Calculus without limits ## Arboricity of Surfaces

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

## Acyclic Chromatic Number

The Acyclic Chromatic Number is bounded above by the arboricity. We can improve this by one if the Acyclic Chromatic number is even.

## Arboricity of spheres

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).

## Three Trees Suffice

We work on the result that any 2-sphere can be covered by 3 trees.
In other words: three trees suffice.

## Arboricity, Dimension, Category

We aim to show that the Lusternik-Schnirelmann category of a graph is bound by the augmented dimension. We can try to prove this by using tree coverings and so look at arboricity.

## Discrete Vector Fields

A notion of a discrete vector should work for theorems like Poincare-Hopf and also produce a dynamics as classically, a vector field F, a smooth section of the tangent bundle on a manifold produces a dynamics . A directed graph does not give a dynamics without telling how to go …

## Glass Theorem

In 1973 Leon Glass proved a discrete Poincare Hopf theorem for directed graphs embedded in a 2 dimensional manifold. Kate Perkins has related this to a discrete Poincare Hopf theorem of mine. This is a discussion of the connection.

## Discrete Time Wave Equation

It is a simple but interesting fact that if we look at a wave equation with discrete space, we also need discrete time provided we want causality, a property which every wave equation should have. The reason is that the solution equation in a causal locally finite geometry is an …

## Markov Dynamics on a Complex

Dynamical systems on geometric spaces are very common in mathematics. Mean curvature flows, Ricci flows etc. We can think of a subcomplex K in a simplicial complex G as a geometric object. We can not break randomly elements away as would in general lose the property of having a subcomplex. …

## Four Papers Relevant to the Fusion Inequality

We write a bit more about four papers which are relevant for the Fusion inequality on the Betti numbers of an open and closed pair in a simplicial complex.

## Projective Tales

An update about the scattering problem when an open and closed set merge. During the process the harmonic forms on K and U merge to harmonic forms on G. The open problem is to prove that no new harmonic forms can appear. This is the content of the fusion inequality. b(U) + b(K) – b(G) being non-negative.

## The 5 lines of Cohomology

Over the weekend, I gave a glimpse on some code which allows to compute the cohomology of an open or closed set in a simplicial complex. Here are 5 lines for cohomology. We see how elegant this can all be. Simplicity, Clarity and Generality. This code could in principle compute …

## A multi-particle energy theorem

A finite abstraact simplicial complex or a finite simple graph comes with a natural finite topological space. Some quantities like the Euler characteristic or the higher Wu characteristics are all topological invariants. One can also reformulate the Lefschetz fixed point theorem for continuous maps on finite topological spaces.

## Morse, Lusternik and Schnirelmann

The Arnold conjecture in symplectic geometry has been tackled with two different approaches. One of them is Morse theory, an other one is Lusternik-Schnirelmann theory.

## Bosonic and Fermionic Features in Mathematics

Even or odd, symmetric or anti-symmetric, integer or half integer, measures or de Rham currents, densities or differential forms, undirected or directed, orientation oblivious, or orientation sensitive, primes of the form 4k+1 or primes of the form 4k-1, permanents or determinants: there are many notions of mathematics which can be …

## Updates to the Cauchy Central Limit

There were two updates on the Cauchy central limit theorem telling that if the Cauchy mean and risk of a random variable X with PDF f is finite and non-zero, then any IID random process with that distribution has normalized sums which converge in distribution to the Cauchy distribution. There …

## A central limit theorem for high risk

There were two triggers for this: first of all the KAM fixed point equation is for c=0 a renormalization map which matters in probability theory, in particular for the Cauchy distribution which is a fixed point under the map which gets from a distribution the distribution of [X+Y]/2, where X,Y …

## A KAM Challenge

There are various versions of the implicit function theorem. We look first at the soft implicit function theorem, then a theorem of John Neuberger, then at the twist map theorem which uses a hard implicit function theorem.

## Incidence and Intersection

Barycentric and Connection graphs Barycentric graphs depend on incidence, connection graphs on intersection. Here are some examples from this blog. Both graphs have as the vertex set the complete subgraphs of the graph. In the connection graph, we take the intersection, in the Barycentric case, we take incidence. Here are …

## Exploring machine graphs

All data are finite. A computer can only represent finitely many real numbers in an interval. We can connect two such numbers if they are considered equal by the computer. This produces a graph,l the machine graph. It is a special Vietoris-Rips graph. The story can also illustrate non-standard notions.

## From Pebbles to Networks

Leonhard Euler wrote in 1767 a “complete guide to algebra”. It might appear strange that one of the most productive and innovative mathematicians of all time would “waste” his time with writing a textbook for school children. But Euler knew how important algebra is. Not only for school but also …

## Do we need infinity?

In the new Netflix documentary “a trip to infinity” the question of quantum space comes up. It is interesting to see Brian Green (the famous TV star covering about quantum gravity, big bang, fabric of the cosmos elegant universe or string theory) now seems have been converted to the “finite …

## More Mandelstuff

A bit more about Mandelbulb, Mandelbrot, Hopfbrot and discrete Mandelstuff. Some slides [PDF]. To be pure to the discrete, we also have a short part on Mandelbrot sets in finite rings … Update September 15, 2022: Mandelbulbs in Mathematica in 3 lines. I worked this morning to get this down …

## Mandelstuff

The White-Nylander Mandelbulb is one of the most beautiful objects in mathematics. It extends the Mandelbrot set since one slice is the Mandelbrot set.

## Cool Inverse Problems

We look at two inverse problems in mathematics. The first asks to reconstruct a function from its level curves. The second asks to reconstruct a graph from the quiver spectra.

## Riemann Roch for a Single Point

The current trend in mathematics is to become more and more abstract and strange. We go here the opposite way and look at Riemann-Roch in the simplest possible case, when space is a point!

## Graph Products

The video of August 20 also gave a bit of an overview of graph products. I should have mentioned the Lexicographic product (introduced by Hausdorff in 1914) mentioned in my article on Graphs, Groups and Geometry. Here is an abbreviation of the story of that video: for finite simple graphs, …

## Delta Sets for Quivers

A new video recorded on August 13, 2022. We look at how Delta sets can be used to define a natural Euler characteristic for quivers. Delta sets have been introduced in 1950 by Samuel Eilenberg and Joseph Zilber. Delta sets are for me much more intuitive than simplicial sets. Look …

## Euler Characteristic of Quivers

About the problem of defining an Euler characteristic for a multi-graph or quiver. While for finite simple graphs we have a natural associated finite abstract simplicial complex, we need for multi-graphs a more general structure. Delta sets do the job and allow to define a good Euler characteristic.

## More Quiver Math

The paper [PDF] on the upper bound is updated a bit. It will also be updated on the ArXiv. In the video below, there is an update. Both for quivers as well as non-magnetic quivers (quivers without multiple connections), we can use induction to prove results. One of the amendments …

## Schroedinger operators on Graphs

Quivers allow to model Schroedinger operators on graphs. With loops we can modify scalar values on vertices, with multiple connections, we can modify the magnetic part. This makes quivers attractive as well as theorems about quivers more interesting. An example was . In the talk last Saturday, I also mentioned …

## Spectra of Quivers

Quiver Terminology Quivers are finite graphs (V,E) in which also multiple connections or loops are allowed. V is a finite set and E is just a finite set of pairs (a,b) with a,b in V. Now (a,a) loops are allowed as are multiple occurrences of the same entry (a,b). The …

## The Upper bound Spectral Theorem

In my theorem telling that the k-th largest eigenvalue for the Kirchhoff Laplacian of a finite simple graph satisfies the bound I use a lemma which generalizes the Andrson-Morley super symmetry result of 1985. That paper had provided a ground breaking upper bound for the spectral radius . The proof …

## Soft and hard manifolds

The definition of soft manifoldss now fixed. It is only fitting to call the old discrete manifolds hard manifolds. Unlike the later, soft manifolds can be multipied. They define a subring of the Shannon ring.

## Soft Manifolds

What is space? In Spring 2021, prompted by work on graph complements of circular graphs, I started to think more about discrete manifolds. One can see the definition on this video on discrete homotopy manifolds. In the spring of 2022, the definition got slightly modified and the name homotopy manifold …

## Bounds on Graph Spectra

An update on tree and forest indices which measure the exponential growth rate of the number of spanning trees or forests when doing Barycentric refinement. This needs an upper bound estimate of the eigenvalues of the Kirchhoff Laplacian.

## Tree Forest Ratio

The tree forest ratio of a finite simple graph is the number of rooted spanning forests divided by the number of rooted spanning trees. By the Kirchhoff matrix tree theorem and the Chebotarev-Shamis matrix forest theorem this is where Det is the pseudo determinant and K the Kirchhoff matrix the …

## The Maze Theorem

The maze theorem tells that the number of spanning trees in a 2-sphere is equal to the number of spanning trees in the dual graph.

## More on Analytic Torsion

We report on some progress on analytic torsion A(G) for graphs. A(G) is a positive rational number attached to a network. We can compute it for contractible graphs or spheres.

## Analytic torsion for graphs

We comment on analytic torsion for graphs. We prove here a conjecture voiced in the video (discovered experimentally in 2015) that the analytic torsion of a 2-dimensional sphere is |V|/|F|.

## Lamplighter Group

This still belongs to the framework of natural groups. The Lamplighter group as a wreath product or semi-direct product is a prototype group which illustrates some mathematics. First of all, the group, like the integers, is not a natural group. Given a metric structure invariant under the group, one can …

## The quantum plane riddle

This is a bit of an update on the problem to find the limiting law in the Barycentric central limit theorem. (See some older slides.) The distribution has first experimentally been found in the PeKeNePaPeTe paper in 2012. I proved universality in 2015 using a modification of the Lidski theorem …

## Cayley, Zig-Zag and Rubik

We continue to look at natural groups.
There is some progress in seeing that the Rubik cube group is natural: semi direct products can be represented by zig-zag products of Cayley graphs.

## Group Extensions and Naturalness

A bit about group theory triggered by the observation that some of the non-natural groups are non-simple groups which do not split. The later problem is a central issue in the classification problem of finite groups. While the finite simple groups have been classified in a monster effort and finitalized …

## Graphs, Groups and Geometry

A bit more update on the project of natural spaces. Which groups are natural, which metric spaces are natural, which graphs are natural?

## Towards Ultra-Finite Taylor

A discrete ultra-finite Taylor theorem is a perfect expansion producing exact numbers (data fitting) with the property that evaluating the value of the sum at every point is a finite sum.

## Dihedral time

An update about natural spaces, something about time and the irreversibility riddle, a dihedral time postulate and something about natural graphs.

## Ultra finite calculus

These days, I look a bit at an elementary problem. Deform the algebra of polynomials and derivatives so that all Taylor series are finite sums when evaluated at some point. This is a task which an ultra finitist is interested in. Ultra finitism is a branch of mathematics in which …

## Discrete Cauchy Integral Formula

The discrete Cauchy integral theorem can be formulated in the discrete so that it looks like in the continuum.

## More on Graph Arithmetic

A few more remarks [PDF] about graph arithmetic. Now on the ArXiv. (Previous documents are here (June 2017 ArXiv), here (August 2017 ArXiv) and here (May 2019 ArXiv). The talk below on youtube was used for me to get organized a bit. It does not look like much has changed, …

## Topology of Manifold Coloring

Last summer I have had some fun with codimension 2 manifolds M in a purely differential geometric setting: a positive curvature d-manifold which admits a circular action of isometries has a fixed point set K which consists of even codimension positive curvature manifold. The Grove-Searle situation https://arxiv.org/abs/2006.11973 is when K …

## Coloring Discrete Manifolds

It is now 7 years, since I started to think about coloring discrete manifolds, graphs for which unit spheres are (d-1)-spheres, where d-spheres are d-manifolds which after a removal of a vertex become contractible. The problem has not gained any attention which is a good thing because there are many …

Last week, I practiced a bit more enhanced talk presentation style in which, rather than with slides, the content is spoken and then enhanced in the video using additonal illustrations. The presentation deals with some things I have done in graph theory which I consider as part of quantum calculus …

## Homotopy Manifolds

The problem of discretization It is a question which probably was pondered first by philosophers like Democrit or Archimedes. What is the nature of space? It is made of discrete stuff or is it a true continuum? As Plato already noted, such questions border to being pointless as we live …

## Discrete Real Analysis

One dimension I have always also fun teaching single variable calculus. This year, the course site is here. One of the great pioneers in real analysis was Bernard Bolzano. He was not recognized enough during his life but he was one of the first to realize that theorems about continuous …

## Graph Complements of Cyclic Graphs

Graph complements of cylic graphs are homotopic to spheres or wedge sums of spheres. Their unit spheres are graph complements of path graphs and have Gauss-Bonnet curvature which converges to a limit.

## Calculating with Networks

Summary of update of current work Since a few weeks I work more on the arithmetic of graphs and especially the strong ring of simplicial complexes which can be extended to a unital commutative Banach algebra. This line of research had started late 2016, while preparing for a workshop on …

## Energy relation for Wu characteristic

The energy theorem for Euler characteristic X= sum h(x)was to express it as sum g(x,y)of Green function entries. We extend this to Wu characteristic w(G)= sum h(x) h(y) over intersecting sets. The new formula is w(G)=sum w(x) w(y) g(x,y)2, where w(x) =1 for even dimesnional x and w(x)=-1 for odd dimensional x.

## Mathematicians who did not die

As Hardy once pointed out in his apology, “languages die but mathematical ideas do not”. Hardy added that a mathematician “]has the best chance of becoming immortal whatever the word immortal might mean”. Let me illustrate this with three mathematicians, all teachers of mine, who have earned more than immortality. …

## Complex energized complexes

The energy theorem for simplicial complexes equipped with a complex energy comes with some surpises.

## A Sine Gordon Theme

Abstract for this post Analysis had not been my favorite topic at first. The subject appeared to me quite technical and tedious even when sweetened like in analytic number theory. The Sine-Gordon topic and other rather concrete topics in dynamical systems theory changed that a bit and I ended up …

## Physics on finite sets of sets?

Introduction The idea to base physics on the evolution finite set of sets is intriguing. It has been tried as an approach to quantum gravity. Examples are causal dynamical triangulation models or spin networks. It is necessary to bring in some time evolution as otherwise, a model has little chance …

## The Hopf Conjectures

The Hopf conjectures were first formulated by Hopf in print in 1931. The sign conjecture claims that positive curvature compact Riemannian 2d-manifolds have positive Euler characteristic and that negative curvature compact Riemannian 2d-manifolds have Euler characteristic with sign (-1)d . The product conjecture claims there is no positive curvature metric …

## Poincare-Hopf for Vector Fields on Graphs

The question In discrete Poincare-Hopf for graphs the question appeared how to generalize the result from gradient fields to directed graphs. The paper mentions already the problem what to do in the case of the triangle with circular orientation. The triangle has Euler characteristic 1. An integer index on vertices …

## Mickey Mouse Sphere Theorem

The Mickey mouse theorem assures that a connected positive curvature graph of positive dimension is a sphere.

## Energized Simplicial Complexes

If a set of set is equipped with an energy function, one can define integer matrices for which the determinant, the eigenvalue signs are known. For constant energy the matrix is conjugated to its inverse and defines two isospectral multi-graphs.

## The counting matrix of a simplicial complex

The counting matrix of a simplicial complex has determinant 1 and is isospectral to its inverse. The sum of the matrix entries of the inverse is the number of elements in the complex.

## Poincare-Hopf and the Clique Problem

The parametrized poincare-hopf theorem allows to see the f-vector of a graph in terms of the f-vector s of parts of the unit spheres of the graph.

## Small Dehn-Sommerville Spaces

Dehn-Sommerville spaces are generalized spheres as they share many properties of spheres: Euler characteristic and more generally Dehn-Sommerville symmetries.

## On Numbers and Graphs

We have calculated with graphs from the very beginning: Humans computed with pebbles like in this scene of the Clan of the Cave Baer (1986)) or with line graphs when writing with tally sticks (see this lecture). In all of these cases, the addition of graphs is the disjoint union …

## Gauss-Bonnet for the f-function

The f-function of a graph minus 1 is the sum of the antiderivatives of the f-function anti-derivatives evaluated on the unit spheres.

## Dehn-Sommerville

Dehn-Sommerville relations are a symmetry for a class of geometries which are of Euclidean nature.

## Branko GrÃ¼nbaum

Branko GrÃ¼nbaum (1929-2018) passed away last September. Here is the obituary from the university of Washington. One of his master pieces is the book “Tilings and patterns”, written with G.C. Shephard. The well illustrated book is considered the bible on Tilings. Here is a page from that book: Links: Personal …

## Discrete Calculus etc

Some update about recent activities: a new calculus course, the Cartan magic formula and some programming about the coloring algorithm.

## Euler Game

We prove that any discrete surface has an Eulerian edge refinement. For a 2-disk, an Eulerian edge refinement is possible if and only if the boundary length is divisible by 3

## The Hamiltonian Manifold Theorem

We prove that connected combinatorial manifolds of positive dimension define finite simple graphs which are Hamiltonian.

## 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.

## 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.

## 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.

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 on the imaginary axes. This is the Dyadic Riemann Hypothesis.

## 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 J
a checkerboard matrix. The eigenvector has components whicih are
Wu intersection numbers.

## 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 remains also a fundamental problem still in general relativity: a Gedanken experiment in which the particles in the sun suddenly transition to particles without mass …

## Quest for a Green Function Formula

A simplicial complex G, a finite set of non-empty sets closed under the operation of taking finite non-empty subsets, has the Laplacian $L(x,y) = {\rm sign}(|x \cap y|)$. It is natural as it is always unimodular so that its inverse $g(x,y)$ is always integer valued. In a potential theoretical setup, …

## 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 …

## 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 …

## 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 …

## 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?

## 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, …

## 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 and a connection operator L. Both are related in
various ways to topology.

## The Dirac operator of Products

Implementing the Dirac operator D for products of simplicial complexes without going to the Barycentric refined simplicial complex has numerical advantages. If G is a finite abstract simplicial complex with n elements and H is a finite abstract simplicial complex with m elements, then is a strong ring element with …

## 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 strong ring is a “geometric space” carrying cohomology (simplicial, and more general interaction cohomologies) and has nice spectral properties (like McKean Singer) and a “counting …

## 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 …

## 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 …

## 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 …

## 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 …

## Tensor Products Everywhere

The tensor product is defined both for geometric objects as well as for morphisms between geometric objects. It appears naturally in connection calculus.

## The Helmholtz Hamiltonian System

As we have an internal energy for simplicial complexes and more generally for every element in the Grothendieck ring of CW complexes we can run a Hamiltonian system on each geometry. The Hamiltonian is the Helmholtz free energy of a quantum wave.

## Energy theorem for Grothendieck ring

Energy theorem The energy theorem tells that given a finite abstract simplicial complex G, the connection Laplacian defined by L(x,y)=1 if x and y intersect and L(x,y)=0 else has an inverse g for which the total energy is equal to the Euler characteristic with . The determinant of is the …

## Helmholtz free energy for simplicial complexes

Over spring break, the Helmholtz paper [PDF] has finished. (Posted now on “On Helmholtz free energy for finite abstract simplicial complexes”.) As I will have little time during the rest of the semester, it got thrown out now. It is an interesting story, relating to one of the greatest scientist, …

## Energy, Entropy and Gibbs free Energy

Energy U and Entropy S are fundamental functionals on a simplicial complex equipped with a probability measure. Gibbs free energy U-S combines them and should lead to interesting minima.

## Shannon Entropy and Euler Characteristic

Entropy is the most important functional in probability theory, Euler characteristic is the most important functional in topology. Similarly as the twins Apollo and Artemis displayed above they are closely related. Introduction This blog mentions some intriguing analogies between entropy and combinatorial notions. One can push the analogy in an …

## A Gauss-Bonnet connection

An experimental observation : the sum over all Green function values is the Euler characteristic. There seems to be a Gauss-Bonnet connection.

## Sphere spectrum paper

The sphere spectrum paper is submitted to the ArXiv. A local copy. It is an addition to the unimodularity theorem and solves part of the riddle about the Green function values, the diagonal elements of the inverse of the matrix 1+A’ where A’ is the adjacency matrix of the connection …

## Spectra of Sums of Networks

What happens with the spectrum of the Laplacian if we add some graphs or simplicial complexes? (I owe this question to An Huang). Here is an example, where we sum a circular graph G=C4 and a star graph H=S4. The Laplace spectrum of G is {4,2,2,0}, the spectrum of H …

## A ring of networks

Assuming the join operation to be the addition, we found a multiplication which produces a ring of oriented networks. We have a commutative ring in which the empty graph is the zero element and the one point graph is the one element. This ring contains the usual integers as a subring. In the form of positive and negative complete subgraphs.

## Arithmetic with networks

The join operation on graphs produces a monoid on which one can ask whether there exists an analogue of the fundamental theorem of arithmetic. The join operation mirrors the corresponding join operation in the continuum. It leaves spheres invariant. We prove the existence of infinitely many primes in each dimension and also establish Euclid’s lemma, the existence of prime factorizations. An important open question is whether there is a fundamental theorem of arithmetic for graphs.

## Sphere Spectrum

This is a research in progress note while finding a proof of a conjecture formulated in the unimodularity theorem paper.

## Partial differential equations on graphs

During the summer and fall of 2016, Annie Rak did some URAF (a program formerly called HCRP) on partial differential equations on graphs. It led to a senior thesis in the applied mathematics department. Here is a project page and here [PDF] were some notes from the summer. The research …

## Unimodularity theorem slides

Here are some slides about the paper. By the way, an appendix of the paper contains all the code for experimenting with the structures. To copy paste the code, one has to wait for the ArXiv version, where the LaTeX source is always included. Here is the unimodularity theorem again …

## From the Christmas Theorem to Particle Physics

The Christmas Theorem Because Pierre de Fermat announced his two square theorem to Marin Mersenne in a letter of December 25, 1640 (today exactly 376 years ago) the theorem A prime of the form 4k+1 is the sum of two squares. is also called the Christmas theorem. The converse, the …

## Dimension: from discrete to general metric spaces

This is an informal overview over definitions of dimension, both in the continuum as well as in the discrete. It also contains suggestions for generalizations to general metric spaces.

## Euler and Fredholm

The following picture illustrates the Euler and Fredholm theme in the special case of the prime graphs introduced in the Counting and Cohomology paper. The story there only dealt with the Euler characteristic, an additive valuation (in the sense of Klain and Rota). Since then, the work on the Fredholm …

## Quantum calculus talk of 2013

Just uploaded a larger version of my 2013 Pecha-Kucha talk “If Archimedes knew functions…”. The Pecha-Kucha format of presenting 20 slides with 20 seconds time each is fantastic to keep talks concise and to the point. The video has been produced by Diane Andonica from the Bok Center for teaching …

## On Bowen-Lanford Zeta Functions

Zeta functions are ubiquitous in mathematics. One of the many zeta functions, the Bowen-Lanford Zeta function was introduced by my Phd dad Oscar Lanford and Rufus Bowen. I am in the process to wrap up a proof of a theorem which is so short that its statement can be done …

## The Kustaanheimo prime

Paul Kustaanheimo (1924-1997) was a Finnish astronomer and mathematician. In celestial mechanics, his name is associated with the Kustaanheimo-Stiefel transform or shortly KS transform which allows to regularize the Kepler problem using Clifford algebras. In this elegant picture, the motion of the two bodies becomes a rotation in three dimensions …

## Particles and Primes, Counting and Cohomology

Source: Pride and Prejudice, 2005 Judy Dench plays the role of Lady Catherine de Bourgh. I recently posted a “Particles and Primes” as well as a “Counting and Cohomology” article on the ArXiv, because, as it is a truth universally acknowledged, that an article in possession of a good result, …

## Counting and Cohomology

There are various cohomologies for finite simplicial complexes. If the complex is the Whitney complex of a finite simple graph then many major results from Riemannian manifolds have discrete analogues. Simplicial cohomology has been constructed by Poincaré already for simplicial complexes. Since the Barycentric refinement of any abstract finite simplicial complex is always the Whitney complex of a finite simple graph, there is no loss of generality to study graphs instead of abstract simplicial complexes. This has many advantages, one of them is that graphs are intuitive, an other is that the data structure of graphs exists already in all higher order programming languages. A few lines of computer algebra system allow so to compute all cohomology groups. The matrices involved can however become large, so that alternative cohomologies are desired.

## Quaternions and Particles

The standard model of particle physics is not so pretty, but it is successful. Many lose ends and major big questions remain: is there a grand unified gauge group? Why are there three generations of particles? Why do neutrini oscillate? How is general relativity included? (See for example page 540 …

## Bosonic and Fermionic Calculus

Traditional calculus often mixes up different spaces, mostly due to pedagogical reasons. Its a bit like function overload in programming but there is a prize to be payed and this includes confusions when doing things in the discrete. Here are some examples: while in linear algebra we consider row and …

## Interaction cohomology

[Update, March 20, 2018: see the ArXiv text. See also an update blog entry with some Mathematica code. More mathematica code can be obtained from the TeX Source of the ArXiv article.]. Classical calculus we teach in single and multi variable calculus courses has an elegant analogue on finite simple …

## Wu Characteristic

Update: March 8, 2016: Handout for a mathtable talk on Wu characteristic. Gauss-Bonnet for multi-linear valuations deals with a number in discrete geometry. But since the number satisfies formulas which in the continuum need differential calculus, like curvature, the results can be seen in the light of quantum calculus. Here …

## Barycentric refinement

A finite graph has a natural Barycentric limiting space which can serve as the geometry on which to do quantum calculus or physics. The holographic picture has universal spectral properties.

## Why quantum calculus?

Quantum calculus is easy to learn, allows experimentation with small worlds and allows to use the same notation we are used to classically.

## Calculus on graphs

Calculus on graphs is a natural coordinate free frame work for discrete calculus.

## Exponential Function

We have seen that $f'(x)=Df(x) = (f(x+h)-f(x))/h$ satisfies $D[x]^n = n [x]^{n-1}$.
We will often leave the constant $h$ out of the notation and use terminology like $f'(x) = Df(x)$ for the “derivative”. It makes sense not to simplify $[x]^n$ to $x^n$ since the algebra structure is different.

Define the exponential function as $exp(x) = \sum_{k=0}^{\infty} [x]^k/k!$. It solves the equation $Df=f$. Because each of the approximating polynomials $exp_n(x) = \sum_{k=0}^{n} [x]^k/k!$ is monotone and positive also $exp(x)$ is monotone and positive for all $x$. The fixed point equation $Df=f$ reads $f(x+h) = f(x) + h f(x) = (1+h) f(x)$ so that for $h=1/n$ we have $f(x+1) = f(x+n h) = (1+h)^n f(x) = e_n f(x)$
where $e_n \to e$. Because $n \to e_n$ is monotone, we see that the exponential function $\exp(x)$ depends in a monotone manner on h and that for $h \to 0$ the graphs of $\exp(x)$ converge to the graph of $\exp(x)$ as $h \to 0$.

Since the just defined exponential function is monotone, it can be inverted on the positive real axes. Its inverse is called $\log(x)$. We can also define trigonometric functions by separating real and imaginary part of $\exp(i x) = \cos(x) + i \sin(x)$. Since $D\exp=\exp$, these functions satisfy $D\cos(x) = - \sin(x)$ and $D\sin(x) = \cos(x)$ and are so both solutions to $D^2 f = -f$.