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.

摘要奖：DMS-0505968 首席研究员：Helmut H. Hofer 和 Dusa McDuff 这个教学研讨会由 DFG（德国相当于 NSF）共同主办，将汇集辛几何领域的初级和高级研究人员。该领域起源于哈密顿动力学和几何光学，并具有许多基础应用。例如，基于辛思想（辛积分器）的数值方法用于计算卫星和其他天体的轨道。历史亮点是 KAM 理论（柯尔莫哥洛夫-阿诺德-莫泽），它证明了我们太阳系的稳定性，以及无限维可积系统理论，它是光纤的数学基础，而光纤是一项处于核心技术的技术。我们所有的通信网络。最近，快速发展的伪全纯曲线理论引发了对三维和四维空间结构的深刻新见解，以及几何学和物理学之间意想不到的新联系。 令人惊讶的是，相同的基本思想也适用于几何光学、动力学和最抽象的物理学（即弦理论）。本次教学研讨会的目的是详细描述辛场论（SFT）的构造。 SFT是辛不变量的综合理论，包括Floer理论、Gromov-Witten理论和接触同调等著名理论。它是通过测量伪全纯曲线的模空间来构造的。其结构的丰富性来自于模空间有边界和奇异性以及无限多个模空间相互作用。 这些结构可以通过新颖的非线性 Fredholm 理论来捕获，该理论有两个事实。周围空间并不具有通常意义上的平滑结构，甚至具有局部变化的尺寸。然而，存在横向性（或规律性）的概念，并且在规则点处，解集是具有边界和角的平滑轨道。 此外，该理论精确地解释了无限多个 Fredholm 算子之间相互作用的含义，这导致了所谓的“Fredholm 算子理论”。这个抽象的 Fredholm 理论是通过其在 SFT 中的应用来发展和说明的。所讨论的问题与各种重要问题相关，可以预期研讨会上提出的想法将对许多数学领域及其应用产生影响。