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Update of May 27, 2017: I dug out some older unpublished slides authored in 2015 and early 2016.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

Energy theorem for Grothendieck ring

Energy theorem The energy theorem tells that given a finite abstract simplicial complex G, the connection Laplacian defined by L(x,y)=1 if x and y intersect and L(x,y)=0 else has an inverse g for which the total energy is equal to the Euler characteristic with . The determinant of is the Fermi characteristic . In the “spring 2017 linear algebra Mathematica … ….

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 … ….