A gauge-invariant form of the nonlinear Hodge equations is studied. 1991 MSC: 58E15 (Classical field theory)

Mixed finite element methods solve a PDE using two or more variables. The theory of Discrete Exterior Calculus explains why the degrees of freedom associated to the different variables should be stored on both primal and dual domain meshes with a discrete Hodge star used to transfer information between the meshes. We show through analysis and examples that the choice of discrete Hodge star is e...

§0. Introduction §1. The Étale Integral Structure on the Universal Extension §2. The Étale Integral Structure for an Ordinary Elliptic Curve §2.1. Some p-adic Function Theory §2.2. The Verschiebung Morphism §3. Compactified Hodge Torsors §4. The Étale Integral Structure on the Hodge Torsors §4.1. Notation and Set-Up §4.2. Degenerating Elliptic Curves §4.3. Ordinary Elliptic Curves §4.4. The Gen...

This is an elementary introduction to the Hodge Laplacian on a graph, a higher-order generalization of the graph Laplacian. We will discuss basic properties including cohomology and Hodge theory. At the end we will also discuss the nonlinear Laplacian on a graph, a nonlinear generalization of the graph Laplacian as its name implies. These generalized Laplacians will be constructed out of coboun...

In the present paper, which is the second in a series of four papers, we study the Kummer theory surrounding the Hodge-Arakelovtheoretic evaluation — i.e., evaluation in the style of the scheme-theoretic HodgeArakelov theory established by the author in previous papers — of the [reciprocal of the l-th root of the] theta function at l-torsion points, for l ≥ 5 a prime number. In the first paper ...

Let π : X → T be a family of smooth projective complex varieties. Assume X , π, T are defined over Q. An immediate consequence of the fact that there are only countably many components of the relative Hilbert scheme for π, and that the relative Hilbert scheme (with fixed Hilbert polynomial) is defined over Q, is the following: if the Hodge conjecture is true, the components of the Hodge locus i...

It is shown that any compact Kähler manifold M gives canonically rise to two strongly homotopy algebras, the first one being associated with the Hodge theory of the de Rham complex and the second one with the Hodge theory of the Dolbeault complex. In these algebras the product of two harmonic differential forms is again harmonic. If M happens to be a Calabi-Yau manifold, there exists a third st...