Curvature and Topology

小行星9803188 研究了流形的曲率与拓扑的关系. 特别是，他继续他的工作有关的存在和分类（一致性）ofmetrics的正标量曲率。 他早期的一些工作表明，这些问题归结为确定一个阿贝尔群R，它只依赖于所考虑的流形M的维数，它的基本群，以及它的第一和第二Stiefel-Whitney类（例如，这个群在M上的正阶曲率度量集上自由地和传递地作用。 R的作用是让人想起的roleof墙的手术梗阻组在分类的流形高达cross-morphisms。 研究者希望证明，R的一个新版本包含与数据相关的C*-代数的K-理论作为直接和项。 他希望找到不变量的“非周期性”的一部分，R利用最小超曲面方法研究正标量曲率度量。 关于其他类型的曲率，研究者正在进行他的猜想：在第一Pontryagin类为零的自旋流形M上存在正Ricci曲率的度量蕴涵着M的维滕亏格为零。 他希望通过在M的自由圈空间上构造一个Dirac型算子并分析其Weitzenbock公式来证明这个猜想。 将流形M的几何特征与其拓扑联系起来是整体几何的中心问题。 这里的“几何”是指流形的数量特征;例如，将一个真实的数s（x）与M中任意一点x相关联的标量曲率;这个数是衡量x周围小球的体积与欧几里得空间中相同半径的球的体积相比的尺度。 M的“拓扑”捕捉了定性的特征（如洞的数量）;当M "变形“（想象一下挤压气球）时，这些特征不会改变。通常，就像在标量曲率的情况下一样，流形的几何和拓扑之间的基本联系是通过研究M上的微分方程来建立的。 一个令人兴奋的可能性是，通过研究M中无限维空间上的类似的微分方程，可能会得到M的几何和拓扑之间的新关系;这与随机分析和物理学（“弦理论”）有联系。

9803188Stolz The investigator studies the relationship between curvature andthe topology of manifolds. In particular, he continues his workconcerning the existence and the classification (up to concordance) ofmetrics of positive scalar curvature. Some of his earlier work showsthat these questions boil down to determining an abelian group R thatdepends only on the dimension of the manifold M under consideration,its fundamental group, and its first and second Stiefel-Whitney classes(e.g., this group acts freely and a transitively on the set of positivescalar curvature metrics on M). The role of R is reminiscent of the roleof Wall's surgery obstruction group in the classification of manifolds upto diffeomorphisms. The investigator hopes to show that a periodicversion of R contains as a direct summand the K-Theory of a C*-algebraassociated to the data. He hopes to find invariants for the `non-periodic'part of R by exploiting the minimal hypersurface method for studyingpositive scalar curvature metrics. Concerning other types of curvature,the investigator is pursuing his conjecture that the existence of ametric of positive Ricci curvature on a spin manifold M with vanishingfirst Pontryagin class implies that the Witten genus of M is zero. Hehopes to prove this conjecture by constructing a Dirac type operator onthe free loop space of M and analysing its Weitzenbock formula. Relating geometric features of a manifold M to its topology is thecentral problem in global geometry. Here `geometric' refers toquantitative features of the manifold; e.g., the scalar curvature thatassociates a real number s(x) to any point x in M; this number is ameasure for how the volume of small balls around x compares to thevolume of a ball of the same radius in Euclidean space. The `topology'of M captures the qualitative features (like the number of holes);these don't change when M is `deformed' (think of squeezing a balloon).Often, as in the case of scalar curvature, the basic connection betweenthe geometry and the topology of a manifold is made by studyingdifferential equations on M. It is an exciting possibility that newrelations between the geometry and the topology of M might be obtainedby studying similar differential equations on the infinite dimensionalspace of loops in M; this makes contact with stochastic analysis andwith physics (`string theory').***