Mathematical Sciences: Topology and Manifolds

数学科学：拓扑与流形

• 批准号: 9401443
• 负责人: Dusa McDuff
• 金额: $ 34.5万
• 依托单位: SUNY at Stony Brook
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1994
• 资助国家: 美国
• 起止时间: 1994-07-01 至 1998-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9401443&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Topology; Manifolds

9401443 McDuff Dusa McDuff: The study of symplectic topology has made great progress in the past decade, and is now sufficiently developed to have significant impact in other fields, for example the use of Floer homology in studying 3- and 4-dimensional manifolds, and, more recently, the application of J-holomorphic curves in the study of mirror symmetry and quantum cohomology. In recent years, McDuff has devoted her energies to developing the tools of symplectic topology with the main aim of understanding the topological structure of compact symplectic manifolds. She intends to continue working in the same general direction in the next three years, in an attempt to solve some of the very interesting problems which still remain. Lowell Jones: Jones will continue his collaborative efforts with F. T. Farrell, investigating rigidity properties of the classical closed spherical manifolds. In prior work they have shown that closed nonpositively curved manifolds and closed double coset manifolds are topologically rigid, in dimensions 5. In work in progress they show that any closed complete affine flat manifold is topologically rigid, in dimensions 5. They have also constructed many examples of strictly negatively curved closed manifolds which are homeomorphic to one another but not diffeomorphic, thus verifying that the class of nonpositively curved manifolds does not enjoy smooth rigidity. Whether or not the class of closed double coset manifolds, or the class of closed complete affine flat manifolds, are smoothly rigid remains an open question which Jones intends to pursue. Dusa McDuff: A symplectic structure is a very basic kind of geometric structure which underlies all the equations of classical and quantum physics. In the last ten years, new tools have been developed which allow mathematicians for the first time to gain some understanding of the global meaning of this kind of structure. One focus of McDuff's work is the geometric explora tion of the concept of energy. In physics this is the driving force of movement (of systems ranging from the planetary system to plasmas). The new methods of symplectic topology make it possible in some cases to estimate the minimum energy required to accomplish a certain movement. This work is very basic and should eventually have important implications in the study of physical systems. Another focus of her work is the attempt to understand what a symplectic structure really is by investigating the geometric properties of the spaces which are symplectic. The idea is that these spaces are the embodiment of the abstract idea of a symplectic structure. Lowell Jones: In 1829 and 1832 the mathematicians N. Lobachevsky and J. Bolyai independently solved a two thousand year old mathematical problem by verifying that Euclid's fifth axiom for plane geometry is independent from the first four of Euclid's axioms. They did this by investigating a new type of geometry having "negative curvature." Today the 2-dimensional geometries of Lobachevsky and Bolyai are important special cases of the larger class of geometric structures having "non-positive curvature," which can occur in all the higher dimensions, and which have been the focus of a tremendous amount of research over the last forty years. An important theme in this research has been that each such geometric structure has an associated algebraic structure (called the "fundamental group") which acts as a sort of genetic code for the underlying geometric structure. Although it is far from true that the fundamental group always determines the underlying geometric structure, this is true in some special cases. Jones has recently verified that the fundamental group always does completely determine the "topological nature" of the underlying geometric structure. To paraphrase this, Jones has shown that the fundamental group determines the rough outline of the underlying geometric structure, even though it does not de termine all the details of the geometric structure. ***

9401443 McDuff Dusa McDuff：辛拓扑的研究在过去的十年中取得了很大的进展，现在已经发展到足以对其他领域产生重大影响，例如在研究3维和4维流形中使用Floer同调，以及最近在研究镜像对称和量子上同调中的j全纯曲线的应用。近年来，McDuff致力于辛拓扑工具的开发，主要目的是理解紧辛流形的拓扑结构。她打算在接下来的三年里继续沿着同样的大方向工作，试图解决一些仍然存在的非常有趣的问题。Lowell Jones: Jones将继续与F. T. Farrell合作，研究经典闭球流形的刚性特性。在之前的工作中，他们已经证明了闭合非正弯曲流形和闭合双辅助集流形在5维上是拓扑刚性的。在正在进行的工作中，他们证明了任何封闭完全仿射平面流形在5维上都是拓扑刚性的。他们还构造了许多彼此同胚但不微分同胚的严格负弯曲闭流形的例子，从而验证了一类非正弯曲流形不具有光滑刚性。无论是闭双辅集流形，还是闭完全仿射平面流形是光滑刚性的，这仍然是琼斯想要研究的一个悬而未决的问题。杜萨·麦克杜夫：辛结构是一种非常基本的几何结构，它是所有经典和量子物理方程的基础。在过去的十年里，新的工具被开发出来，使数学家们第一次对这种结构的整体意义有了一些了解。McDuff工作的一个重点是对能量概念的几何探索。在物理学中，这是运动（从行星系统到等离子体系统）的驱动力。辛拓扑的新方法使得在某些情况下估计完成某一运动所需的最小能量成为可能。这项工作是非常基础的，最终将对物理系统的研究产生重要影响。她工作的另一个重点是试图通过研究辛空间的几何性质来理解辛结构到底是什么。这个想法是，这些空间是辛结构的抽象概念的体现。洛厄尔·琼斯：在1829年和1832年，数学家n·罗巴切夫斯基和j·波耶通过验证欧几里得平面几何的第五个公理独立于欧几里得的前四个公理，独立地解决了一个两千年前的数学问题。他们通过研究一种具有“负曲率”的新型几何来做到这一点。今天，Lobachevsky和Bolyai的二维几何是具有“非正曲率”的更大一类几何结构的重要特例，它可以出现在所有高维中，并且在过去的四十年中一直是大量研究的焦点。这项研究的一个重要主题是，每一个这样的几何结构都有一个相关的代数结构（称为“基本群”），它作为一种潜在几何结构的遗传密码。虽然基本群并不总是决定底层几何结构，但在某些特殊情况下这是正确的。Jones最近证实，基本群总是完全决定底层几何结构的“拓扑性质”。换句话说，琼斯已经证明，基本群决定了底层几何结构的大致轮廓，尽管它并不能决定几何结构的所有细节。＊＊＊