# Category: discrete geometry

## Graphs, Groups and Geometry

A bit more update on the project of natural spaces. Which groups are natural, which metric spaces are natural, which graphs are natural?

## Product Formula For Curvature

The Curvature of graphs multplies under the Shannon product (strong product).

## More on Graph Arithmetic

A few more remarks [PDF] about graph arithmetic. Now on the ArXiv. (Previous documents are here (June 2017 ArXiv), here (August 2017 ArXiv) and here (May 2019 ArXiv). The talk below on youtube was used for me to get organized a bit. It does not look like much has changed, …

## Topology of Manifold Coloring

Last summer I have had some fun with codimension 2 manifolds M in a purely differential geometric setting: a positive curvature d-manifold which admits a circular action of isometries has a fixed point set K which consists of even codimension positive curvature manifold. The Grove-Searle situation https://arxiv.org/abs/2006.11973 is when K …

## More theorems about graphs

Last week, I practiced a bit more enhanced talk presentation style in which, rather than with slides, the content is spoken and then enhanced in the video using additonal illustrations. The presentation deals with some things I have done in graph theory which I consider as part of quantum calculus …

## 10 theorems on discrete manifolds

10 theorems about discrete manifolds were featured in a youtube video.

## Graph Complements of Cyclic Graphs

Graph complements of cylic graphs are homotopic to spheres or wedge sums of spheres. Their unit spheres are graph complements of path graphs and have Gauss-Bonnet curvature which converges to a limit.

## 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 things up more generally and there is also some mentioning in the article: allowing general rings and not just division algebras extends the …

## Energy relation for Wu characteristic

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.

## Complex energized complexes

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

## Physics on finite sets of sets?

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 causal dynamical triangulation models or spin networks. It is necessary to bring in some time evolution as otherwise, a model has little chance …

## Poincare-Hopf for Vector Fields on Graphs

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 what to do in the case of the triangle with circular orientation. The triangle has Euler characteristic 1. An integer index on vertices …

## Mickey Mouse Sphere Theorem

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

## The joy of sets of sets

Finite sets of sets can be seen as a combintarorial oddity. There is a lot of mathematical structure available on such simple finitist constructs.

## 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.

## 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.

## Interaction Cohomology (II)

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 started more than 2 years ago and got delayed when the unimodularity of the connection Laplacian took over. There was an announcement [PDF] …

## The Hydrogen Relation

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.

## Cohomology in six lines

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. The input is a simplicial complex, the out put is the basis for $H^0,H^1,H^2 etc$. The length of the code compares in complexity …

## Green Star Formula

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.

## A Dyadic Riemann hypothesis

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.

## Quest for a Green Function Formula

A simplicial complex G, a finite set of non-empty sets closed under the operation of taking finite non-empty subsets, has the Laplacian $L(x,y) = {\rm sign}(|x \cap y|)$. It is natural as it is always unimodular so that its inverse $g(x,y)$ is always integer valued. In a potential theoretical setup, …

## Hearing the shape of a simplicial complex

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?

## Discrete Atiyah-Singer and Atiyah-Bott

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. The classical theorems from the sixties are heavy, as they involve virtually every field of mathematics. By searching for analogues in the discrete, …