Calculus without limits ## On the Manifold Playground

Here is the table shown in part 2 of the presentation showing some of the toys. Here is the manifold page of Frank H. Lutz mentioned in the clip. About the left hand side with some history pointers of manifolds: it is interesting that what we teach in multi-variable calculus …

## Morse Sard and Small Physics

One of the nice things in mathematics is that one can play with models which do not necessarily have to do directly with the real world, whatever the later means. We can look at abstract objects, like finite simple groups, number theory in some number field or topology in 1001 …

## Foliage inequalities

Arboricity and Chromatic number are linked in various ways. The topic also links to difficult NP complete problems. We muse about how often it is the case that for manifolds the question is easy. An example is the Hamiltonian path problem which is linked to Peg Solitaire

## Arboricity and 4-manifolds

One of the nice things to work in a subject not having grown up in is to be in steep learning curves. I have thought about the arboricity of manifolds for a while now but the fact that the arboricity can be arbitrary large for d-manifolds with d larger than …

## Arboricity of spheres

We explain why the arboricity of 3 spheres can take values between 4 and 7 and mention that for 3 manifolds the upper bound is 9 (but believed to be 7).

## 3 trees and 4 colors

The three tree theorem follows from an upgrade of the 4 color theorem.

## The Three Tree Theorem

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

## Discrete Vector Fields

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 . A directed graph does not give a dynamics without telling how to go …

## Cup Length of a Graph

We discuss briefly how to make the cohomology space of a graph into a cohomology ring. In other words, how to define the cup product on the kernel of the Hodge Laplacian.

## On Knots and Cohomology and Dowker

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

## A Topological Topos

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

## Relative Cohomology

A youtube presentation of May 13, 2023. We point out that we have a relatively simple approach to Eilenberg-Steenrod.

## Fusion Rules for Cohomology

Over the winter break I started to look at Mayer-Vietoris type rules when looking at cohomology of subsets of a simplicial complex. See January 28, 2023 (Youtube) , and February 4th 2023 (Youtube) and most recently on February 19, 2023 (Youtube). Classically, cohomology is considered for simplicial complexes and especially …

## A multi-particle energy theorem

A finite abstraact simplicial complex or a finite simple graph comes with a natural finite topological space. Some quantities like the Euler characteristic or the higher Wu characteristics are all topological invariants. One can also reformulate the Lefschetz fixed point theorem for continuous maps on finite topological spaces.

## More on Analytic Torsion

We report on some progress on analytic torsion A(G) for graphs. A(G) is a positive rational number attached to a network. We can compute it for contractible graphs or spheres.

## 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 …

## Complex energized complexes

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

## 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.