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 … ….
Classical calculus we teach in single and multi variable calculus courses has an elegant analogue on finite simple graphs. The discrete theory is completely analogue, Stokes theorem is almost tautological and simplicial cohomology is easy to compute just by finding the nullity of some Laplacians. Taylor expansions on graphs are just wave equation solutions on graphs. Any partial differential equation … ….
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 are some slides: So, why is the Wu characteristic an … ….
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.