Workshop on Symplectic Field Theory; May 14-20, 2005; Leipzig, Germany

辛场论研讨会；

• 批准号: 0505968
• 负责人: Helmut Hofer
• 金额: $ 2万
• 依托单位: New York University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-03-01 至 2007-02-28
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0505968&HistoricalAwards=false
• 关键词: Workshop; Symplectic; Field; Theory; May

AbstractAward: DMS-0505968Principal Investigator: Helmut H. Hofer and Dusa McDuffThis instructional workshop, co-sponsored by the DFG (the Germanequivalent of the NSF) will bring together junior and seniorresearchers in the field of symplectic geometry. The field hasits origins in Hamiltonian dynamics and geometric optics and hasmany fundamental applications. For example, numerical methodsbased on symplectic ideas (symplectic integrators) are used tocompute the orbits of satellites and other celestialbodies. Historic highlights are KAM-Theory(Kolmogorov-Arnold-Moser), which leads to a proof of thestability of our solar system, and the theory ofinfinite-dimensional integrable systems, which is themathematical underpinning of fiber optics, a technology that liesat the heart of all of our communication networks. More recently,the rapidly developing theory of pseudoholomorphic curves has ledto deep new insights into the structure of three andfour-dimensional space, and to unexpected new connections betweengeometry and physics. It is a surprising fact that the sameunderlying ideas apply to geometric optics, to dynamics and tothe most abstract physics (i.e. string theory).It is the purpose of this instructional workshop to describe theconstruction of symplectic field theory (SFT) in detail. SFT is acomprehensive theory of symplectic invariants, including suchwell-known theories as Floer theory, Gromov-Witten theory andcontact homology. It is constructed by measuring moduli spaces ofpseudoholomorphic curves. The richness of its structure comesfrom the fact that the moduli spaces have boundaries andsingularities and that infinitely many moduli spaces interactwith each other. These structures can be captured by a novelnonlinear Fredholm theory which is distinguished by twofacts. The ambient spaces do not carry smooth structures in theusual sense and even have locally varying dimensions. Howeverthere is a notion of transversality (or regularity), and atregular points the solution sets are smooth orbifolds withboundaries and corners. Moreover the theory makes precise whatit means for infinitely many Fredholm operators to interact witheach other, which leads to a so-called "Fredholm Theory withOperations". This abstract Fredholm Theory is developed andillustrated by its application to SFT. The issues addressed arepertinent to a variety of important problems, and one can expectthat the ideas presented at the workshop will have implicationsin a number of mathematical fields and their applications.

Abstractaward：DMS-0505968原理研究人员：Helmut H. Hofer和Dusa McDuffthis教学研讨会，由DFG（NSF的杰出人物）共同赞助，将在Symplectic Geemetry领域将Junior和SeniorreSearcher聚集在一起。 HASITS起源于哈密顿动力学和几何光学和Hasmany基本应用。例如，使用基于符号思想（符号积分器）的数值方法用于卫星和其他天体的轨道。历史悠久的亮点是KAM理论（Kolmogorov-Arnold-Moser），它导致了我们太阳能系统的可疑性的证明，以及Infinite-Dimensionalable的可二维集成系统的理论，这是光纤的基础，这是我们通信网络中心的技术的基础。最近，迅速发展的假酚形态曲线理论使人们对三个和四维空间的结构进行了深刻的新见解，并引起了地几何和物理学之间意外的新连接。 令人惊讶的事实是，相同的思想适用于几何光学，动力学和大多数抽象的物理学（即弦理论）。这是该教学研讨会的目的，以详细描述符号现场理论（SFT）的结构。 SFT是对符合性不变性的理论，包括诸如浮动理论，gromov-witten理论和接触同源性等知名理论。它是通过测量模量曲线的模量空间来构建的。其结构的丰富性来自模量空间具有边界和奇异性，并且无限的许多模量空间相互作用。 这些结构可以通过新颖的弗雷德霍尔姆理论来捕获，该理论以两种方式区分。环境空间在同性意义上没有光滑的结构，甚至没有局部变化的维度。但是，有一个横向性（或规律性）的概念，而溶液集的截面点是光滑的，带有横向和角落。 此外，该理论对许多弗雷德·霍尔姆（Fredholm）经营者与其他人进行互动的含义确实是什么意义，这导致了所谓的“弗雷德霍尔姆理论合作”。这种抽象的弗雷霍尔姆理论是通过其应用于SFT的。解决的问题涉及各种重要问题，人们可以期望在研讨会上提出的想法将在许多数学领域及其应用中产生影响。