This paper gives an account of the unitary representations of the braid group that arise via the Hodge theory of cyclic branched coverings of P, highlighting their connections with ergodic theory, complex reflection groups, moduli spaces of 1-forms and open problems in surface topology.

This is a continuation of [10], where it was shown that K-equivalent complex projective manifolds have the same Betti numbers by using the theory of p-adic integrals and Deligne’s solution to the Weil conjecture. The aim of this note is to show that with a little more book-keeping work, namely by applying Faltings’ p-adic Hodge Theory, our p-adic method also leads to the equivalence of Hodge nu...

Hodge theory is a beautiful synthesis of geometry, topology, and analysis, which has been developed in the setting of Riemannian manifolds. On the other hand, spaces of images, which are important in the mathematical foundations of vision and pattern recognition, do not fit this framework. This motivates us to develop a version of Hodge theory on metric spaces with a probability measure. We bel...

In the present paper, which is the second in a series of four papers, we study theKummer theory surrounding the Hodge-Arakelov-theoretic evaluation — i.e., evaluation in the style of the scheme-theoretic Hodge-Arakelov theory established by the author in previous papers — of the [reciprocal of the lth root of the] theta function at l-torsion points [strictly speaking, shifted by a suitable 2-to...

Abstract We develop a Fredholm theory for the Hodge Laplacian in weighted spaces on ALG ∗ manifolds dimension four. then give several applications of this theory. First, we show existence harmonic functions with prescribed asymptotics at infinity. A corollary is non-existence result non-negative Ricci curvature having group <m:mi ma...