# Quantum Calculus

Calculus without limits
##### Cohomology in six lines
Here is the code to compute a basis elements of the cohomology...
##### 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
Classical calculus we teach in single and multi variable calculus courses has...
##### 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...
##### Calculus on graphs
Calculus on graphs is a natural coordinate free frame work for discrete...
##### 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)...
##### 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...

## 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 there is also arithemetic. We remark that the multiplicative primes in the ring are the simplicial complexes. The Sabidussi theorem imlies that additive primes (particles) have a unique prime factorization (into elementary particles).

## 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 Ln‘ with itself has a mass gap in the limit n to infinity. The picture below shows this product graph for n=13, and to the right s part of the spectrum near 0 for n=40. … ….

## One ring to rule them all

Arithmetic with networks The paper “On the arithmetic of graphs” is posted. (An updated PDF). The paper is far from polished, the document already started to become more convoluted as more and more results were coming in. There had been some disappointment early June when realizing that the Zykov multiplication (which I had been proud of discovering in early January) … ….

## Three Kepler Problems

Depending on scale, there are three different Kepler problems: the Hydrogen atom, the Newtonian Kepler problem as well as the binary Blackhole problem. The question whether there is a unifying model which covers all of them is part of the quest of finding a quantum theory of gravity.

## More about the ring of networks

The dual multiplication of the ring of networks is topological interesting as Kuenneth holds for this multiplication and Euler characteristic is a ring homomorphism from this dual ring to the ring of integers.

## Unique prime factorization for Zykov addition

We give two proofs that the additive Zykov monoid on the category of finite simple graphs has unique prime factorization. We can determine quickly whether a graph is prime and also produce its prime factorization.

## Hardy-Littlewood Prime Race

The Hardy-Littlewood race has been running now for more than a year on my machine. The Pari code is so short that it is even tweetable. Here are some slides which also mention Gaussian Goldbach: What do primes have to do with quantum calculus? First of all, analytic number theory is all about calculus. But as mentioned in other places … ….

## The Hydrogen trace of a complex

Motivated by the Hamiltonian of the Hydrogen atom, we can look at an anlogue operator for finite geometries and study the spectrum. There is an open conjecture about the trace of this operator.

## The quantum plane

Update of May 27, 2017: I dug out some older unpublished slides authored in 2015 and early 2016. I added something about the quantum gap and something on the quantum plane at the very end. Here is the presentation, just spoken now. The quantum line In one dimension, there is a natural compact metric space D on which one has … ….

## A mass gap in the Barycentric limit

The Barycentric limit of the density of states of the connection Laplacian has a mass gap.