Momentum maps in symplectic, algebraic and discrete geometry

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Tara Holm's principal focus in this proposal is the role of group actions and quotients in symplectic and algebraic geometry. While the questions that Holm and her collaborators plan to address have origins in symplectic and algebraic geometry, their solutions will involve methods from and have applications to algebraic topology, mathematical physics, and combinatorics. A fundamental tool in all of these problems is the momentum map, which relates the geometry and topology of a Hamiltonian system to the discrete geometry of the momentum polytope. The PI will use this tool to create a general framework for Lam, Schilling and Shimozono's combinatorial K-theory calculations of the affine Grassmannian. Holm and collaborator S. Tolman will use the momentum map to enhance our understanding of the integral cohomology rings of symplectic quotients. Together with A. Bertiger and K. Taiplae, Holm will investigate quantum invariants of symplectic quotients, with applications to combinatorics and algebraic geometry. There are many examples of spaces endowed with group actions that are not Hamiltonian systems, but enjoy many of their topological properties. Together with M. Harada, N. Ray and G. Williams, Holm will study invariants of orbifolds in the context of toric topology. Holm and A. Pires will investigate the topology of toric origami manifolds. These are classified by origami templates, whose combinatorics should provide insight into the topological invariants of the original manifolds. Finally, Holm and Y. Karshon are writing a research monograph that will disseminate the powerful methods from equivariant symplectic geometry that have applications across mathematics to a wide audience of research mathematicians.Symplectic geometry is the mathematical framework for describing phenomena in mathematical physics, from classical mechanics to string theory. The momentum map is an important tool that translates the symmetries of a physical system into discrete data. An expert in symplectic geometry, Holm will achieve a deeper understanding of the relationship between the geometry of a symplectic manifold and the combinatorics of the moment map data. Together with collaborators, she will investigate questions whose origins are in symplectic and algebraic geometry, and whose solutions will have wide applications. The proposed activities will advance our knowledge in the fields of symplectic geometry, algebraic geometry, algebraic topology, combinatorics and mathematical physics. Holm's broader objectives include increasing the participation and visibility of women in research mathematics, and enhancing the undergraduate experience in mathematics.