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 calculus” in which Euler characteristic is the most natural functional. … ….

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).

One ring to rule them all

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 more convoluted as more and more results were coming in. There had been some disappointment early June when realizing that the Zykov multiplication (which I had been proud of discovering in early January) … ….

Hardy-Littlewood Prime Race

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 … ….

The quantum plane

Update of May 27, 2017: I dug out some older unpublished slides authored in 2015 and early 2016.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 … ….