyun Bai | Institution: Massachusetts Institute of Technology, CAMBRIDGE, MA | Award Amount: $244,913 View on NSF Award Search: https://www.nsf.gov/awardsearch/show-award/?AWD_ID=2540393 View on Research.gov: https://www.research.gov/awardapi-service/v1/awards/2540393.html