## More on Ringed Complexes

The results mentioned in the slides before are now written down. This document contains a proof of the energy relation . There are several reason for setting … ….

The results mentioned in the slides before are now written down. This document contains a proof of the energy relation . There are several reason for setting … ….

The energy theorem for Euler characteristic X= sum h(x)was to express it as sum g(x,y)of Green function entries. We extend this to Wu characteristic w(G)= sum h(x) h(y) over intersecting sets. The new formula is w(G)=sum w(x) w(y) g(x,y)^{2}, where w(x) =1 for even dimesnional x and w(x)=-1 for odd dimensional x.

The energy theorem for simplicial complexes equipped with a complex energy comes with some surpises.

Introduction The idea to base physics on the evolution finite set of sets is intriguing. It has been tried as an approach to quantum gravity. Examples are … ….

The question In discrete Poincare-Hopf for graphs the question appeared how to generalize the result from gradient fields to directed graphs. The paper mentions already the problem … ….

The Mickey mouse theorem assures that a connected positive curvature graph of positive dimension is a sphere.

The simplest construct in mathematics is probably a finite set of sets. Unlike a simple set alone, it has natural algebraic, geometric, analytic and order structures built … ….

If a set of set is equipped with an energy function, one can define integer matrices for which the determinant, the eigenvalue signs are known. For constant energy the matrix is conjugated to its inverse and defines two isospectral multi-graphs.

The counting matrix of a simplicial complex has determinant 1 and is isospectral to its inverse. The sum of the matrix entries of the inverse is the number of elements in the complex.

The f-function of a graph minus 1 is the sum of the antiderivatives of the f-function anti-derivatives evaluated on the unit spheres.

Dehn-Sommerville relations are a symmetry for a class of geometries which are of Euclidean nature.

Some update about recent activities: a new calculus course, the Cartan magic formula and some programming about the coloring algorithm.

We prove that any discrete surface has an Eulerian edge refinement. For a 2-disk, an Eulerian edge refinement is possible if and only if the boundary length is divisible by 3

We prove that connected combinatorial manifolds of positive dimension define finite simple graphs which are Hamiltonian.

The beautiful Alexander duality theorem for finite abstract simplicial complexes.

We compute the quadratic interaction cohomology in the simplest case.

This is an other blog entry about interaction cohomology [PDF], (now on the ArXiv), a draft which just got finished over spring break. The paper had been … ….

For a one-dimensional simplicial complex, the sign less Hodge operator can be written as L-g, where g is the inverse of L. This leads to a Laplace equation shows solutions are given by a two-sided random walk.

Here is the code to compute a basis of the cohomology groups of an arbitrary simplicial complex. It takes 6 lines in mathematica without any outside libraries. … ….

We found a formula of the green function entries g(x,y). Where g is the inverse of the connection matrix of a finite abstract simplicial complex. The formula involves the Euler characteristic of the intersection of the stars of the simplices x and y, hence the name.

When replacing the circle group with the dyadic group of integers, the Riemann zeta function becomes an explicit entire function for which all roots are on the imaginary axes. This is the Dyadic Riemann Hypothesis.

A simplicial complex G, a finite set of non-empty sets closed under the operation of taking finite non-empty subsets, has the Laplacian $L(x,y) = {\rm sign}(|x \cap … ….

A finite abstract simplicial complex has a natural connection Laplacian which is unimodular. The energy of the complex is the sum of the Green function entries. We see that the energy is also the number of positive eigenvalues minus the number of negative eigenvalues. One can therefore hear the Euler characteristic. Does the spectrum determine the complex?

As a follow-up note to the strong ring note, I tried between summer and fall semester to formulate a discrete Atiyah-Singer and Atiyah-Bott result for simplicial complexes. … ….