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.
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.
This is a research in progress note while finding a proof of a conjecture formulated in the unimodularity theorem paper.
During the summer and fall of 2016, Annie Rak did some URAF (a program formerly called HCRP) on partial differential equations on graphs. It led to a senior thesis in the applied mathematics department. Here is a project page and here [PDF] were some notes from the summer. The research of Annie mostly dealt with advection models on directed graphs … ….
Here are some slides about the paper. By the way, an appendix of the paper contains all the code for experimenting with the structures. To copy paste the code, one has to wait for the ArXiv version, where the LaTeX source is always included. Here is the unimodularity theorem again in a nutshell: Given a finite abstract simplicial complex G, … ….
The Christmas Theorem Because Pierre de Fermat announced his two square theorem to Marin Mersenne in a letter of December 25, 1640 (today exactly 376 years ago) the theorem A prime of the form 4k+1 is the sum of two squares. is also called the Christmas theorem. The converse, the fact that a prime p of the form 4k+3 is … ….
The proof of the unimodularity theorem is finished.
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.