The investigator studies the relationship between curvature and the topology of manifolds. In particular, he continues his work concerning the existence and the classification (up to concordance) of metrics of positive scalar curvature. Some of his earlier work shows that these questions boil down to determining an abelian group R that depends 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 positive scalar curvature metrics on M). The role of R is reminiscent of the role of Wall's surgery obstruction group in the classification of manifolds up to diffeomorphisms. The investigator hopes to show that a periodic version of R contains as a direct summand the K-Theory of a C*-algebra associated to the data. He hopes to find invariants for the `non-periodic' part of R by exploiting the minimal hypersurface method for studying positive scalar curvature metrics. Concerning other types of curvature, the investigator is pursuing his conjecture that the existence of a metric of positive Ricci curvature on a spin manifold M with vanishing first Pontryagin class implies that the Witten genus of M is zero. He hopes to prove this conjecture by constructing a Dirac type operator on the free loop space of M and analysing its Weitzenbock formula. Relating geometric features of a manifold M to its topology is the central problem in global geometry. Here `geometric' refers to quantitative features of the manifold; e.g., the scalar curvature that associates a real number s(x) to any point x in M; this number is a measure for how the volume of small balls around x compares to the volume 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 between the geometry and the topology of a manifold is made by studying differential equations on M. It is an exciting possibility that new relations between the geometry and the topology of M might be obtained by studying similar differential equations on the infinite dimensional space of loops in M; this makes contact with stochastic analysis and with physics (`string theory'). ***
