Given a q-manifold, a valuation X like Euler characteristic or volume and an integer , we can look at the variational problem to maximize or minimize X(H) among all submanifolds of X. A submanifold is given by a function . This is a finite problem so that we can in …
We still have not yet implemented the construction of geodesic manifolds in the computer. The local constructions work. What happens however is that when building up the manifold, we will have to rename the vertices as we locally do identifications which globally do not hold. In the case k=1, when …
Given a finite abstract simplicial complex G that is a q-manifold, we consider a k-simplex y in an oriented maximal simplex x as an element in the Grassmannian G(k,q). Of course we are in finite geometry and have no vector spaces but a maximal simplex serves as a q-dimensional frame …
In the evening of the 13th of September after giving the talk below, it dawned me after doing lots more experiments that the connection dynamical connection Laplacian and the dynamical connection Green function that were written down on the right hand side of the black board work together if one …
Before we start, here is the code which generated the two matrices on the board of the talk. 5 lines for the Hodge Laplacian, 1 line for the Connection Laplacian. This works for any simplicial complex. Not only in physics, also in mathematics, there is a fundamental distinction between Fermionic …
The elements of Euclid of Byrne (internet archive) are a nice example also in how to illustrate mathematics. The tools to illustrate mathematics have multiplied since Byrne’s time. Yesterday, I wanted to visualize the identity w(B(x))=w(U(x))-w(S(x)) for quadratic (Wu) characteristic which comes after linear (Euler) characteristic. In the video, the …
We continue the quest to define a sectional curvature for q-manifolds. A good notion should produce classical theorems like that if sufficiently pinched manifolds are spheres. Asking all embedded wheel graphs to have positive curvature was much too rigid and produced only spheres, so small that I called this the …
This is just an update on two topics looked over during the summer. In the case of the QR flow and Toda flow equivalence, I have had a hard time finding it due to some strange ways how Mathematica computes the QR composition. You can try yourself: the diagonal entries …
Explanations of the Casimir effect using common physics intuition like “polarization” (it originally was studied in the context of van der Waals forces) or “pressure” do not work. The reason in the case of the Casimir effect is that in the case of two planes or two cylinders the Casimir …
In this presentation, there is a bit of advertisement for finite geometry and delta sets in particular. I also tried to get a bit into the history of finitist ideas in geometry and physics(starting with Riemann). One usually thinks about finite projective spaces when talking about “finite geometries”. I like …
While linear cohomology deals with functions on simplices, quadratic cohomology deals with functions on pairs of simplices that intersect. Linear cohomology is to Euler characteristic what quadratic cohomology is to Wu characteristic $w(G) = \sum_{x,y, x \cap y \in G} w(x) w(y)$. If the simplicial complex is split into a …
The discrete Sard theorem in the simplest case (which I obtained in 2015) that a function from a discrete d-manifold to {-1,1} has level sets that are (d-1) manifolds or empty. (See here for the latest higher generalization to higher codimension.) A simplicial complex is a d-manifold if every unit …
Brevity contributes both to clarity and simplicity. Surprisingly, it often contributes to generality. I myself am obsessed with brevity. I especially love short code. A short program is like a poem. If it is also effective, it can also be used as building blocks of larger programs. The Unix philosophy …
Dynamical systems on geometric spaces are very common in mathematics. Mean curvature flows, Ricci flows etc. We can think of a subcomplex K in a simplicial complex G as a geometric object. We can not break randomly elements away as would in general lose the property of having a subcomplex. …
We write a bit more about four papers which are relevant for the Fusion inequality on the Betti numbers of an open and closed pair in a simplicial complex.
We prove that the spectrum of the Hodge Laplacian dd* +d*d depends in a monotone way on the simplicial complex.
A finite abstraact simplicial complex or a finite simple graph comes with a natural finite topological space. Some quantities like the Euler characteristic or the higher Wu characteristics are all topological invariants. One can also reformulate the Lefschetz fixed point theorem for continuous maps on finite topological spaces.
We found a formula of the green function entries g(x,y). Where g is the inverse of the connection matrix of a finite abstract simplicial complex. The formula involves the Euler characteristic of the intersection of the stars of the simplices x and y, hence the name.
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 …
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?
