This research project explores the discovery, by the PI and his collaborators, of surprising interconnections between seemingly disparate questions in mathematics. As always with such discoveries, this understanding facilitates dialogue between previously disconnected fields, with the consequence that well understood structures in one area may be profitably translated into another. Such confluences also offer ideal training grounds for graduate students: there are many new, exciting, and unexplored questions on which to work; at the same time, the answers clarify old and established fields of knowledge.
In particular the project centers around a conjecture (due to the PI and collaborators) equating certain algebrao-geometric invariants of plane curve singularities with topological invariants of knots determined by the singularities. Specifically, the former is the local contribution of the singular curve to motivic or categorified Donaldson-Thomas theory, and the latter are the Khovanov-Rozansky homologies. The research project aims at a proof by passing through two other subjects: the theory of Legendrian knots and the non-abelian Hodge correspondence. This intersection of algebraic geometry, symplectic geometry, and low dimensional topology is the only place in mathematics where the quantum SU(n) invariants and their categorifications have ever been seen as coming from the geometry of a knot in space, as opposed to from its planar projections. In addition, the research touches meaningfully on many subjects of present mathematical interest: enumerative geometry as influenced by string theory, the Hitchin system, Khovanov-Rozansky homology, the Cherednik algebra and its representations, legendrian knots and symplectic field theory, and 3-manifold invariants. It may be expected to advance some of these separately by exploiting their connections. In particular the project aims to make progress on the "P = W" conjecture in nonabelian Hodge theory, and on the question in symplectic field theory of whether all representations of the Chekanov-Eliashberg dga come from geometry.
The award is co-funded by the Algebra and Number Theory and the Topology programs.
