Quantum Calculus

Given a finite abstract simplicial complex G that is a q-manifold, we consider a k-simplex y in an oriented maximal simplex x as an element in the Grassmannian G(k,q).

At the end of the wave front calculus paper, I added some pictures about curvature defined by wave fronts.

A first draft is now available [PDF].

In my project picked up again this winter, I had looked at the deformation of exterior derivative df in a q-manifold given by and saw that it satisfies the modified wave equation , where is the Hodge Laplacian.

It usually takes a few decades until some mathematical object slides into Pop culture and eventually becomes kitsch.

The Polarization identity for quadratic forms like inner products is also known as the parallelogram law.

In the video below I gave a bit of an overview about the “wave calculus project”.

I only learned recently thought the “not even wrong” blog that Konrad Osterwalder passed away last December.

The three dimensional space is important because we live in it.

This is a continuation.

We discuss the exterior derivative on a compact Riemannian manifold (M,g) in dimension 1.

Here is a bit of calculus in dimension q.

The dispute of 1689 between Newton and Huygens was won by Newton (he was by far the more important person) but in reality, Newton was wrong (*).

If P is a light source on a cube we can look at , the wave front.

This winter, I started to revive some seeds which have been placed in the winter of 2010.

We currently make some more experiments for the density of wave fronts project.

Here is again some Mathematica code.

A geometry is a space on which one has a derivative and notion of integration.

Causality Principle One of the most exciting principles in physics is the Huygens principle relating the geodesic motion with the wave dynamics.

This is a bit a blast from the past.

For slides and first remarks about the P-NP problem, see this page.

Having talked a bit about GR (general relativity) and the SM (standard model), lets talk a bit about QM (quantum mechanics).

In 1960, Regge formulated a discrete calculus as a numerical scheme for relativity.

The number theory of quaternion integers has connections to many beautiful parts of mathematics.

The topic is a combination of the “particles and primes” and geodesics in discrete manifolds” topics, I had worked on before.

AXIOMS: Most basic mathematical structures are heavenly in the sense that they would be discussed in any part of the universe.

We have seen last time that describing many particles moving on a manifold is best done by keeping on each cell a tag telling how many particles there are there.

On October 15, 2025, Sourav Chatterjiee gave the second of the Millenium prize lectures.

This is a bit about getting back to the story of geodesics.

Manifolds are in the following two dimensional, compact, connected and without boundary.

If K(t) is a function in a Lie algebra then the Frenet equations Q'(t)=K(t) Q(t), Q(0)=1 defines a path in the corresponding Lie group.

The arrow of time is probably one of the most written fundamental topics in physics.

A periodic Frenet curve in is up to isometry uniquely defined by a periodic curve in where the later describes the motion of the Frenet frame.

Maybe it is a good idea to add current “discovery code” also with the Dirac part.

Update October 7, 2025: Kitsch is already produced by professional mathematicians .

Let (G,T) be a finite dynamical system.

Yesterday was 9/16/25.

Here is my “discovery code” used over the weekend to detect first experimentally that there are dynamical connection and green function matrices which are inverse of each other.

In the evening of the 13th of September after giving the talk below, it dawned me after doing lots more experiments that the connection dynamical connection Laplacian and the dynamical connection Green function that were written down on the right hand side

When I had been studying mathematics at ETHZ, taking two semesters of calculus and at the same time taking two semesters of linear algebra had been mandatory.

When trying to translate everything that has been done in mathematics to the finite world it is good first to look at the most important theorems.

For a finite abstract simplicial complex, we can look at the Dirac Zeta function, the Connection Zeta function and the Lefschetz zeta function.

Here is an other “poem” that has been added to the “math poetry page”.

During a nice 23 K run on Friday, I was thinking a bit about Lefschetz.

A conference earlier this year had the effect that some main stream media started to report about it.

Given a finite abstract simplicial complex G, there are two amazing matrices which are two sides of a coin.

Before we start, here is the code which generated the two matrices on the board of the talk. 5 lines for the Hodge Laplacian, 1 line for the Connection Laplacian.

Theorem: if a graph G=(V,E) with n=|V| vertices and m=|E| edges and maximal vertex degree d satisfies , then G satisfies the Brouwer conjecture .

Let G=(V,E) be a graph with n vertices and m edges.

If K is the Kirchhoff matrix of a finite simple graph with n vertices and m edges and eigenvalues and edge degrees .