Holomorphic Curves and Two-Dimensional Gauge Theory

摘要奖：DMS-0605097首席研究员克里斯托弗·伍德沃德国际和平研究所将开展全纯曲线和二维规范理论相互作用的研究项目。WithK.Wehheim和S.Mau将研究拉格朗日响应在Floer-Fukaya理论中的作用，特别是证明与拉格朗日响应相关的函子的组成是与该组成相关的函子。他还将发展霍尔哈精确三角形的镜像类比。他将把这些结果应用于构造具有边界的三维和四维流形的新不变量，可能包含纠缠。他将与C.Teleman一起证明曲线上丛的模空间的Chern类上的Newstead-Ramananan猜想，并研究商栈的K-理论Gromov-Witten不变量。他将与冈萨雷斯一起研究具有哈密顿群作用的辛流形的Gromov-Witten不变量，推广了二维Yang-Mills理论的拓扑极限。他将为本科生提供一些研究经验，并改进系的本科生研讨会计划。这些项目将促进对辛几何的理解，辛几何是经典力学的数学语言，以及范畴理论、表示理论和量子物理之间的关系。这项研究还有望促进有限和无限维李群理论的进步，这些李群代表着许多科学领域的对称性。

AbstractAward: DMS-0605097Principal Investigator: Christopher WoodwardThe PI will carry out research projects on the interplay ofholomorphic curves and two-dimensional gauge theory. WithK. Wehrheim and S. Mau the PI will study the role of Lagrangiancorrespondences in Floer-Fukaya theory, and in particular provethat the composition of functors associated to Lagrangiancorrespondences is the functor associated to the composition. Hewill also develop the mirror analogue of Horja's exact triangle.He will apply these results to the construction of new invariantsof three and four-manifolds with boundary, possibly containingtangles. With C. Teleman he will prove the Newstead-Ramananconjectures on Chern classes of the moduli space of bundles on acurve and investigate K-theoretic Gromov-Witten invariants ofquotient stacks. With E. Gonzalez he will investigateGromov-Witten invariants for symplectic manifolds withHamiltonian group actions, generalizing the topological limit oftwo-dimensional Yang-Mills theory. He will run several researchexperiences for undergraduates, and improve the department'sundergraduate seminar program.These projects will advance the understanding of symplecticgeometry, which is the mathematical language for classicalmechanics, and the relationship between category theory,representation theory, and quantum physics. The research is alsoexpected to lead to advances in the theory of finite- andinfinite-dimensional Lie groups, which represent symmetries inmany areas of science.