Mathematical Sciences: Topology and Manifolds

9401443 McDuff Dusa McDuff：辛拓扑的研究在过去十年中取得了很大的进展，现在已经足够发展到在其他领域产生重大影响，例如在研究三维和四维流形中使用Floer同调，以及最近J-全纯曲线在镜像对称和量子上同调研究中的应用。近年来，McDuff致力于发展辛拓扑工具，主要目的是了解紧辛流形的拓扑结构。她打算在未来三年继续朝着同样的大方向努力，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，研究经典闭合球面流形的刚性性质。在以前的工作中，他们已经证明了闭的非正曲线流形和闭的双陪集流形在5维上是拓扑刚性的。在工作中，他们证明了任何闭的完备仿射平坦流形在5维上是拓扑刚性的。他们还构造了许多严格负曲线闭流形的例子，它们彼此同胚但不同胚，从而证明了这类非正曲线流形不具有光滑刚性。闭双陪集流形类或闭完全仿射平坦流形类是否光滑刚性仍然是Jones打算追求的一个悬而未决的问题。Dusa McDuff：辛结构是一种非常基本的几何结构，它是经典物理和量子物理所有方程的基础。在过去的十年里，新的工具已经被开发出来，使数学家们第一次能够对这种结构的全球意义有一些了解。麦克达夫的工作重点之一是对能量概念的几何解释。在物理学中，这是运动的驱动力(从行星系统到等离子体的各种系统)。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，最终应该会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什么是辛结构。我们的想法是，这些空间是辛结构抽象概念的体现。Lowell Jones：在1829和1832年间，数学家N.Lobachevsky和J.Bolyai通过验证欧几里得关于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。他们通过研究一种新型的具有“负曲率”的几何学来做到这一点。今天，Lobachevsky和Bolyai的二维几何是具有“非正曲率”的更大类几何结构的重要特例，这种几何结构可以出现在所有更高的维度上，并且在过去的40年里一直是大量研究的焦点。这项研究的一个重要主题是，每个这样的几何结构都有一个相关的代数结构(称为“基本群”)，它充当潜在几何结构的一种遗传密码。虽然基本群总是决定基本几何结构这一点远非正确，但在某些特殊情况下这是正确的。琼斯最近证实，基本群总是完全决定基本几何结构的“拓扑性”。为了解释这一点，琼斯已经证明了基本群决定了基本几何结构的大致轮廓，尽管它并不决定几何结构的所有细节。***

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. ***