Quantum Calculus

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 …

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 …

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.

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 …

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 …

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 …

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?

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 …

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, …

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).

The finitist bunker

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.

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, …

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 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.

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 …

The Unimodularity Theorem for CW Complexes

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.

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 …

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.

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 …

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.

Fundamental Theorem

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 support. They are zero outside some large interval. No smoothness is required of course. Here is the simplest version of the fundamental theorem …