Finite geometric categories: graphs – simplicial complexes -simplicial sets – delta sets Delta sets were originally called semi-simplicial sets by Samuel Eilenberg and Joseph Zilber in 1950. Similarly than semi-rings are more general than rings or semi-groups are more general than groups, also delta sets are more general than simplicial …
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 …
Here are some links to the articles mentioned in the talk: It surprisingly often happens that a big conjecture tumbles at around the same time. In the case of the Arnold conjecture, several approaches, spear headed by Conley-Zehnder, Eliashberg and Floer have reached the goal. But also almost always with …
The last geometric theorem of Poincare was conceived shortly before the death of Poincare. Poincare had a prostate problem when he was 58 and went to surgery in 1912 which he did not survive. Fortunately his last theorem was sent to an Italian journal two weeks before he died, but …
Finite topological spaces are only interesting if non-Hausdorff. The reason is that every Hausdorff finite topological space is just the boring discrete topology. The topology from a simplicial complex is an example of a nice and interesting topology because it produces the right connectivity and dimension on the complex without …
It is not quite yet a poem, but here, as promised in the movie, some code to generate both the Betti vector and Wu betti vector of a random submanifold in a given manifold. It is 25 lines without any additional libraries, so not yet quite a poem, but it …
My experiments so far indicate that the Wu cohomology of a d-manifold G can be read off from the usual cohomology. If is the Betti vector of G then (0,\dots,0,b_d,b_{d-1},\dots,b_1,b_0)$ appears to be the Wu Betti vector. So far, this is only a conjecture. In the talk, the case , …
[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 …
The video below is an attempt to get back to an older story of Wu characteristic. One of the things which still needs to be explored badly is the Wu cohomology of the complement K of knots H and more generally of the complement K of k-dimensional manifolds H in …
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 …
In my paper “Manifolds from Partitions”, I stated that that the case of empty graphs can not occur, but did not prove it. It is indeed not true. Here is an update [PDF] with an additional section. It is very rare although that a surjective map produces still an empty …
My experience from my Schweizer Jugend Forscht adventure was not only invaluable from the scientific point of view, I also met some other young aspiring scientists (here is the book with all the participants (PDF)) which was published in 1983 (when I was already a second year ETH student), and …
A nice thing about mathematics is that it has no dogmas, statements which have to be taken on good faith. Axioms come closest, but by nature also, they come with an honest warning that one can either accept them or not. Already Euclid fought with the parallel axiom. Today it …
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 …
Happy new year 2024. Here is the code displayed on the right upper corner of the board written this morning when wondering how frequent the situation is that the year is divisible by the year modulo 1000 minus 1. This happens for 2024 as it is divisible by 23. The …
First about Sard: (a write-up [PDF] ). I also display a bit my hobbies: Panorama photography (since 1999, a time when panoramas were still stichted). Later with a mirror camera. Then with GoPro Max, Iphone and more recently with the insta 360 camera (I for strange tech enhousiastic reasons pride …
In the program to get rid of any notion of infinity, one necessarily has to demonstrate that very classical and entrenched notions like topics appearing in a contemporary multi-variable calculus course can be replaced and used. Artificial discretisations do not help much in that; they serve as numerical schemes but …
There are lots of finite geometric structures. Graphs are probably the most clear ones. Simplicial complexes can not be beaten in simplicity. And delta sets can not be surpassed by generality. So, they are a geometric incarnation of the paradigm “Simplicity, Clarity and Generality”, which appeared on the book cover …
Here is the table shown in part 2 of the presentation showing some of the toys. It had been generated by Mathematica. It uses manifolds from the manifold page of Frank H. Lutz mentioned in the clip. About the left hand side (of the chalkboard) with some history pointers of …
One of the nice things in mathematics is that one can play with models which do not necessarily have to do directly with the real world, whatever the later means. We can look at abstract objects, like finite simple groups, number theory in some number field or topology in 1001 …
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.
Arboricity and Chromatic number are linked in various ways. The topic also links to difficult NP complete problems. We muse about how often it is the case that for manifolds the question is easy. An example is the Hamiltonian path problem which is linked to Peg Solitaire
One of the nice things to work in a subject not having grown up in is to be in steep learning curves. I have thought about the arboricity of manifolds for a while now but the fact that the arboricity can be arbitrary large for d-manifolds with d larger than …
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).
We look at the problem to find the possible arboricities which a 3-sphere can have.
The three tree theorem tells that any discrete 2-sphere has arboricity exctly 3.
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 …
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.
About the cohomology of measurable sets in a probability space equipped with an automorphism.
Something about knots and something about topological data analysis and something about the general frame work to do mathematics in a finite setting.
A bit the bigger picture about the mathematical and data structures which come in when working on these finite geometries.
A youtube presentation of May 13, 2023. We point out that we have a relatively simple approach to Eilenberg-Steenrod.
Over the winter break I started to look at Mayer-Vietoris type rules when looking at cohomology of subsets of a simplicial complex. See January 28, 2023 (Youtube) , and February 4th 2023 (Youtube) and most recently on February 19, 2023 (Youtube). Classically, cohomology is considered for simplicial complexes and especially …
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 report on some progress on analytic torsion A(G) for graphs. A(G) is a positive rational number attached to a network. We can compute it for contractible graphs or spheres.
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 simplicial complexes equipped with a complex energy comes with some surpises.
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.
