Hodge-de Rham theory on fractal graphs and fractals

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,∞}

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...

On Quantum De Rham Cohomology Theory

We define the quantum exterior product ∧h and quantum exterior differential dh on Poisson manifolds. The quantum de Rham cohomology, which is a deformation quantization of the de Rham cohomology, is defined as the cohomology of dh. We also define the quantum Dolbeault cohomology. A version of quantum integral on symplectic manifolds is considered and the corresponding quantum Stokes theorem is ...

Hodge-deRham Theory of K-Forms on Carpet Type Fractals

We outline a Hodge-deRham theory of K-forms ( for k=0,1,2) on two fractals: the Sierpinski Carpet(SC) and a new fractal that we call the Magic Carpet(MC), obtained by a construction similar to that of SC modified by sewing up the edges whenever a square is removed. Our method is to approximate the fractals by a sequence of graphs, use a standard Hodge-deRham theory on each graph, and then pass ...