Hodge theory in combinatorics

Hodge theory in combinatorics

If G is a finite graph, a proper coloring of G is a way to color the vertices of the graph using n colors so that no two vertices connected by an edge have the same color. (The celebrated four-color theorem asserts that if G is planar, then there is at least one proper coloring of G with four colors.) By a classical result of Birkhoff, the number of proper colorings of G with n colors is a poly...

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.

Combinatorics of binomial decompositions of the simplest Hodge integrals

We reduce the calculation of the simplest Hodge integrals to some sums over decorated trees. Since Hodge integrals are already calculated, this gives a proof of a rather interesting combinatorial theorem and a new representation of Bernoulli numbers.

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...