Mathematical Sciences: Curvature and Topology

9504418主要研究者Stolz研究了哪些流形具有正标量(分别为Ricci)曲率的黎曼度量的问题。根据Gromov-Lawson-Rosenberg猜想，自旋流形接纳正标量曲率的度规当且仅当“指数”障碍为零。对于基本群具有周期上同调的自旋流形，这一猜想已被主要研究者和他的合著者证明。此外，对于具有有限基本群的流形，已经证明了该猜想的“稳定版本”(在“稳定版本”中，流形允许用其具有足够多副本的“Bott-流形”的乘积来代替该流形，“Bott-流形”是代表实K-理论中的周期元素的八维自旋流形)。主要研究人员致力于证明基本群满足Baum-Connes猜想的所有自旋流形的“稳定”猜想。此外，他怀疑Gromov-Lawson-Rosenberg猜想并不普遍成立，并试图找到阻碍正标量曲率度量存在的新的“不稳定”障碍。关于正Ricci曲率，调查者正在寻求对他的猜想的一个证明，即在具有消失的第一Pontryagin类的自旋流形上存在正Ricci曲率度量意味着它的Witten亏格的消失。这涉及到狄拉克算符在这个流形的自由环空间上的“Weitzenboeck公式”。这些项目符合试图将流形的拓扑(关于其全局形状的定性信息)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(如球或轮胎的表面)，一种很好的分类早已为人所知：两个这样的表面具有相同的拓扑(即，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”(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，如果一个曲面具有正曲率，即每个边为测地线(最短曲线)的三角形的角度和大于180度，则该曲面具有与球的曲面相同的拓扑。将这些结果推广到更高维的流形是现代数学的一个主要目标(例如，我们的宇宙是一个3维的流形，爱因斯坦的时空有4维，10维的流形。26，在试图统一四种基本力量的理论物理学中发挥关键作用)。***

9504418 Stolz The principal investigator studies the question of which manifolds admit Riemannian metrics of positive scalar (respectively Ricci) curvature. According to the Gromov-Lawson-Rosenberg conjecture, a spin manifold admits a metric of positive scalar curvature if and only if an `index' obstruction vanishes. This conjecture has been proved (by the principal investigator and his coauthors) for spin manifolds whose fundamental groups have periodic cohomology. Moreover, a "stable version" of this conjecture has been proved for manifolds with finite fundamental groups (in the "stable version" one allows the manifold to be replaced by its product with sufficiently many copies of the "Bott-manifold," an eight-dimensional spin manifold that represents the periodicity element in real K-theory). The principal investigator works on proving the "stable" conjecture for all spin manifolds whose fundamental groups satisfy the Baum-Connes conjecture. Moreover, he suspects that the Gromov-Lawson-Rosenberg conjecture does not hold in general, and tries to find new "unstable" obstructions to the existence of positive scalar curvature metrics. 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 its Witten genus. This involves a "Weitzenboeck formula " for the Dirac operator on the free loop space of this manifold. 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). For 2-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 (i.e., they can be deformed into each other if we think of them as being made of thin rubber) if and only if they have the same number of "holes" (the surface of a ball has 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 (e.g., our universe is a manifold of dimension 3, Einstein's space-time has dimension 4, and manifolds of dimension 10, resp. 26, play a crucial role in the theoretical physics of the attempted unification of the four fundamental forces). ***