# Quantum Calculus

Calculus without limits
##### 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...
##### Wenjun Wu, 1919-2017
According to Wikipedia, the mathematician Wen-Tsun Wu passed away earlier this year. I encountered some mathematics developed by Wu when working on Wu characteristic. See...
##### 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...
##### Aspects of Discrete Geometry
The area of discrete geometry is a maze. There are various flavors.
##### Symmetry via Ergodic Theory
One of the attempts to quantize space without losing too much symmetry is ergodic theory. Much of my thesis belongs to this program. It is...
##### What is geometry?
In the context of quantum calculus one is interested in discrete structures like graphs or finite abstract simplicial complexes studied primarily in combinatorics or combinatorial...
##### Jones Calculus
The mathematics of evolving fields with two complex components is known already in Jones calculus.
##### A quaternion valued elliptic complex
This blog entry delivers an other example of an elliptic complex which can be used in discrete Atiyah-Singer or Atiyah-Bott type setups as examples. We...
##### Discrete Atiyah-Singer and Atiyah-Bott
As a follow-up note to the strong ring note, I tried between summer and fall semester to formulate a discrete Atiyah-Singer and Atiyah-Bott result for...
##### Strong Ring of Simplicial Complexes
The strong ring is a category of geometric objects G which are disjoint unions of products of simplicial complexes. Each has a Dirac operator D...
##### The Dirac operator of Products
Implementing the Dirac operator D for products $latex G \times H$ of simplicial complexes $latex G,H$ without going to the Barycentric refined simplicial complex has...
##### Do Geometry and Calculus have to die?
In the book 'This Idea Must Die: Scientific Theories That Are Blocking Progress', there are two entries which caught my eye because they both belong...
##### The Two Operators
The strong ring The strong ring generated by simplicial complexes produces a category of geometric objects which carries a ring structure. Each element in the...
##### Space and Particles
Elements in the strong ring within the Stanley-Reisner ring still can be seen as geometric objects for which mathematical theorems known in topology hold. But...
##### Graph limits with Mass Gap
The graph limit We can prove now that the graph limit of the connection graph of Ln x Ln which is the strong product of...

## More Green Function Values

We have seen that for a finite abstract simplicial complex $G$, the connection Laplacian L has an inverse g with integer entries and that $g(x,x) = 1-X(S(x))$, where $S(x)$ is the unit sphere of $x$ in the graph $G_1=(V,E)$, where $V=G$ and where (a,b) in E if and only if $a \subset b$ or $b \subset a$. We have also … ….

## 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 wondered whether there is an argument proving that that there … ….

## Wenjun Wu, 1919-2017

According to Wikipedia, the mathematician Wen-Tsun Wu passed away earlier this year. I encountered some mathematics developed by Wu when working on Wu characteristic. See the Slides and the paper on multi-linear valuations. There is an other paper on this in preparation, especially dealing with the cohomology belonging to Wu characteristics. Just as a reminder, the Wu characteristic of a … ….

## 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?

## Aspects of Discrete Geometry

The area of discrete geometry is a maze. There are various flavors.

## Symmetry via Ergodic Theory

One of the attempts to quantize space without losing too much symmetry is ergodic theory. Much of my thesis belongs to this program. It is a flavor of quantum calculus, as “no limits” are involved. The story is closely related to Jacob Feldman, one of my heroes of my graduate and postdoc time. I write this blog entry after having … ….

## What is geometry?

In the context of quantum calculus one is interested in discrete structures like graphs or finite abstract simplicial complexes studied primarily in combinatorics or combinatorial topology. Are they geometry? Are they calculus? What is geometry? In MathE320 I try to use the following definition: Geometry is the science of shape, size and symmetry. The symmetry statement is borrows from Klein’s … ….

## Jones Calculus

The mathematics of evolving fields with two complex components is known already in Jones calculus.

## A quaternion valued elliptic complex

This blog entry delivers an other example of an elliptic complex which can be used in discrete Atiyah-Singer or Atiyah-Bott type setups as examples. We had seen that when deforming an elliptic complex with an integrable Lax deformation, we get complex elliptic complexes. We had wondered in that blog entry whether a complex can lead to quaternion-valued fields. The discussion … ….

## Discrete Atiyah-Singer and Atiyah-Bott

As a follow-up note to the strong ring note, I tried between summer and fall semester to formulate a discrete Atiyah-Singer and Atiyah-Bott result for simplicial complexes. The classical theorems from the sixties are heavy, as they involve virtually every field of mathematics. By searching for analogues in the discrete, I hoped to get a grip on the ideas. (I … ….