Given a manifold and a Riemannian metric, the investigator has defined and intends to study a real-valued linking number of a pair of cycles generalizing the ordinary integer topological linking number. This pairing is a "lift" to the real numbers of the Abel-Jacobi homomorphism in Algebraic Geometry. The initial definition is by differential forms with suitable singularities (going back to Gauss in Euclidean 3-space). The investigator studies an exactly analogous complex linking number (originally due to Beilinson and Bloch) called the (Archimedean) Height Pairing. This pairing enters into their conjectures relating cycles and values of L-functions, but it has been possible to calculate it in extremely few cases. The investigator will consider an equivalent construction of the pairings based on the kernel of the heat operator. It turns out that this new approach is well-suited for obtaining concrete formulas in some cases, e.g., cycles introduced by B. Gross and C. Schoen involving correspondences on a product of three curves. The investigator would like to use this approach in many more cases coming from Algebraic and Riemannian Geometry, and also to find relations with index theorems and with Quantum Field Theory. Basically, the investigator and his student Bin Wang will attempt to make more accessible a previously defined invariant of Riemannian manifolds. Their approach will be two-fold. First, they will explore an alternative to the former abstract definition, one which is rooted in geometric and physical concepts and so may provide more insight. Second, they will use this approach to calculate the invariant explicitly in more cases, again in the hope of assisting insight into its true meaning.
