Curves, Knots, Hilbert schemes, and the Hitchin system

这个研究项目探索了PI和他的合作者在看似不相干的数学问题之间惊人的相互联系。与此类发现一样，这种认识促进了以前互不相关的领域之间的对话，其结果是，一个地区的良好理解结构可能会有效地转化为另一个地区。这样的融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、尚未探索的问题需要研究；与此同时，这些答案澄清了旧的和已确立的知识领域。特别是，该项目围绕着一个猜想（由于PI和合作者）将平面曲线奇点的某些代数几何不变量与奇点决定的结的拓扑不变量等同起来。具体来说，前者是奇异曲线对动机或分类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结理论和非阿贝尔Hodge对应。代数几何、辛几何和低维拓扑的交集是数学中唯一一个量子SU(n)不变量及其分类被视为来自空间结的几何形状，而不是来自其平面投影的地方。此外，该研究还涉及了许多当前数学感兴趣的主题：受弦理论影响的枚举几何，Hitchin系统，Khovanov-Rozansky同调，Cherednik代数及其表示，legendrian结和辛场论，以及3流形不变量。它可能会通过利用它们之间的联系来单独推进其中的一些。特别地，本项目旨在研究非阿贝尔霍奇理论中的“P = W”猜想，以及辛场论中关于Chekanov-Eliashberg ga的所有表示是否都来自几何的问题。该奖项由代数和数论以及拓扑学项目共同资助。

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.