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Motivated by our attempt to recast Cartans work on Lie pseudogroups in a more global and modern language, we are brought back to the question of understanding the linearization of multiplicative forms on groupoids and the corresponding integrability problem. From this point of view, the novelty of this paper is that we study forms with coefficients. However, the main contribution of this paper is conceptual: the finding of the relationship between multiplicative forms and Cartans work, which provides a completely new approach to integrability theorems for multiplicative forms. Back to Cartan, the multiplicative point of view shows that, modulo Lies functor, the Cartan Pfaffian system (itself a multiplicative form with coefficients!) is the same thing as the classical Spencer operator.
We define and study multiplicative connections in the tangent bundle of a Lie groupoid. Multiplicative connections are linear connections satisfying an appropriate compatibility with the groupoid structure. Our definition is natural in the sense that
A differential 1-form $alpha$ on a manifold of odd dimension $2n+1$, which satisfies the contact condition $alpha wedge (dalpha)^n eq 0$ almost everywhere, but which vanishes at a point $O$, i.e. $alpha (O) = 0$, is called a textit{singular contact
It is shown that the theory of real symmetric second-order elliptic operators in divergence form on $Ri^d$ can be formulated in terms of a regular strongly local Dirichlet form irregardless of the order of degeneracy. The behaviour of the correspondi
Extracting parton distribution functions (PDFs) from lattice QCD calculation of quasi-PDFs has been actively pursued in recent years. We extend our proof of the multiplicative renormalizability of quasi-quark operators in Ref. [1] to quasi-gluon oper
These notes are the first chapter of a monograph, dedicated to a detailed proof of the equivariant index theorem for transversally elliptic operators. In this preliminary chapter, we prove a certain number of natural relations in equivariant cohomolo