# Quantum Calculus

Calculus without limits
##### The beginnings of Shellability
About the origin of the definitino of shellability.
##### Connection Duality
A simplicial complex G defines the connection matrix L which is L(x,y)=1...
##### Combinatorial Alexander Duality
The beautiful Alexander duality theorem for finite abstract simplicial complexes.
##### Interaction cohomology Example
We compute the quadratic interaction cohomology in the simplest case.
##### The dunce hat and Lusternik-Schnirelmann
The interaction cohomology of the dunce hat is computed. We then comment...
##### Interaction Cohomology (II)
This is an other blog entry about interaction cohomology [PDF], (now on...
##### The Hydrogen Relation
For a one-dimensional simplicial complex, the sign less Hodge operator can be...
##### Cohomology in six lines
Here is the code to compute a basis of the cohomology groups...
##### Green Star Formula
We found a formula of the green function entries g(x,y). Where g...
When replacing the circle group with the dyadic group of integers, the...
##### A Perron-Frobenius Vector to Wu Characteristic
The Wu characteristic of a simplicial complex is the eigenvalue of an...
##### Is there physics for the connection Laplacian?
The classical potential $V(x,y) = 1/|x-y|$ has infinite range which violently clashes...
##### Quest for a Green Function Formula
A simplicial complex G, a finite set of non-empty sets closed under...
##### More Green Function Values
We have seen that for a finite abstract simplicial complex $G$, the...
##### Isospectral Simplicial Complexes
One can not hear a complex! After some hope that some kind...
##### Wenjun Wu, 1919-2017
According to Wikipedia, the mathematician Wen-Tsun Wu passed away earlier this year....
##### Hearing the shape of a simplicial complex
A finite abstract simplicial complex has a natural connection Laplacian which is...
##### 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...
##### What is geometry?
In the context of quantum calculus one is interested in discrete structures...
##### Jones Calculus
The mathematics of evolving fields with two complex components is known already...
##### A quaternion valued elliptic complex
This blog entry delivers an other example of an elliptic complex which...
##### Discrete Atiyah-Singer and Atiyah-Bott
As a follow-up note to the strong ring note, I tried between...
##### Strong Ring of Simplicial Complexes
The strong ring is a category of geometric objects G which are...
##### The Dirac operator of Products
Implementing the Dirac operator D for products $latex G \times H$ of...
##### Do Geometry and Calculus have to die?
In the book 'This Idea Must Die: Scientific Theories That Are Blocking...
##### The Two Operators
The strong ring The strong ring generated by simplicial complexes produces a...
##### Space and Particles
Elements in the strong ring within the Stanley-Reisner ring still can be...
##### Graph limits with Mass Gap
The graph limit We can prove now that the graph limit of...
##### One ring to rule them all
Arithmetic with networks The paper "On the arithmetic of graphs" is posted....
##### Three Kepler Problems
Depending on scale, there are three different Kepler problems: the Hydrogen atom,...
##### More about the ring of networks
The dual multiplication of the ring of networks is topological interesting as...
##### Unique prime factorization for Zykov addition
We give two proofs that the additive Zykov monoid on the category...
##### Hardy-Littlewood Prime Race
The Hardy-Littlewood race has been running now for more than a year...
##### The Hydrogen trace of a complex
Motivated by the Hamiltonian of the Hydrogen atom, we can look at...
##### The quantum plane
Update of May 27, 2017: I dug out some older unpublished slides...
##### A mass gap in the Barycentric limit
The Barycentric limit of the density of states of the connection Laplacian...
##### Tensor Products Everywhere
The tensor product is defined both for geometric objects as well as...
##### Functional integrals on finite geometries
We look at examples of functional integrals on finite geometries.
##### The finitist bunker
As Goedel has shown, mathematics can not tame the danger that some...
##### The Helmholtz Hamiltonian System
As we have an internal energy for simplicial complexes and more generally...
##### Energy theorem for Grothendieck ring
Energy theorem The energy theorem tells that given a finite abstract simplicial...
##### From Affinity over Vis Viva to Energy
The history of the developent of energy and entropy is illustrated. This...
##### Helmholtz free energy for simplicial complexes
Over spring break, the Helmholtz paper [PDF] has finished. (Posted now on...
##### Who is this famous person?
A rather unfamiliar picture of a famous mathematician/physisist.
##### Energy, Entropy and Gibbs free Energy
Energy U and Entropy S are fundamental functionals on a simplicial complex...
##### Shannon Entropy and Euler Characteristic
Entropy is the most important functional in probability theory, Euler characteristic is...
##### A Gauss-Bonnet connection
An experimental observation : the sum over all Green function values is...
##### Sphere spectrum paper
The sphere spectrum paper is submitted to the ArXiv. A local copy....
##### Spectra of Sums of Networks
What happens with the spectrum of the Laplacian $latex L$ if we...
##### A ring of networks
Assuming the join operation to be the addition, we found a multiplication...
##### Arithmetic with networks
The join operation on graphs produces a monoid on which one can...
##### Sphere Spectrum
This is a research in progress note while finding a proof of...
##### Partial differential equations on graphs
During the summer and fall of 2016, Annie Rak did some URAF...
##### Unimodularity theorem slides
Here are some slides about the paper. By the way, an appendix...
##### From the Christmas Theorem to Particle Physics
The Christmas Theorem Because Pierre de Fermat announced his two square theorem...
##### The unimodularity theorem proof
The proof of the unimodularity theorem is finished.
##### Dimension: from discrete to general metric spaces
This is an informal overview over definitions of dimension, both in the...
##### Euler and Fredholm
The following picture illustrates the Euler and Fredholm theme in the special...
##### The Unimodularity Theorem for CW Complexes
The unimodularity theorem equates a fredholm determinant with a product of indices....
##### Quantum calculus talk of 2013
Just uploaded a larger version of my 2013 Pecha-Kucha talk "If Archimedes...
##### On Bowen-Lanford Zeta Functions
Zeta functions are ubiquitous in mathematics. One of the many zeta functions,...
##### The Kustaanheimo prime
Paul Kustaanheimo (1924-1997) was a Finnish astronomer and mathematician. In celestial mechanics,...
##### Particles and Primes, Counting and Cohomology
Source: Pride and Prejudice, 2005 Judy Dench plays the role of Lady...
##### Counting and Cohomology
There are various cohomologies for finite simplicial complexes. If the complex is...
##### Quaternions and Particles
The standard model of particle physics is not so pretty, but it...
##### Bosonic and Fermionic Calculus
Traditional calculus often mixes up different spaces, mostly due to pedagogical reasons....
##### Interaction cohomology
[Update, March 20, 2018: see the ArXiv text. See also an update...
##### Wu Characteristic
Update: March 8, 2016: Handout for a mathtable talk on Wu characteristic....
##### Barycentric refinement
A finite graph has a natural Barycentric limiting space which can serve...
##### Level surfaces and Lagrange
How to define level surfaces or solve extremization problems in a graph....
##### Why quantum calculus?
Quantum calculus is easy to learn, allows experimentation with small worlds and...

Quantum calculus deals with alternative flavors of calculus and especially studies calculus on discrete or even finite sets. Examples of non-traditional calculus flavors are non-standard analysis, difference calculus or calculus on graphs. All these flavors of calculus can be seen as extensions of traditional calculus or calculus on Riemannian manifolds as the usual calculus can be seen as a special or limiting case.

One idea is to build notions which look close to the known notions of calculus. An ideal generalization of traditional calculus to the discrete does not change language at all but just changes the meaning of the operations, possibly after an extension of language. This has already been realized in the form of internal set theory IST of Nelson. But this flavor of calculus is not easy to manage and teach and due to the reliance on more logic a bit more risky: it is easy to make mistakes. One can imagine historical circumstances, in which quantum calculus have been first developed and the calculus we know later been derived and have the status of generalized function theory or geometric measure theory. In the same way that we could in principle for practical purposes work with rational numbers alone, one could look exclusively at discrete geometries and look at continuum geometries as limiting cases. Any computation on a device is by nature finite as there is only a finite amount of memory available.In some sense, already numerical analysis is already a quantized calculus. However, the theory as told in numerical analysis books is much less elegant than the actual theory so that this is not the right approach.

##### What is the main idea?

One of the useful ideas of calculus is to look at “rates of changes” in order to “predict the future”. This idea is everywhere: look at the sequence 4, 15, 40, 85, 156, 259, 400, 585 for example. How does it continue? In order to figure that out, we take derivatives 11, 25, 45, 71, 103, 141, 185, then again derivatives 14, 20, 26, 32, 38, 44 and again derivatives 6,6,6,6,6,6. We see now a pattern and can integrate the three times starting with 6,6,6,6,6,6,6 always adjusting the constant. This gives us the next term 820. We can predict the future by analyzing the past.

More generally, any geometric theory with a notion of exterior derivative on “forms” and integration of “forms” is a calculus flavor. A nice example is Riemannian geometry, which generalizes calculus in flat space. The frame work allows then to define notions like curvature or geodesics which are so central in modern theories of gravity. These notions can be carried over to discrete spaces.

But it is not only physics which motivates to look at calculus. A big “customer” of calculus ideas is computer science. Discrete versions of notions like gradient, curvature, surface etc allow a computer to “see” or to build new objects, never seen. All movies using some kind of CGI make heavy use of calculus ideas.

##### Why do we study calculus?

The power, richness and applicability of calculus are all reasons why we teach it. Calculus is a wonderful and classical construct, rich of historical connections and related with many other fields. Here is a 30 second spot. We hardly have to mention all the applications of calculus (the last 2 minutes of this 15 minute review for single variable calculus give a few). We have barely scratched the surface of what is possible when extending calculus and how it can be applied in the future.

The traditional exterior derivatives (like div,curl grad) or curvature notions based on differential forms define a traditional calculus. Integration and derivatives lead to theorems like Stokes, Gauss-Bonnet, Poincaré-Hopf or Brouwer-Lefschetz. In classical calculus, the basic building blocks of space are simplices. Differential forms are functions of these simplices. In the continuum, one can not see these infinitesimal simplices. To remedy this, sheaf theoretical constructs like tensor calculus have been developed, notably by Cartan. In the discrete, when looking at graphs, the structure is simple and transparent. The theorems become easy. Here, here and here are some write ups.

##### Calculus on graphs

Calculus on graphs is probably the simplest quantum calculus with no limits: everything is finite and combinatorial. I prefer to work on graphs but one can work also with finite abstract simplicial complexes. It turns out that the category of complexes and especially graphs is quite powerful, despite its simplicity. Graphs are quite an adequate language because the Barycentric refinement of an abstract finite simplicial complex is always the Whitney complex of a finite simple graph. Most topological considerations therefore can be done on graphs. Having “trivialized calculus”, one can look at more complex constructs.

##### Interaction calculus

In interaction calculus, functions on pairs (or more generally k-tuples) of interacting (=intersecting) simplices in the graph are at the center of attention. Exterior derivatives again lead to cohomology. While in classical calculus, the derivative is df(x)=f(dx) (which is Stokes theorem as dx is the boundary chain), in second level interaction calculus, the derivative is df(x,y)=f(dx,y)+(-1)dim(x) f(x,dy). The chain complex is bigger. The cohomology groups more interesting.

##### Interaction cohomology

The analogue of Euler characteristic is the more general Wu characteristic. There are generalized versions of the just mentioned theorems. At the moment (2016), only a glimpse of the power of interaction calculus is visible. There are indications that it could be powerful: it allows to distinguish topological spaces, which traditional calculus does not: here is a case study. And here is a preprint on Interaction cohomology.

##### Open mind

Its not very helpful to look only for analogies to the continuum. Very general principles (numerical and computer science demonstrate constantly how finite machines mode things) show that the continuum can be emulated very well in the discrete. We also have to look out for new things. A surprise for example is that there is a Laplacian for discrete geometries which is always invertible: discovered in February 2016 and proven in the fall 2016, it leads to invariants and potential theory different from the usual Hodge Laplacian. Unlike the Hodge Laplacian which is the square of the Dirac Laplacian, this new Laplacian has a quantized and completely finite potential theory, no super symmetry. The total energy of the geometry is the Euler characteristic.

##### Project page

For more, see the project page on my personal web page. The entries here are more like drafts, sometimes expository, sometimes research logs which I leave as it is for me also interesting to see later, where and how I was stuck. So, it can happen that initially in an entry, the story has not yet been clear but at the end been understood.

With limited time at hand, comments are not turned on as this would require time for moderation. This might change at some later when more material has been added. At the moment there is not much yet here.

##### The beginnings of Shellability
About the origin of the definitino of shellability.
##### Connection Duality
A simplicial complex G defines the connection matrix L which is L(x,y)=1 if and only if x and y intersect. The dual matrix is K(x,y)=1...
##### Combinatorial Alexander Duality
The beautiful Alexander duality theorem for finite abstract simplicial complexes.
##### Interaction cohomology Example
We compute the quadratic interaction cohomology in the simplest case.
##### The dunce hat and Lusternik-Schnirelmann
The interaction cohomology of the dunce hat is computed. We then comment on the discrete Lusternik-Schnirelmann theorem.
##### Interaction Cohomology (II)
This is an other blog entry about interaction cohomology [PDF], (now on the ArXiv), a draft which just got finished over spring break. The paper...
##### The Hydrogen Relation
For a one-dimensional simplicial complex, the sign less Hodge operator can be written as L-g, where g is the inverse of L. This leads to...
##### Cohomology in six lines
Here is the code to compute a basis of the cohomology groups of an arbitrary simplicial complex. It takes 6 lines in mathematica without any...
##### 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....
When replacing the circle group with the dyadic group of integers, the Riemann zeta function becomes an explicit entire function for which all roots are...
##### A Perron-Frobenius Vector to Wu Characteristic
The Wu characteristic of a simplicial complex is the eigenvalue of an eigenvector to a matrix L J, where L is the connection Laplacian and...
##### Is there physics for the connection Laplacian?
The classical potential $V(x,y) = 1/|x-y|$ has infinite range which violently clashes with relativity. Solving this problem had required a completely new theory: GR. It...
##### 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...

## The beginnings of Shellability

About the origin of the definitino of shellability.

## Connection Duality

A simplicial complex G defines the connection matrix L which is L(x,y)=1 if and only if x and y intersect. The dual matrix is K(x,y)=1 if and only if x and y do not intersect. It is the adjacency matrix of the dual connection graph.

## Combinatorial Alexander Duality

The beautiful Alexander duality theorem for finite abstract simplicial complexes.

## Interaction cohomology Example

We compute the quadratic interaction cohomology in the simplest case.

## The dunce hat and Lusternik-Schnirelmann

The interaction cohomology of the dunce hat is computed. We then comment on the discrete Lusternik-Schnirelmann theorem.

## Interaction Cohomology (II)

This is an other blog entry about interaction cohomology [PDF], (now on the ArXiv), a draft which just got finished over spring break. The paper had been started more than 2 years ago and got delayed when the unimodularity of the connection Laplacian took over. There was an announcement [PDF] which is now included as an appendix. [Not to appear … ….

## The Hydrogen Relation

For a one-dimensional simplicial complex, the sign less Hodge operator can be written as L-g, where g is the inverse of L. This leads to a Laplace equation shows solutions are given by a two-sided random walk.

## Cohomology in six lines

Here is the code to compute a basis of the cohomology groups of an arbitrary simplicial complex. It takes 6 lines in mathematica without any outside libraries. The input is a simplicial complex, the out put is the basis for $H^0,H^1,H^2 etc$. The length of the code compares in complexity with computations in basic planimetric computations in a triangle (Example … ….

## 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.