Calculus without limits

## Stability of the Vacuum

Explanations of the Casimir effect using common physics intuition like “polarization” (it originally was studied in the context of van der Waals forces) or “pressure” do not work. The reason in the case of the Casimir effect is that in the case of two planes or two cylinders the Casimir …

## Geometry of Delta Sets

In this presentation, there is a bit of advertisement for finite geometry and delta sets in particular. I also tried to get a bit into the history of finitist ideas in geometry and physics(starting with Riemann). One usually thinks about finite projective spaces when talking about “finite geometries”. I like …

## Fusion Inequality for Quadratic Cohomology

While linear cohomology deals with functions on simplices, quadratic cohomology deals with functions on pairs of simplices that intersect. Linear cohomology is to Euler characteristic what quadratic cohomology is to Wu characteristic $w(G) = \sum_{x,y, x \cap y \in G} w(x) w(y)$. If the simplicial complex is split into a …

## Sard for delta sets

The discrete Sard theorem in the simplest case (which I obtained in 2015) that a function from a discrete d-manifold to {-1,1} has level sets that are (d-1) manifolds or empty. (See here for the latest higher generalization to higher codimension.) A simplicial complex is a d-manifold if every unit …

## Algorithmic Poetry

Brevity contributes both to clarity and simplicity. Surprisingly, it often contributes to generality. I myself am obsessed with brevity. I especially love short code. A short program is like a poem. If it is also effective, it can also be used as building blocks of larger programs. The Unix philosophy …

## Markov Dynamics on a Complex

Dynamical systems on geometric spaces are very common in mathematics. Mean curvature flows, Ricci flows etc. We can think of a subcomplex K in a simplicial complex G as a geometric object. We can not break randomly elements away as would in general lose the property of having a subcomplex. …

## 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.

## 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.

## Isospectral Simplicial Complexes

One can not hear a complex! After some hope that some kind of algebraic miracle allows to recover the complex from the spectrum (for example by looking for the minimal polynomial which an eigenvalue has and expecting that the factorization reflects some order structure in the abstract simplicial complex), I …

## 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?