Curvature and Topology

DMS-0104077 Stephan Stolz主要研究者提出研究哪些流形允许正标量黎曼度量和Ricci曲率黎曼度量的问题。关于第一个问题，主要研究者以前的工作表明，这些问题归结为计算只依赖于维数的阿贝尔群，这些度量存在于流形的基本群和前两个Stiefel-Whitney类中，它们在正标量曲率度量的存在性/分类问题中所起的作用类似于Wall的外科手术阻塞群在正标量曲率度量的存在性/分类问题中所起的作用。光滑结构的分类问题 主要研究者希望通过研究各种映射来获得对这些群的代数/函子理解，这些映射将这些群与群的分类空间的同调/K-理论/连接K-理论以及相关群C*-代数的K-理论联系起来。关于正的Ricci曲率，研究者正在寻求证明他的猜想：在第一Pontryagin类为零的自旋流形上存在正的Ricci曲率度量意味着它的Witten亏格为零。这个不变量来自弦理论的考虑。启发性地，它是一个尚未严格定义的“狄拉克算子”在这个流形的自由循环空间的指数。迈克霍普金斯描述了同伦理论的方式来定义维滕属的家庭这样的流形生活在椭圆上同调的参数空间。用弦理论家分析的对象来表达椭圆上同调和维滕亏格族，从而给出霍普金斯构造的几何解释，这是一个非常有趣的挑战，这些项目符合试图将流形的拓扑（关于其整体形状的定性信息）与其几何（关于其局部形状的定量信息）联系起来的一般框架。对于二维流形（就像球或轮胎的表面一样），一个很好的分类已经知道很久了：两个这样的曲面具有相同的拓扑（也就是说，如果我们认为它们是由薄橡胶制成的，它们可以相互变形）当且仅当它们具有相同数量的“孔”时（球的表面没有孔，轮胎或杯子的表面有一个孔，椒盐卷饼有两个孔）。 此外，如果一个表面具有“正曲率”，在这个意义上说，在每个三角形的角度总和，其边缘是测地线（最短曲线）是大于180度，那么这个表面具有相同的拓扑结构作为一个球的表面。现代数学的一个主要目标是将这些结果推广到高维流形。 例如，我们的宇宙是三维的流形，爱因斯坦的时空是四维的，而流形的维度分别是10，26在试图统一四种基本力的理论物理学中起着至关重要的作用。

DMS-0104077Stephan Stolz The principal investigator proposes to study the questionwhich manifolds admit Riemannian metrics of positive scalarand Ricci curvature, respectively. Concerning thefirst question, previous work of the principalinvestigator shows that these questions boil down tocomputing abelian groups which depend only on thedimension, the fundamental group and the first twoStiefel-Whitney classes of the manifold these metricslive on. The role these groups play in the existence/classification problem for positive scalarcurvature metrics is analogous of the role of Wall'ssurgery obstruction groups in the existence/classificationproblem for smooth structures. The principal investigator hopes to gain an algebraic/functorial understanding of these groups by studying various maps that relate these groups with homology/K-theory/connective K-theory of classifying spaces of groups, and with the K-theory of theassociated group C*-algebras. Concerning positive Ricci curvature, the investigator is pursuing a proof of his conjecture that the existence of a positive Ricci curvature metric on a spin manifold with vanishing first Pontryagin class implies the vanishing of itsWitten genus. This invariant arose from considerations in string theory. Heuristically it is the index of a yet to be rigorously defined "Dirac operator" on the free loop space of this manifold. Mike Hopkins has described a homotopy theoretic way to define the Witten genus of a family of such manifolds which lives in the elliptic cohomology of the parameter space. It is a very interesting challenge to express the elliptic cohomology and thefamily Witten genus in terms of the objects that string theorists are analysing and thus to give a geometric interpretation of Hopkins construction.These projects fit in the general framework of trying to relate the topology of a manifold (qualitative information about its global shape) and its geometry (quantitative information about its local shape). For2-dimensional manifolds (like the surface of a ball or a tire), a nice classification has been known for a long time: Two such surfaces have the same topology (that is, they can be deformed into each other if we think ofthem as being made of thin rubber) if and only if they have the same number of `holes' (the surface of a ball no holes, the surface of a tire or a cup has one hole, and a pretzel has two holes). Moreover, if a surface has`positive curvature' in the sense that the angle sum in each triangle whose edges are geodesics (shortest curves) is larger than 180 degrees, then this surface has the same topology as the surface of a ball. It is a major goal of modern day mathematics to generalize these results to higher dimensional manifolds. For example, our universe is a manifold of dimension 3, Einstein's space-time has dimension 4, and manifolds of dimension 10, respectively, 26 play a crucial role in the theoretical physics of the attempted unification of the four fundamental forces.