Curvature and Topology

[803188]研究者研究流形的曲率与拓扑之间的关系。特别地，他继续他关于正标量曲率的度量的存在性和分类（直到一致性）的工作。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，它只取决于所考虑的流形M的维数，它的基本群，以及它的第一和第二Stiefel-Whitney类(例如。，这个群自由地、传递地作用于M)上的正标量曲率度量集。R的作用让人联想到Wall's手术阻塞群在分类流形直至微分同态中的作用。研究者希望证明R的周期性包含与数据相关的C*-代数的k理论的直接求和。他希望通过利用最小超曲面方法研究正标量曲率度量来找到R的“非周期”部分的不变量。对于其他类型的曲率，研究者正在继续他的猜想，即具有消失第一庞特里亚金类的自旋流形M上正Ricci曲率的度规的存在意味着M的Witten格为零。他希望通过在M的自由环空间上构造狄拉克型算子并分析其Weitzenbock公式来证明这一猜想。将流形M的几何特征与其拓扑联系起来是全局几何中的核心问题。这里的“几何”是指流形的数量特征；例如，将实数s(x)与M中的任意点x相关联的标量曲率；这个数字是测量x周围小球的体积与欧几里德空间中半径相同的小球的体积的比较。M的“拓扑”捕获了定性特征（如孔的数量）；当M“变形”（想象挤压气球）时，它们不会改变。通常，如在标量曲率的情况下，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。通过研究M中无限维空间上的环路上的类似微分方程，可以获得M的几何和拓扑之间的新关系，这是一种令人兴奋的可能性；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').***