Principal Investigator: Todor Milanov
The long term goal of these projects is to understand the topology of moduli spaces of holomorphic curves in symplectic manifolds in terms of the theory of integrable systems. The first project aims to construct an integrable hierarchy which governs the topology of the moduli spaces of degree d stable holomorphic maps from a genus-g, nodal Riemann surface to a variety X which is assumed to be a toric complete intersection. The topological information is encoded in a formal power series called the total descendant potential of the toric variety, and the goal is to show that this potential is a solution to an integrable hierarchy. The second project in this research program is dedicated to an interaction of integrable systems with symplectic field theory of a compact Kaehler manifold Y, developing a potential related to the total descendant potential described above. Relationships are expected to emerge between Gromov-Witten invariants for Y and for a projective line bundle over Y, with the relationships described via transformations of integrable systems.
Symplectic geometry is the structure underlying the Hamiltonian formalism of classical mechanics, in which the behavior of a mechanical system is determined by an energy-like function. Geometric spaces carrying such structures can be high-dimensional and structurally complicated, and much recent work in the area is devoted to exploring symplectic structures through associated spaces of well-imbedded two-dimensional subsurfaces of them. Two-dimensional surfaces have been studied for more than 150 years and our detailed understanding of them leads to constraints upon the associated invariants of high-dimensional symplectic manifolds. The principal investigator's work explores new constraints of this kind that take the form of well-behaved differential equations, the integrable systems of the proposal title.
