Quantum Calculus

An abstract delta set (G,D,R) is a finite set G with n elements, a selfadjoint Dirac matrix with and a dimension vector defining a partition and Hilbert spaces called the spaces of k-forms.

This is a presentation from Saturday, July 13, 2024.

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.

In this presentation, there is a bit of advertisement for finite geometry and delta sets in particular.

I looked at a quadratic cohomology example.

While linear cohomology deals with functions on simplices, quadratic cohomology deals with functions on pairs of simplices that intersect.

Delta sets are very general.

Quivers are graph in which multiple connections and loops are allowed.

If is a finite abstract simplicial complex, a finite set of non-empty sets closed under the operation of taking non-empty subsets, we can ask about what f-vectors can occur if counts the number of sets of cardinality k in G.

The smallest open sets in a finite topological space form the atom of space.

Historically, geometry started in Euclidean spaces.

Here are three catchy open problems in differential geometry.

Finite geometric categories: graphs – simplicial complexes -simplicial sets – delta sets Delta sets were originally called semi-simplicial sets by Samuel Eilenberg and Joseph Zilber in 1950.

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.

Here are some links to the articles mentioned in the talk: It surprisingly often happens that a big conjecture tumbles at around the same time.

The last geometric theorem of Poincare was conceived shortly before the death of Poincare.

Finite topological spaces are only interesting if non-Hausdorff.

It is not quite yet a poem, but here, as promised in the movie, some code to generate both the Betti vector and Wu betti vector of a random submanifold in a given manifold.

My experiments so far indicate that the Wu cohomology of a d-manifold G can be read off from the usual cohomology.

[Update 3/5/2024: given that one knows now the optimal Moebius strip, one can wonder about the much easier question of what the smallest simplicial complex producing a cylinder or Moebius strip is.

The video below is an attempt to get back to an older story of Wu characteristic.

If G is a d-manifold and is an arbitrary map, the discrete Sard theorem tells that is an open set in G that is a (d-k)-manifold.

Here is some code illustrating the story.

Coordinates are function values.

The updated document “manifolds from partitions” with more code.

Again a little bit of a flashback to my earliest steps in doing mathematical research.

In my paper “Manifolds from Partitions”, I stated that that the case of empty graphs can not occur, but did not prove it.

My experience from my Schweizer Jugend Forscht adventure was not only invaluable from the scientific point of view, I also met some other young aspiring scientists (here is the book with all the participants (PDF)) which was published in 1983 (when I was…

A nice thing about mathematics is that it has no dogmas, statements which have to be taken on good faith.

Brevity contributes both to clarity and simplicity.

Eugene Wigner in 1939 associated elementary particles with irreducible representations of groups, especially the Poincare group.

Happy new year 2024.

First about Sard: (a write-up [PDF] ).

In the program to get rid of any notion of infinity, one necessarily has to demonstrate that very classical and entrenched notions like topics appearing in a contemporary multi-variable calculus course can be replaced and used.

There are lots of finite geometric structures.

Here is the table shown in part 2 of the presentation showing some of the toys.

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.

What is {f=0,g=0} for two functions f,g on a graph G.

We start tp experiment with more general versions of Morse-Sard on graphs.

We look at four different features or properties which can be invoked when looking at a mathematical problem.

Arboricity and Chromatic number are linked in various ways.

One of the nice things to work in a subject not having grown up in is to be in steep learning curves.

We look at the arboricity of 3-manifolds.

The arboricity of a non-sphere surface is 4.

We aim to write down a short proof of the statement that a planar graph has arboricity 3 or less.

The Acyclic Chromatic Number is bounded above by the arboricity.

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).

We look at the problem to find the possible arboricities which a 3-sphere can have.

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

We asked some of the AI's about Arboricity and got some really terrible answers.