Holomorphic Curves and Two-Dimensional Gauge Theory

项目编号：dms -0605097项目负责人：Christopher woodwardpi将开展全纯曲线与二维规范理论相互作用的研究项目。WithK。Wehrheim和S. Mau the PI将研究拉格朗日对应在flower - fukaya理论中的作用，特别是证明与拉格朗日对应相关的函子的组合是与该组合相关的函子。他还将开发与Horja的三角形完全相同的镜像模拟。他将把这些结果应用到具有边界（可能包含缠结）的三流形和四流形的新不变量的构造中。他与C. Teleman一起证明了曲线上束模空间的Chern类上的newsted - ramanan猜想，并研究了商堆的k理论Gromov-Witten不变量。他将与E. Gonzalez一起研究具有hamilton群作用的辛流形的gromov - witten不变量，推广二维Yang-Mills理论的拓扑极限。他将为本科生开展几项研究经验，并改进系里的本科生研讨会项目。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，以及范畴论、表示理论和量子物理学之间的关系。这项研究还有望导致有限维和无限维李群理论的进步，它们代表了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.