Collaborative Research: NSF-BSF: Equivariant Symplectic Geometry

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Symplectic geometry is an area of mathematics that has roots in the mathematical framework for classical mechanics. Many physical systems exhibit a great deal of symmetry. For example, the dynamics of a top that is spinning on a flat table is not affected by the position of the table in the room; changes in the table's position amount to a symmetry of the system. The momentum map translates the symmetries of a physical system into discrete data. The research supported by this award uses the momentum map to address questions about symmetries and invariants in symplectic geometry. This research program is largely driven by the interplay between deeply probing examples and advancing more general and abstract theory. The key is to find examples simple enough to be tractable and complex enough to exhibit complicated phenomena. Through their work, the investigators will create and strengthen bridges between symplectic geometry and other areas such as algebraic geometry, equivariant topology, and mathematical physics. Continuing their strong mentoring record, the investigators will advise graduate students and postdoctoral fellows from all over the world, both as supervisors in their institutions, and in informal research settings during visits to other institutions and at the conferences they organize and attend. More broadly, the activities supported by this award will lead to new projects for students and postdocs. Traveling between their institutions will enrich the students' and postdocs' experiences and advance their future careers: they will learn new methods, their mathematical perspective will broaden, and they will make valuable connections with different mathematical communities. Finally, the investigators have all been engaged in and plan to continue the outreach to K-12 and college students, with particular attention to students from underserved groups. Hamiltonian group actions give rise to the momentum map. This allows for the construction of the symplectic reduction, which can also be described algebraically using geometric invariant theory. The research funded here uses the momentum map to address questions about group actions and invariants in symplectic geometry, focusing on: the equivariant geometry of symplectic four-manifolds; applications of equivariant theory to symplectic topology; complexity one Hamiltonian torus actions; completely integrable systems; and the geometry and topology of momentum maps. Each of the investigators is an outstanding communicator and has been invited to lecture at top conferences in symplectic geometry and many related fields, including algebraic geometry, mathematical physics, and combinatorics. They build bridges between their work in symplectic geometry and foundational questions in these related fields. Through this research, Holm, Karshon, Kessler, and Tolman will achieve a deeper understanding of the relationship between the geometry of a Hamiltonian system and the combinatorics of the momentum image. The supported activities will advance our knowledge in the fields of symplectic geometry and combinatorics, with applications to algebraic geometry, algebraic topology and mathematical physics.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.