The hodge de rham theory of relative malcev completion1

Lecture Notes on De Rham–Hodge Theory

These are lecture notes for a 1–semester undergraduate course (in mathematics, physics, engineering, chemistry or biology) on the De Rham–Hodge theory. This landmark theory of the 20th Century mathematics gives a rigorous foundation to modern field and gauge theories in physics, engineering and physiology. The only necessary background for comprehensive reading of these notes is Green’s theorem...

Lectures on the Hodge-de Rham Theory of the Fundamental Group of P − {0, 1,∞}

Hodge Theory in the Sobolev Topology for the De Rham Complex

The authors study the Hodge theory of the exterior differential operator d acting on q-forms on a smoothly bounded domain in R, and on the half space R + . The novelty is that the topology used is not an L 2 topology but a Sobolev topology. This strikingly alters the problem as compared to the classical setup. It gives rise to a boundary value problem belonging to a class of problems first intr...

De Rham Cohomology and Hodge Decomposition for Quantum Groups

ISTVÁN HECKENBERGER and AXEL SCHÜLER Abstra t Let Γ = Γτ,z be one of the N -dimensional bicovariant first order differential calculi for the quantum groups GLq(N), SLq(N), SOq(N), or Spq(N), where q is a transcendental complex number and z is a regular parameter. It is shown that the de Rham cohomology of Woronowicz’ external algebra Γ coincides with the de Rham cohomologies of its left-coinvar...

Semi-Infinite de Rham Theory

I shall start by explaining just what “semi-infinite” means and how it gives rise to interesting structures in geometry. These structures are studied as part of String Theory and are closely related to Floer Theory. Manifolds which carry a “semi-infinite” structure include loop spaces (C∞(S1, M) for M a closed manifold) and the theory seems to have particular simplicities when the original mani...