Curves, Knots, Hilbert schemes, and the Hitchin system

这个研究项目探索了PI和他的合作者发现的数学中看似不同的问题之间令人惊讶的相互联系。与此类发现一样，这种理解促进了先前不相关的领域之间的对话，其结果是，一个领域中已被充分理解的结构可能会被有益地转化为另一个领域。 这样的汇合也为研究生提供了理想的训练场地：有许多新的，令人兴奋的，和未探索的问题，工作;同时，答案澄清了旧的和建立的知识领域。特别是该项目围绕一个猜想（由于PI和合作者）等同于某些代数几何不变量的平面曲线奇点与拓扑不变量的奇点所确定的结。具体地说，前者是奇异曲线对motivic或范畴化的Donaldson-Thomas理论的局部贡献，后者是Khovanov-Rozansky同调。该研究项目旨在通过另外两个主题进行证明：Legendrian knots理论和非阿贝尔Hodge对应。这种代数几何、辛几何和低维拓扑的交叉是数学中唯一一个量子SU（n）不变量和它们的重叠被认为来自空间中的一个结的几何，而不是来自它的平面投影的地方。此外，研究触及有意义的许多主题，目前的数学兴趣：枚举几何的影响弦理论，希钦系统，Khovanov-Rozansky同源，的Cherednik代数及其表示，奥德里德结和辛场理论，和3流形不变量。可以预期，通过利用它们之间的联系，将分别推动其中一些进程。特别是该项目的目标是在非交换霍奇理论中的“P = W”猜想，以及辛场论中的问题上取得进展，即Chekanov-Eliashberg dga的所有表示是否都来自几何。该奖项由代数和数论以及拓扑计划共同资助。

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.