An update about natural spaces, something about time and the irreversibility riddle, a dihedral time postulate and something about natural graphs.
Is the dyadic group of integers natural in the sense that there is a metric space which forces the group structure on it?
A natural metric space defines a unique group structure on the same set such that all group operations are isometries. The integers are not natural but the dihedral numbers are. The half integers are in this sense more natural than the integers.
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 … ….
The discrete Cauchy integral theorem can be formulated in the discrete so that it looks like in the continuum.
Two videos about discrete symplectic geometry and discrete complex analysis.
The Curvature of graphs multplies under the Shannon product (strong product).
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 … ….
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 … ….
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 … ….
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 … ….
10 theorems about discrete manifolds were featured in a youtube video.
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 … ….
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 … ….
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.
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 … ….
The results mentioned in the slides before are now written down. This document contains a proof of the energy relation . There are several reason for setting … ….
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.
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 … ….
We look at random circle valued fields on a simplicial complex and the spectra of the corresponding connection laplacian.
The energy theorem for simplicial complexes equipped with a complex energy comes with some surpises.
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 … ….
Being in the process of wrapping up the latest summer calculus course, here are some thoughts about the frame work of calculus and more generally of classical … ….
A major open problem in Riemannian geometry is the classification of even dimensional positive curvature manifolds with symmetry. There is a reduction theory which produces a periodic system of elements. This picture has affinity with gauge bosons in physics.
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 … ….
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 … ….
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 … ….
The Mickey mouse theorem assures that a connected positive curvature graph of positive dimension is a sphere.
Finite sets of sets can be seen as a combintarorial oddity. There is a lot of mathematical structure available on such simple finitist constructs.
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 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.
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.
Dehn-Sommerville spaces are generalized spheres as they share many properties of spheres: Euler characteristic and more generally Dehn-Sommerville symmetries.
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 … ….
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 relations are a symmetry for a class of geometries which are of Euclidean nature.
Branko Grünbaum (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”, … ….
Some update about recent activities: a new calculus course, the Cartan magic formula and some programming about the coloring algorithm.
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
We prove that connected combinatorial manifolds of positive dimension define finite simple graphs which are Hamiltonian.
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 beautiful Alexander duality theorem for finite abstract simplicial complexes.
We compute the quadratic interaction cohomology in the simplest case.
The interaction cohomology of the dunce hat is computed. We then comment on the discrete Lusternik-Schnirelmann theorem.
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 … ….
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.
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. … ….
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.
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.
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 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 … ….
We have seen that for a finite abstract simplicial complex $G$, the connection Laplacian L has an inverse g with integer entries and that $g(x,x) = 1-X(S(x))$, … ….
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 … ….
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 … ….
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?
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 … ….
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 … ….
The mathematics of evolving fields with two complex components is known already in Jones calculus.
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 … ….
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 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.
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 … ….
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 to interests … ….
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 … ….
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).
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 … ….
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 … ….
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.
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.
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.
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 … ….
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.
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 … ….
The Barycentric limit of the density of states of the connection Laplacian has a mass gap.
The tensor product is defined both for geometric objects as well as for morphisms between geometric objects. It appears naturally in connection calculus.
We look at examples of functional integrals on finite geometries.
As Goedel has shown, mathematics can not tame the danger that some inconsistency develops within the system. One can build bunkers but never will be safe. But the danger is not as big as history has shown. Any crisis which developed has been very fruitful and led to new mathematics. (Zeno paradox->calculus, Epimenids paradox ->Goedel, irrationality crisis ->number fields etc.
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 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 … ….
The history of the developent of energy and entropy is illustrated. This page is a picture book featuring some of the people involved shaping the concept of energy.
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 … ….
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.
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 … ….
An experimental observation : the sum over all Green function values is the Euler characteristic. There seems to be a Gauss-Bonnet connection.
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 … ….
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, … ….
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.
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.
This is a research in progress note while finding a proof of a conjecture formulated in the unimodularity theorem paper.
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 … ….
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 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) … ….
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.
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 … ….
The unimodularity theorem equates a fredholm determinant with a product of indices. It originally was formulated for graphs or simplicial complexes. It turns out to be valid for more general structures, generalized cellular complexes. While for discrete CW complexes, the fredholm determinant is 1 or -1, in general it can now take more general values but the structures are also more strange: in the continuum much more general than CW complexes as the attached cells do not need to be bound by spheres but can be rather arbitrary.
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 … ….
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. … ….
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 … ….
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 … ….
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.
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 … ….
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 … ….
[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 … ….
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 … ….
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.
How to define level surfaces or solve extremization problems in a graph. A comment on a recent paper on Sard.
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 is a natural coordinate free frame work for discrete calculus.
We have seen that 
![D[x]^n = n [x]^{n-1}](https://s0.wp.com/latex.php?latex=D%5Bx%5D%5En+%3D+n+%5Bx%5D%5E%7Bn-1%7D&bg=ffffff&fg=000000&s=0 "D[x]^n = n [x]^{n-1}")
We will often leave the constant $h$ out of the notation and use terminology like 
![[x]^n](https://s0.wp.com/latex.php?latex=%5Bx%5D%5En&bg=ffffff&fg=000000&s=0 "[x]^n")
Define the exponential function as
![exp(x) = \sum_{k=0}^{\infty} [x]^k/k!](https://s0.wp.com/latex.php?latex=exp%28x%29+%3D+%5Csum_%7Bk%3D0%7D%5E%7B%5Cinfty%7D+%5Bx%5D%5Ek%2Fk%21&bg=ffffff&fg=000000&s=0 "exp(x) = \sum_{k=0}^{\infty} [x]^k/k!")
![exp_n(x) = \sum_{k=0}^{n} [x]^k/k!](https://s0.wp.com/latex.php?latex=exp_n%28x%29+%3D+%5Csum_%7Bk%3D0%7D%5E%7Bn%7D+%5Bx%5D%5Ek%2Fk%21&bg=ffffff&fg=000000&s=0 "exp_n(x) = \sum_{k=0}^{n} [x]^k/k!")
where $e_n \to e$. Because $n \to e_n$ is monotone, we see that the exponential function 
Since the just defined exponential function is monotone, it can be inverted on the positive real axes. Its inverse is called 
Let denote the discrete derivative of a continuous function f on the real line. In this post, I assume that all functions are continuous of have compact … ….
Assume f is a continuous function of one real variable. Lets call a point p a critical point of f if Df(p)=0 where Df(x) = f(x+1)-f(x) is … ….
