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

摘要奖：DMS-0505968首席研究员：赫尔穆特·H·霍费尔和杜萨·麦克达夫这个由德国国家科学基金会(DFG)共同赞助的教学研讨会将把辛几何领域的初级和高级研究人员聚集在一起。该领域起源于哈密顿动力学和几何光学，并有许多基本的应用。例如，基于辛思想的数值方法(辛积分器)被用来计算卫星和其他天体的轨道。历史性的亮点是Kam理论(Kolmogorov-Arnold-Moser)，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，是我们所有通信网络的核心技术。最近，快速发展的伪全纯曲线理论为三维和四维空间的结构带来了新的见解，并在测量学和物理学之间产生了意想不到的新联系。令人惊讶的是，同样的基本思想也适用于几何光学、动力学和最抽象的物理(即弦理论)。本教学工作坊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，包括著名的Floer理论、Gromov-Witten理论和接触同调理论。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点，以及无穷多个模空间相互作用。这些结构可以用一种新颖的非线性Fredholm理论来捕捉，该理论由两个事实区分。周围的空间并不具有通常意义上的平滑结构，甚至有局部不同的维度。然而，有一个横截性(或正则性)的概念，在正则点上，解集是光滑的或有边界和角点的二叉形。此外，该理论精确地说明了无限多个Fredholm算子相互作用的确切含义，这导致了所谓的“Fredholm带运算的理论”。这一抽象的Fredholm理论得到了发展，并通过它在SFT中的应用得到了说明。所讨论的问题与各种重要问题有关，人们可以预期，在研讨会上提出的想法将在一些数学领域及其应用中产生影响。

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.