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.
Category: discrete geometry
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 … ….
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 … ….
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 … ….
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
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 … ….
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 … ….
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. … ….
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 … ….
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. … ….