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 only dealt with the Euler characteristic, an additive valuation (in the sense of Klain and Rota). Since then, the work on the Fredholm characteristic has made more progress and is now understood. The … ….
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 fantastic to keep talks concise and to the point. The video has been produced by Diane Andonica from the Bok Center for teaching and learning. The event had been organized by Curt Mc … ….
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 in 140 characters: A finite simple graph G=(V,E) defines H=(W,F) … ….
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 rendering therefore the Kepler motion into a harmonic oscillator. The … ….
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, must be in want of a place to be read. … ….
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 gauge group? Why are there three generations of particles? Why do neutrini oscillate? How is general relativity included? (See for example page 540 in Woit’s online monograph). When experimenting with quaternion primes, especially … ….
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 column vectors, in multivariable calculus, we only look at one … ….