Mathematical Sciences: Curvature and Topology

Stolz首席研究员研究了流形是否承认正标量（分别为里奇）曲率的黎曼度量的问题。根据Gromov-Lawson-Rosenberg猜想，当且仅当“指标”障碍消失时，自旋流形允许一个正标量曲率度规。对于基本群具有周期上同调的自旋流形，这个猜想已经被（由首席研究员和他的合作者）证明了。此外，对于具有有限基本群的流形，已经证明了这个猜想的“稳定版本”（在“稳定版本”中，人们允许流形被它的乘积取代，该乘积具有足够多的“bot流形”副本，“bot流形”是一种代表实际k理论中周期性元素的八维自旋流形）。首席研究员致力于证明所有基本群满足Baum-Connes猜想的自旋流形的“稳定”猜想。此外，他怀疑Gromov-Lawson-Rosenberg猜想在一般情况下并不成立，并试图寻找新的“不稳定”障碍，以阻止正标量曲率度量的存在。关于正Ricci曲率，研究者正在寻求证明他的猜想，即在一个第一Pontryagin类消失的自旋流形上存在一个正Ricci曲率度量意味着它的Witten属的消失。这涉及到该流形的自由循环空间上的狄拉克算子的“Weitzenboeck公式”。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）和几何（关于其局部形状的定量信息）联系起来的一般框架。二维流形(如球的表面或轮胎),一个不错的分类已经认识很长一段时间:两个这样的表面具有相同的拓扑结构(也就是说,他们可以变形为彼此,如果我们认为他们是用橡胶制成的薄)当且仅当他们有相同数量的“洞”(一个球的表面没有漏洞,轮胎的表面或杯有一个洞,和椒盐卷饼有两个孔)。此外，如果一个曲面具有“正曲率”，即每个三角形的边是测地线（最短曲线）的角和大于180度，则该曲面具有与球表面相同的拓扑结构。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，我们的宇宙是一个维数为3的流形，爱因斯坦的时空是一个维数为4的流形，以及维数为10的流形）。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)。* * *

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). ***