Award: DMS-1105837 Principal Investigator: Sikimeti Mau
A-infinity algebra structures have recently emerged in symplectic topology and will be investigated and extended by these projects. The principal investigator intends to study these algebraic structures in concrete examples based on Hilbert schemes of points on complex curves or complex surfaces. Part of the goal will be to extend the structures to higher categorical structures, using symplectic constructions closely related to the string diagrams of physicists. The extended algebraic structures are motivated by constructions in algebraic geometry for the same Hilbert schemes, which should have symplectic analogues by mirror symmetry. Two examples in particular that have been well studied on the symplectic side, and can function as guides, are the Heegaard Floer theory developed by low-dimensional topologists, and Seidel-Smith's symplectic Khovanov homology, a symplectically constructed invariant of knots and links. The short-term objective is to find concrete illustrations, and potential applications, of a new theory that is largely abstract, but has the potential to explain algebraic phenomena in these fields. The broader goal is to describe as much of the algebraic structure of Lagrangian Floer theory as possible in a single algebraic language coming from "quilts", a recent technique in symplectic topology due to Wehrheim and Woodward.
A symplectic structure is the geometric face of Hamiltonian mechanics, in which the position and momentum coordinates of a system of moving particles are tracked and used to write out equations of motion that correspond to Newton's laws. Spaces that carry such structures are always even-dimensional, and their underlying geometry is about two-dimensional area and higher-dimensional volume rather than about length and angle, which are at the root of much of familiar geometry. New methods are coming into symplectic geometry from other subjects such as low--dimensional topology, and it appears that an algebraic formalism can be devised to carry a number of these new constructions and to reveal useful properties of them. ˇ