Mickey Mouse Sphere Theorem
The Mickey mouse theorem assures that a connected positive curvature graph of positive dimension is a sphere.
Category: quantum calculus
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.
Poincare-Hopf and the Clique Problem
The parametrized poincare-hopf theorem allows to see the f-vector of a graph in terms of the f-vector s of parts of the unit spheres of the graph.
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
Interaction cohomology Example
We compute the quadratic interaction cohomology in the simplest case.
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 with computations in basic planimetric computations in a triangle (Example … ….
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.
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?
Symmetry via Ergodic Theory
One of the attempts to quantize space without losing too much symmetry is ergodic theory. Much of my thesis belongs to this program. It is a flavor of quantum calculus, as “no limits” are involved. The story is closely related to Jacob Feldman, one of my heroes of my graduate and postdoc time. I write this blog entry after having … ….