Finite Geometries and the Fundamental Theorem of Calculus

A finite geometry is defined by a set of points and certain subsets of this set, corresponding to lines, planes, and so on. An arbitrary subset is called independent if it is in general position with respect to these lines, planes, etc. This talk is about the relation between finite geometries and multivariate versions of the fundamental theorem of calculus. The points of such a geometry correspond to the set of coordinate directions. For each such geometry there is a version of the fundamental theorem of calculus that expresses the value of a function of several variables as a sum of multiple integrals of mixed partial derivatives. The terms in the sum are indexed by the independent subsets of the geometry. In the application to mathematical physics the points of the geometry correspond to the edges of a graph. There are at least two important geometries associated with the graph. Fundamental theorem of calculus formulas obtained in the graphical context play a role in proving convergence of cluster expansions. In particular, they include the fundamental result of Brydges-Kennedy-Abdesselam-Rivasseau.