## Energized Simplicial Complexes

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

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.

## Gauss-Bonnet for the f-function

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

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

## Discrete Calculus etc

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

## Euler Game

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

## The Hamiltonian Manifold Theorem

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

## The beginnings of Shellability

About the origin of the definitino of shellability.

## Combinatorial Alexander Duality

The beautiful Alexander duality theorem for finite abstract simplicial complexes.

## Interaction cohomology Example

We compute the quadratic interaction cohomology in the simplest case.