Curvature and Topology

Stolz研究流形的曲率与拓扑之间的关系。特别是，他继续了他关于正标量曲率的度量的存在和分类(直到协调)的工作。他早期的一些工作表明，这些问题归结为确定一个只依赖于所考虑的流形M的维度的交换群R，它的基本群，以及它的第一和第二Stiefel-Whitney类(例如，这个群自由地作用于M上的正集和传递的正集合上的数量曲率度量)。R的作用让人想起Wall的手术梗阻组在流形分类中的作用，直到微分同胚。研究者希望证明R的周期版本包含与数据相关的C*-代数的K-理论作为直接求和。他希望通过利用极小超曲面方法来研究正标量曲率度量，从而找到R的“非周期”部分的不变量。关于其他类型的曲率，研究者继续他的猜想，即第一庞特里亚金类为零的自旋流形M上存在正Ricci曲率的参变量意味着M的Witten亏格为零。他希望通过在M的自由环空间上构造一个狄拉克算子并分析它的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整体几何中的中心问题。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联系起来的标量曲率；这个数字是衡量x周围的小球的体积与欧氏空间中相同半径的球的体积的比较。M的“拓扑”捕捉了定性的特征(如孔洞的数量)；当M“变形”时，这些特征不会改变(想想挤压气球)。通常，就像标量曲率的情况一样，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。令人兴奋的是，通过研究M中无限维环空间上的类似的微分方程，可以获得M的几何和拓扑之间的新关系；这使得随机分析和物理(‘弦理论’)相联系。*

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').***