New Directions in Hodge Theory
DescriptionHodge studied fundamental relationships between geometry, topology, and partial differential equations by extending the harmonic analysis of Fourier to higher order linear differential operators on manifolds. For many new applications of mathematics, as in statistics, data analysis, intelligence and learning, broader setting for a new Hodge theory is needed. The purpose of this project is to extend the classical Hodge theory to the quite general and abstract setting of metric spaces with a probability measure. Compared with some previous advances such as combinatorial Hodge theory, our framework and results are new, relying on integral equations to replace the differential equations and moreover apply at each finite fixed scale. We shall establish conditions for the co-boundary operator to have closed image, and study the related full Hodge decompositions.
|Effective start/end date||1/01/11 → 11/03/14|