Gauss-Bonnet tells that integrating curvature K over the geometry G is Euler characteristic X. Curvature K is a quantity attached to points in the geometry and Euler characteristic X is an integer. A consequence of the continuity of the functional getting from K to the integral is continuous, a rigidity …
Here is the proof that is solved by with , the QR decomposition. Proof. We show that is solved . From by $Q_t$ with , we get . Look at the differential equations for Q,R with initial condition . The first equation gives the conjugating orthogonal transformation. Then, . But …
This is a presentation from Saturday, July 13, 2024. Curvatures are usually located on the zero dimensional part of space. I look here at curvature located on one or two dimensional parts of space. In the special case of a triangulation of a 2-dimensional surface, where the usual curvature is …
If is a finite abstract simplicial complex, a finite set of non-empty sets closed under the operation of taking non-empty subsets, we can ask about what f-vectors can occur if counts the number of sets of cardinality k in G. The case of the complete complex with gives a hint …
Here are three catchy open problems in differential geometry. As with any problem, we can look how to formulate discrete versions. The first problem is whether a positive curvature 2d manifold has positive Euler characteristic, the second is whether there is a positive curvature metric on and the third is …
[Update 3/5/2024: given that one knows now the optimal Moebius strip, one can wonder about the much easier question of what the smallest simplicial complex producing a cylinder or Moebius strip is. Below, I use in both cases 6 facets (triangles). For the Moebius strip, one can do with 5 …
If G is a d-manifold and is an arbitrary map, the discrete Sard theorem tells that is an open set in G that is a (d-k)-manifold. We mean with that the graph encoding the incidence of the sets in H is a (d-k) manifold. The sub-manifold H of K is …
Here is some code illustrating the story. We take the 4 manifold (a favorite manifold of Heinz Hopf) and consider two random functions f,g. Now generate the two manifolds and . They are both 2 manifolds. It goes as follows: the sign data of {f,g} are in which are 4 …
Coordinates are function values. On the two functions and , allow to determine every point in the plane. In the continuum coordinates are commutative: is the same than . This commutativity also holds in Riemannmian manifold settings. In the discrete this is no more true. Lets for simplicity assume we …
The updated document “manifolds from partitions” with more code. I then mention the index theorem for simple graphs dealing with the symmetric index . Writing the unit sphere as a union of and and $M_f(v)$, the center manifold. Now . In the interpretation with the joined center manifold , the …
Eugene Wigner in 1939 associated elementary particles with irreducible representations of groups, especially the Poincare group. In a first year algebra course, we learn about representations of finite groups and especially the symmetric group , where there are p(n) irreducible representations, where p(n) is the number of integer partitions of …
What is {f=0,g=0} for two functions f,g on a graph G. If G is a 4-manifold, these are 2-manifolds or empty.
We explain why the arboricity of 3 spheres can take values between 4 and 7 and mention that for 3 manifolds the upper bound is 9 (but believed to be 7).
A notion of a discrete vector should work for theorems like Poincare-Hopf and also produce a dynamics as classically, a vector field F, a smooth section of the tangent bundle on a manifold produces a dynamics . A directed graph does not give a dynamics without telling how to go …
This is a bit of a continuation from a previous video about Lusternik-Schnirelmann and Morse. I would like to have this chapter as elegant as possible. As for the video, there is a bit of overlap with a previous video on this from the fall, when I circled back to …
We discuss briefly how to make the cohomology space of a graph into a cohomology ring. In other words, how to define the cup product on the kernel of the Hodge Laplacian.
Something about knots and something about topological data analysis and something about the general frame work to do mathematics in a finite setting.
We prove that the spectrum of the Hodge Laplacian dd* +d*d depends in a monotone way on the simplicial complex.
A bit more update on the project of natural spaces. Which groups are natural, which metric spaces are natural, which graphs are natural?
The Curvature of graphs multplies under the Shannon product (strong product).
A few more remarks [PDF] about graph arithmetic. Now on the ArXiv. (Previous documents are here (June 2017 ArXiv), here (August 2017 ArXiv) and here (May 2019 ArXiv). The talk below on youtube was used for me to get organized a bit. It does not look like much has changed, …
Last summer I have had some fun with codimension 2 manifolds M in a purely differential geometric setting: a positive curvature d-manifold which admits a circular action of isometries has a fixed point set K which consists of even codimension positive curvature manifold. The Grove-Searle situation https://arxiv.org/abs/2006.11973 is when K …
Last week, I practiced a bit more enhanced talk presentation style in which, rather than with slides, the content is spoken and then enhanced in the video using additonal illustrations. The presentation deals with some things I have done in graph theory which I consider as part of quantum calculus …
10 theorems about discrete manifolds were featured in a youtube video.
Graph complements of cylic graphs are homotopic to spheres or wedge sums of spheres. Their unit spheres are graph complements of path graphs and have Gauss-Bonnet curvature which converges to a limit.
The results mentioned in the slides before are now written down. This document contains a proof of the energy relation . There are several reason for setting things up more generally and there is also some mentioning in the article: allowing general rings and not just division algebras extends the …
The energy theorem for Euler characteristic X= sum h(x)was to express it as sum g(x,y)of Green function entries. We extend this to Wu characteristic w(G)= sum h(x) h(y) over intersecting sets. The new formula is w(G)=sum w(x) w(y) g(x,y)2, where w(x) =1 for even dimesnional x and w(x)=-1 for odd dimensional x.
The energy theorem for simplicial complexes equipped with a complex energy comes with some surpises.
Introduction The idea to base physics on the evolution finite set of sets is intriguing. It has been tried as an approach to quantum gravity. Examples are causal dynamical triangulation models or spin networks. It is necessary to bring in some time evolution as otherwise, a model has little chance …
The question In discrete Poincare-Hopf for graphs the question appeared how to generalize the result from gradient fields to directed graphs. The paper mentions already the problem what to do in the case of the triangle with circular orientation. The triangle has Euler characteristic 1. An integer index on vertices …
The Mickey mouse theorem assures that a connected positive curvature graph of positive dimension is a sphere.
Finite sets of sets can be seen as a combintarorial oddity. There is a lot of mathematical structure available on such simple finitist constructs.
If a set of set is equipped with an energy function, one can define integer matrices for which the determinant, the eigenvalue signs are known. For constant energy the matrix is conjugated to its inverse and defines two isospectral multi-graphs.
The counting matrix of a simplicial complex has determinant 1 and is isospectral to its inverse. The sum of the matrix entries of the inverse is the number of elements in the complex.
The f-function of a graph minus 1 is the sum of the antiderivatives of the f-function anti-derivatives evaluated on the unit spheres.
Dehn-Sommerville relations are a symmetry for a class of geometries which are of Euclidean nature.
Some update about recent activities: a new calculus course, the Cartan magic formula and some programming about the coloring algorithm.
We prove that any discrete surface has an Eulerian edge refinement. For a 2-disk, an Eulerian edge refinement is possible if and only if the boundary length is divisible by 3
We prove that connected combinatorial manifolds of positive dimension define finite simple graphs which are Hamiltonian.
The beautiful Alexander duality theorem for finite abstract simplicial complexes.
We compute the quadratic interaction cohomology in the simplest case.
This is an other blog entry about interaction cohomology [PDF], (now on the ArXiv), a draft which just got finished over spring break. The paper had been started more than 2 years ago and got delayed when the unimodularity of the connection Laplacian took over. There was an announcement [PDF] …
For a one-dimensional simplicial complex, the sign less Hodge operator can be written as L-g, where g is the inverse of L. This leads to a Laplace equation shows solutions are given by a two-sided random walk.
Here is the code to compute a basis of the cohomology groups of an arbitrary simplicial complex. It takes 6 lines in mathematica without any outside libraries. The input is a simplicial complex, the out put is the basis for $H^0,H^1,H^2 etc$. The length of the code compares in complexity …
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.
When replacing the circle group with the dyadic group of integers, the Riemann zeta function becomes an explicit entire function for which all roots are on the imaginary axes. This is the Dyadic Riemann Hypothesis.
A simplicial complex G, a finite set of non-empty sets closed under the operation of taking finite non-empty subsets, has the Laplacian . It is natural, as it is always unimodular so that its inverse matrix is always integer valued. In a potential theoretical setup, the Green function values measure …
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?
As a follow-up note to the strong ring note, I tried between summer and fall semester to formulate a discrete Atiyah-Singer and Atiyah-Bott result for simplicial complexes. The classical theorems from the sixties are heavy, as they involve virtually every field of mathematics. By searching for analogues in the discrete, …
