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). ***
