Hodge Theory for Polymatroids

Hodge Theory for Combinatorial Geometries

The matroid is called loopless if the empty subset of E is closed, and is called a combinatorial geometry if in addition all single element subsets of E are closed. A closed subset of E is called a flat of M, and every subset of E has a well-defined rank and corank in the poset of all flats of M. The notion of matroid played a fundamental role in graph theory, coding theory, combinatorial optim...

Hodge Theory for R- Manifolds

Let X be an R-fold, and let π : E −→ X be a real vector bundle, of rank r, equipped with a positive definite symmetric bilinear form. If e1, . . . , er ∈ π −1(X) are orthonormal, then e1 ∧ · · · ∧ er is a non-trivial vector in ∧r E. Proposition: If f1, . . . , fr is any other orthonormal basis for π −1(X), then e1 ∧ · · · ∧ er = ±f1 ∧ · · · ∧ fr. Proof. Note that fi = g · ei for g ∈ O(r), so de...

Hodge Theory and Representation Theory

Gauge Theory as Hodge Theory

In the framework of an extended BRST formalism, it is shown that the four (3 + 1)-dimensional (4D) free Abelian 2-form (notoph) gauge theory presents an example of a tractable field theoretical model for the Hodge theory.

Hodge-Tate Theory

This thesis aims to expose the amazing sequence of ideas, concerning p-adic representations coming from geometry, that form the heart of what was called Hodge-Tate theory. This subject, initiated by Tate in the late ’60s in analogy to classical Hodge theory, leads in to the now vast and highly fruitful program of p-adic Hodge Theory. The central result of the theory is the Hodge-Tate decomposit...