Quantum Calculus

We work on the result that any 2-sphere can be covered by 3 trees.

The three tree theorem tells that any discrete 2-sphere has arboricity exctly 3.

We aim to show that the Lusternik-Schnirelmann category of a graph is bound by the augmented dimension.

A notion of a discrete vector should work for theorems like Poincare-Hopf and also produce a dynamics as classically, a vector field F, a smooth section of the tangent bundle on a manifold produces a dynamics .

In 1973 Leon Glass proved a discrete Poincare Hopf theorem for directed graphs embedded in a 2 dimensional manifold.

This is a bit of a continuation from a previous video about Lusternik-Schnirelmann and Morse.

We discuss briefly how to make the cohomology space of a graph into a cohomology ring.

About the cohomology of measurable sets in a probability space equipped with an automorphism.

Something about what happens typically in dynamical systems.

The pendulum equation can be solved explicitly using the Weierstrass elliptic function.

Something about knots and something about topological data analysis and something about the general frame work to do mathematics in a finite setting.

A bit the bigger picture about the mathematical and data structures which come in when working on these finite geometries.

A youtube presentation of May 13, 2023.

It is a simple but interesting fact that if we look at a wave equation with discrete space, we also need discrete time provided we want causality, a property which every wave equation should have.