Mathematical Sciences: Curvature and Topology

研究者将继续研究流形的曲率和拓扑之间的关系。特别是，他将继续研究正标量曲率度量的存在性和一致性分类。他最近证明了如果一个闭流形M允许一个正的标量曲率度量，那么这些度量的调和类是双射对应于一个只依赖于M的基本群和前两个Stiefel- Whitney类的bordim群，并且M允许一个正的标量曲率度量当且仅当M在这个bordim群中表示0。如果M的全称覆盖是自旋，那么这个泛群映射到M的基本群的“扭曲”群C*-代数的实k理论，而扭曲是由M的前两个Stiefel- Whitney类决定的。一个乐观的希望是，这实际上是一个同构。研究者打算继续研究正Ricci曲率、椭圆属和椭圆同调之间的关系；特别是，为了追求他的猜想，一个正Ricci曲率流形M的Witten格在M是自旋且它的第一个Pontrjagin类为零的情况下消失。在一个不同的方向上，他希望证明具有四个不同特征值的Dupin超曲面的所有可能的多重性都是通过已知的例子来实现的，即齐次例子和Clifford例子。拓扑学研究的是那些不依赖于距离或角度的几何物体的属性，这些属性是如此基本，以至于在拉伸和弯曲物体后，它们仍然存在，而不会撕裂它。从拓扑学的意义上说，留声机唱片和结婚戒指是一样的，因为没有拓扑学的性质来区分它们。事实上，它们在拓扑结构上都和咖啡杯一样。它们的形状大不相同，但由于它们具有相同的拓扑结构，因此它们具有一些相同的几何特性。著名的高斯-邦尼定理，在电磁理论和微分几何中都很有名，它要求任意曲面的总（高斯）曲率为零。曲率是一个非常非拓扑的性质，局部的曲率在这些曲面上可以有任何值，但是当一个人从一个点移动到另一个点时，这些值不可能是完全无关的，因为定理说如果用面积元素来加权，它们的和都是零。在这个项目寻求的结果中，突出的是这个奇妙定理的现代变体，适用于其他维度和其他曲率的流形。

The investigator will continue to study the relationship between the curvature and the topology of manifolds. In particular, he will continue to study the existence and concordance classification of positive scalar curvature metrics. He has recently shown that if a closed manifold M admits a positive scalar curvature metric, then the concordance classes of such metrics are in bijective correspondence to a certain bordism group which depends only on the fundamental group and the first two Stiefel- Whitney classes of M. Also, M admits a positive scalar curvature metric if and only if M represents zero in this bordism group. If the universal cover of M is spin, this bordism group maps to the real K-theory of the 'twisted' group C*-algebra of the fundamental group of M, the twist being determined by the first two Stiefel- Whitney classes of M. An optimistic hope is that this is in fact an isomorphism. The investigator intends to continue his study of the relations between positive Ricci curvature, elliptic genera and elliptic homology; in particular, to pursue his conjecture that the Witten genus of a positive Ricci curvature manifold M vanishes provided M is spin and its first Pontrjagin class is zero. In a different direction, he hopes to show that all the possible multiplicities of a Dupin hypersurface with four distinct eigenvalues are realized by the known examples, i.e. the homogeneous examples and the Clifford examples. Topology treats those properties of geometric objects which are not dependent upon distances or angles, which are so fundamental that they persist after stretching and bending an object short of tearing it. In a topological sense, a phonograph record and a wedding ring are the same, for no topological properties distinguish one from the other. In fact, each is topologically the same as a coffee cup. Their shapes are wildly different, and yet there are some geometric properties that all will have in common as a result of their common topology. The famous Gauss-Bonnet theorem, well-known in electromagnetic theory as well as differential geometry, requires that the total (Gaussian) curvature of any of their surfaces will be zero. Now curvature is a very non-topological property, and locally it can have any value whatsoever on one of these surfaces, but the values cannot be totally unrelated as one moves from point to point, for the theorem says they will all sum to zero if one weights them by elements of area. Prominent among the results this project seeks are modern variants of this wonderful theorem for manifolds of other dimensions and for other kinds of curvature.