Helmholtz free energy for simplicial complexes

Over spring break, the Helmholtz paper [PDF] has finished. (Posted now on “On Helmholtz free energy for finite abstract simplicial complexes”.) As I will have little time during the rest of the semester, it got thrown out now. It is an interesting story, relating to one of the greatest scientist, Hermann von Helmholtz (1821-1894). It is probably one of the … ….

Shannon Entropy and Euler Characteristic

Entropy is the most important functional in probability theory, Euler characteristic is the most important functional in topology. Similarly as the twins Apollo and Artemis displayed above they are closely related. Introduction This blog mentions some intriguing analogies between entropy and combinatorial notions. One can push the analogy in an other direction and compare random variables with simplicial complexes, Shannon … ….

Sphere spectrum paper

The sphere spectrum paper is submitted to the ArXiv. A local copy. It is an addition to the unimodularity theorem and solves part of the riddle about the Green function values, the diagonal elements of the inverse of the matrix 1+A’ where A’ is the adjacency matrix of the connection graph of the simplicial complex. The paper contains two main … ….

Spectra of Sums of Networks

What happens with the spectrum of the Laplacian if we add some graphs or simplicial complexes? (I owe this question to An Huang). Here is an example, where we sum a circular graph G=C4 and a star graph H=S4. The Laplace spectrum of G is {4,2,2,0}, the spectrum of H is {5,1,1,1,0}. The spectrum of G+H is {9, 9, 9, … ….

Arithmetic with networks

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.

Partial differential equations on graphs

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 … ….