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 are some slides which also mention Gaussian Goldbach: What do primes have to do with quantum calculus? First of all, analytic number theory is all about calculus. But as mentioned in other places … ….
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Update of May 27, 2017: I dug out some older unpublished slides authored in 2015 and early 2016.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 on the quantum plane at the very end. Here is the presentation, just spoken now. The quantum line In one dimension, there is a natural compact metric space D on which one has … ….
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.