Momentum Maps in Symplectic, Algebraic, and Discrete Geometry

Cornell University, Ithaca NY

Investigators

Abstract

Symplectic geometry is a broad and deep subject, with foundations in mathematical physics, that has seen much activity over the last two decades. Symplectic manifolds are spaces that carry a basic geometric structure which assigns an area to each two-dimensional plane. Because symplectic manifolds all locally `look the same', distinguishing two of them involves global properties and measurements, such as how large a ball can be embedded into the space in an area-preserving way (this is an example of a symplectic capacity). Fortunately, many symplectic manifolds have built-in symmetries, and the main focus in this project is the role of symmetries in symplectic geometry (especially, those symmetries that are organized into Hamiltonian group actions) and the quotient spaces that result when the symmetries are used to 'fold up' the manifold. A key tool in studying these symmetries is the momentum map, and in this project the PI will achieve a deeper understanding of the relationship between the geometry of a Hamiltonian system and the combinatorics of the momentum image. More broadly, the activities supported by this award will advance our knowledge in the fields of symplectic geometry and combinatorics, with applications to algebraic geometry, algebraic topology and mathematical physics.  The PI will also continue to train graduate students and mentor postdocs at Cornell, as well as continue activities which have a substantial impact on the broader mathematical community and society as large, such as giving public lectures describing `big ideas' from geometry and topology, and playing significant leadership roles in professional organizations and groups that seek to improve undergraduate mathematics education across the nation. As mentioned above, Hamiltonian group actions give rise to the momentum map. This allows us to construct the symplectic reduction, which can also be described algebraically using geometric invariant theory. Pseudoholomorphic curves provide strong analytic tools to study symplectic invariants. A particularly nice package for using these tools is Hutchings' embedded contact homology (ECH). A  fundamental problem in symplectic geometry is to relate the geometry and topology of a Hamiltonian system to the combinatorics of the momentum polytope, and vice versa.  In the projects supported by this award, the PI will study questions about symplectic embedding problems, including symplectic capacities of toric 4-manifolds and rational ruled surfaces using ECH capacities. The PI's analysis will add to our understanding of topological invariants in equivariant symplectic geometry, including building a surjectivity and formality package for symplectic quotients, and a study of complexity one spaces. She is also writing a graduate textbook  introducing students and researchers to the key aspects of the field. Finally, the PI will address a number of questions in computational toric topology.  The answers will rely on tools from  a variety of fields, including algebraic  geometry, commutative algebra, and algebraic topology. This set of projects includes the study of toric folded symplectic manifolds, a close cousin of symplectic toric manifolds, and  questions about ordinary and stringy invariants of toric orbifolds.