Cohomology in six lines

Here is the code to compute a basis elements of the cohomology groups of an arbitrary simplicial complex. It takes 6 lines in mathematica without any outside libraries. This is so simple, it compares in complexity with computations in basic planimetric computations in a triangle (implementing one of the Euclid constructions on a computer). We just compute the Dirac operator … ….

Is there physics for the connection Laplacian?

The classical potential $V(x,y) = 1/|x-y|$ has infinite range which violently clashes with relativity. This had required a completely new theory. It remains also a fundamental problem still in general relativity: a Gedanken experiment in which the particles in the sun suddenly transition to particles without mass shows this. [This is forbidden by energy conservation but energy conservation is not … ….

Quest for a Green Function Formula

A simplicial complex G, a finite set of non-empty sets closed under the operation of taking finite non-empty subsets, has the Laplacian $L(x,y) = {\rm sign}(|x \cap y|)$. It is natural as it is always unimodular so that its inverse $g(x,y)$ is always integer valued. In a potential theoretical setup, the Green function values $g(x,y)$ measure a potential energy between … ….

More Green Function Values

We have seen that for a finite abstract simplicial complex $G$, the connection Laplacian L has an inverse g with integer entries and that $g(x,x) = 1-X(S(x))$, where $S(x)$ is the unit sphere of $x$ in the graph $G_1=(V,E)$, where $V=G$ and where (a,b) in E if and only if $a \subset b$ or $b \subset a$. We have also … ….

Isospectral Simplicial Complexes

One can not hear a complex! After some hope that some kind of algebraic miracle allows to recover the complex from the spectrum (for example by looking for the minimal polynomial which an eigenvalue has and expecting that the factorization reflects some order structure in the abstract simplicial complex), I wondered whether there is an argument proving that that there … ….

Wenjun Wu, 1919-2017

According to Wikipedia, the mathematician Wen-Tsun Wu passed away earlier this year. I encountered some mathematics developed by Wu when working on Wu characteristic. See the Slides and the paper on multi-linear valuations. There is an other paper on this in preparation, especially dealing with the cohomology belonging to Wu characteristics. Just as a reminder, the Wu characteristic of a … ….

Hearing the shape of a simplicial complex

A finite abstract simplicial complex has a natural connection Laplacian which is unimodular. The energy of the complex is the sum of the Green function entries. We see that the energy is also the number of positive eigenvalues minus the number of negative eigenvalues. One can therefore hear the Euler characteristic. Does the spectrum determine the complex?