The Herman dial circles in the space of bounded measurable matrix cocycles over a dynamical system produces circles on this space. The average Lyapunov exponent over this circle is larger or equal than the Herman number . More generally, we can take a partition of into sets of size and …
Scott Aaronson who is quite outspoken on his blog coined the term “certified organic theorems” and mentioned that the time for such things are over. It is clear that in these days it has become hard to distinguish mathematics that is AI generated (the analog of growing genetically engineered, hormone …
Update October 22: by the Kingman subadditive ergodic theorem, the Lypapunov exponent function is actually already uppersemicontinuous. The upper hull therefore is the Lyapunov exponent function. I used this already during my thesis and had forgotten. So, the construction outlined below only establishes in a different way what I already …
The Math 1.0 and Math 2.0 transition in the terminology of Tao could also be seen as a distinction between a culture of “being” to to a culture of “having”. The dichotomy between “having or being” is a term coined by Erich Fromm, a German philosopher about whom I learned …
In my recent Saturday nostalgia presentation on the entropy problem, I mentioned Alex Honold and Dean Potter. It is related to the question of “taking risks”. It is a fascinating topic for me and very relevant as we live in a society that is obsessed with risk aversion (both for …
Math 2.0 as Terry Tao has called it, gives at first a rather depressive outlook onto the future of mathematics. There is no doubt that we currently experience an enormous shift in how mathematics will be done. This is not different in other parts of life like engineering, literature or …
A rather general positive entropy problem is that Baire generically, a volume preserving diffeomorphism T on a 2-dimensional manifold M has positive entropy. A much bolder conjecture is that every non-integrable volume preserving diffeomorphism on a compact Riemannian 2-manifold has positive entropy. It would mean that the Standard map had …
October 6 is now often called Mathocalypse event. Terry Tao calls it the start of Math 2.0 and sees a positive aspect that the research frontier of mathematics will expand significantly faster. I agree that there is little choice in that. At the moment, there is no realistic way to …
I got Hans Bethe much in my youtube feed recently, probably due to my current interests in group theory. I want to comment on the later but first to Bethe, whose name is in the Bethe Lattice, the Bethe-Slapeter equation, the Bethe Ansatz and more. Most important for science is …
The September 8 event in mathematics was the solution to a millenium problem. Even more brutal (for me at least) is the October 6 announcement of the solution of hundreds of open mathematics problems. It is a slaughter house! One of them is the solution of a problem I had …
The structure theorem is now fully proven thanks to Alex Sutherland. See his paper submitted on September 27, 2026. The structure theorem had been known before only for groups with cardinality less or equal than . See my paper from September 7, 2026. The structure theorem is given in the …
Every abelian group A can be either ‘dihadrated’ or “dicycled”. The technical term is to form the generalized dihedral group Dih(A) or to form the generalized dicyclic group Dic(A,y) with an involution y in A. Let’s leave the superflous “generalized” away as it is clear what we mean in each …
When we were teenagers, we were huge fans of Ruediger Nehberg. It is fascinating to have knowledge to survive outside without any technical tools. It is an art and Nehberg demonstrated such skill not only theoretically but also during his own adventures. Knowing some basic skills is still recommendable today, …
During our high school class reunion I took the following picture (bottom of the page). It shows the abbey of St Katharinental which goes back to the year 1242. The members of that convent collected and illustrated especially graduals, some of which go back even 1000 years from today. Monasteries …
The current crisis in mathematics is unparalleled. Not that there had not been challenges before: the Pythagoreans had to grapple with the existence of irrational numbers. The Galois drama illustrates the crisis, when the realization dawned that explicit solution formulas to polynomial equations do not exist for degree 5 or …
In 2-divisible groups, we can divide by 2.Are all 2-divisible Abelian groups natural? This question appears to be a big cardinality problem. The video below explains this a bit. When learning group theory one can have the impression that “abelian groups are boring”. After all there is simple fundamental theorem …
it is evident that we live in a time of extremely fast change with respect to epistemology, the way how we acquire known and gain new information. As a dinosaur, who has seen several “meteor impacts” (new math, math wars, the rise of the web, the moocs wars and now …
The September 8 event is comparable with the sputnik moment in the late 50ies. This was before I was born, but I felt the repercussions in my own education. We learned about set theory early in primary school. The “new math” revolution had come in the wake of the Sputnik …
In the legendary movie “The 13th worrior”, Helger of the Northmen reminds Ahmed Ibn Fahldlan in latin “propera, ut morti occurras”, which Melchisidek then translates as “Hurry to meet Death, before your place is taken.” I like that movie and mentioned it a couple of times We do not have …
[Update 9/7/2026: as things change so fast, I decided to put up the most recent note on natural groups here. [ArXiv]. As mentioned in the note, the mathematical world will almost certainly look different already next week. We live in a hyper-drive time. Will there be any time again, where …
This Sunday morning, I listened to the german podcast of the Spektrum der Wissenschaft about Hermann Weyl. Weyl is of course important to ETH Zürich. We had many classes and seminars in the Herman Weyl room (I had my thesis defense there for example). All Nachdiplom classes like Ruelle’s, Tromba’s …
The first problem is whether every additive group of every field is natural. Lets call this a field group. All field groups of cardinality not exceeding the continuum are natural. As typical in the naturality habitat, one can see this in various way. Start for example with R, which is …
In my video 3 weeks ago about natural groups (see video link), I had claimed that metric spaces are the champions among all topological structures and that groups are the champions among all algebraic structures with one operation. In order to back this up quantitatively, I initiated two “data analysis …
While updating “Graphs, Groups and Geometry” I also updated other things. One of it was that Larry Nichols, the inventor of the pocket cube (patent 1972, earlier than the Rubik one from 1974) has passed. But as I learned only after uploading the paper to the ArXiv, Nichols might still …
As the new semester starts, I had to wrap up updating the revision of “graphs, groups and geometry”. I worked frantically the last few days on this as it is important to have the correction up also on the ArXiv. There was not only unexpected progress in the case of …
[ Update August 23, 2026: the updated paper looks better now. Lots of stuff previously written has been left but after a few frantic days of working on this, I feel better. Here are the main results Main results are: all connected Lie groups are natural. An arbitrary non-abelian, finitely …
It happened again. A machine has solved a conjecture in linear algebra. it was a beautiful problem, now reduced to junk after letting Chat GPT run through it for 16 hours. Like for an other beautiful conjecture involving polynomials, the Sendov conjecture (Tao wrote about it recently), we see the …
Euler characteristic super sums energy of space. It subtracts the number of odd dimensional parts from the number of even dimensional parts. It can not be defined for an arbitrary metric space in general. Bouquets of spheres like the Hawaiian rings (am infinite union of circles) or the set (an …
Jakub Gismatullin informed me about an issue with the definition of natural groups, I gave here in 2022. It is easy to fix in that we only need one sided invariance assumption (something I seem to have assumed throughout the paper). In the definition however, I assumed that bi-invariance and …
I just learned that Erwin Engeler died on July 28, 2026. It is rather strange to me as I had been thinking about Engeler a lot lately, when working on my “Elements in Finite Geometry” project. Engeler taught us topology in the 3rd semester. See my courses from college. As …
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 …
In my article Remarks about Connection and Dirac matrices, I had asked in question C, whether the connection Laplacian always has a larger spectral radius than its inverse g. Robert J. George has found a counter example. The skeleton complex of K5: Here is some Mathematica code showing that example: …
A write-up on the random manifold theorem is now available as a PDF. (July 28: and now the ArXiv) It contains the following code for the functional version of the theorem that allows to compute the average f-vector of a random manifold. It is again a bit intended as a …
Update: July 26, 2026: A write-up. If is the simplex generating function of a simplicial complex G, then define , the anti-derivative. Gauss-Bonnet is then the elegant formula . It combines all possible Gauss-Bonnet formulas for all valuations. [The simplest of them is the Euler Handshake formula, which can be …
There had been a lot o buzz about Levent Alpoge’s counter example of the Jacobian conjecture. I myself had been introduced to it by Juergen Moser, who was prominently mentioning it in a dynamical systems course. It had been a beautiful conjecture. Unfortunately it has now been gone. There will …
I love short code. My experimentation code for random manifolds had been several dozen lines. I spent the evening yesterday to shorten it to 6 lines. I get a kick of programming on the bottom, without looking anything up. It is like writing a poem. What the program does is …
Define the curvature K(G) of a simplicial complex as . We prove Theorem: Given a (2d+1)-manifold G, then the expectation of the Euler characteristic of a (2d) manifold in G is 2-2K(G). The key is the choice of probability space. Let V denote the set of vertices in G, the …
The presentation on the Lefschetz fixed point theorem reminded me about Riemann Hurwitz, something I discussed with Thomas Tucker in 2012. It deals with an important topic in geometry: what happens if one quotients out a group acting on a geometry. This is very rich because it covers things like …
Epistemology, the study of knowledge starts with a leap of faith. Not of the religious kind. But we have to have faith in that we can think properly, that our memory works reliably. A brain with a neurodegenerative illness is not able to do logical steps any more. We have …
The de Rham theorem equates simplicial cohomology with de Rham cohomology. The correspondence is explicit. Start with a smooth differential k-form f, it defines a discrete differential form by assigning the value to each k-simplex x. This produces so a discrete differential form. A discrete differential form on the other …
Last week we have seen some insensitive administrators praise AI to graduates during commencement speeches. These folks have not read the room, not seen the mood change that happened worldwide during the last couple of months. People start to get sick of it. They are also worried about it. I …
[Update: May 28: I have an expository paper on this uploaded now. Ha a few days for focused writing.] I encountered the Moebius Kantor Graph when trying to get better geodesic sheets in discrete 3-manifolds. For , the smallest 3-sphere, all dual spheres of edges are graphs and the geodesic …
Johann Wolfgang von Goethe wrote in 1797 the “Sourcerer’s apprentice” (der Zauberlehrling). This poem is one of the most famous allegories of all times, especially after being featured in the masterpiece “Fantasia” of Walt Disney in 2000. The story is is very relevant today, for obvious reasons. I got reminded …
The Cylinder and the Moebius strip are a nice paradigm because the classical algebraic topology elements are the same for it: they are both homotopic to the circle, their cohomologies agree, their fundamental groups agree. Their metric properties like diameter or systoles agree (if implemented properly of course). Their main …
In the video below, I tell a few more personal stories related to games. It could have started earlier about group games. As kids we were playing the 15 puzzle a lot. The 15 puzzle is remarkable as the god number 80 has only been computed a couple of years …
This spring, the red-blue button dilemma went viral. In that scenario, every human on earth has to make a decision to push a red or blue button. If more than 50 percent of the populationpush blue, all humanity continues to live. Otherwise, only the ones who have pushed red survive. …
The axiomatic set-up of game theory has some similarity with the setup in statistical mechanics. Due to the shear size of the configuration space one refers to probabilistic set-ups. Getting from the micro-canonical description to the macro-canonical framework is necessary if one can not oversee the full configuration space. A …
The work of John von Neumann and Oskar Morgensten from 1944 is usually considered the start of mathematical game theory. It builds on earlier work, especially of von Neumann in particular the elegant mini-max theorem of Neumann: let be the space of probability vectors in $mathbb R^m$ and let f(x,y) …
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 …
In order to construct the geodesic manifold of dimension k in a q-dimensional manifold, we build up the manifold in the discrete Grassmannian bundle Gr(q,k). We now do this all using the geodesic flow, which is much closer to the continuum. The simplex y in x defines (k+1)! geodesics or …
A good discrete differential geometry should mirror the continuum as closely as possible. Positive curvature manifolds should be bounded, the list of positive curvature manifolds should be identical to the ones we know in the continuum. Last year we had looked at geodesic patches and defined sectional curvature by defining …
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 …
At the end of the wave front calculus paper, I added some pictures about curvature defined by wave fronts. The Bertrand-Diguet-Puiseux formula is remarkable as it shows that one can get curvature of a 2 manifold by measuring the length of wave fronts. Positive curvature means that the wave front …
A first draft is now available [PDF]. The entire topic is a bit off main stream, but that’s why I’m actually quite proud about it, especially because it is not in a field, I’m most familiar with (partial differential equations). It needs courage to work in a new field and …
In my project picked up again this winter, I had looked at the deformation of exterior derivative df in a q-manifold given by and saw that it satisfies the modified wave equation , where is the Hodge Laplacian. Lets call the left hand side . The equation is a Sturm …
It usually takes a few decades until some mathematical object slides into Pop culture and eventually becomes kitsch. The Black-Scholes differential equation origin from the 1970ies. It is derived from stochastic process model. It became Pop culture in 2000, when NOVA made a TV show about it. It became infamous …
The Polarization identity for quadratic forms like inner products is also known as the parallelogram law. It is very important and always a good teaching moment in a multi-variable calculus course. It is more than just an identity. It tells that if know “lengths”, then we can recover “angles”. All …
In the video below I gave a bit of an overview about the “wave calculus project”. At the 5 minute mark , I mention again the fact that we describe a calculus where distances are quantized but where we still have all the symmetries that exist in the continuum. One …
I only learned recently thought the “not even wrong” blog that Konrad Osterwalder passed away last December. He had been department head at ETHZ from 1986-1990 which overlapped quite a bit with my graduate student time there. He then became rector of ETHZ and became later an important figure in …
The three dimensional space is important because we live in it. With the scalar Laplacian in dimension q=3, the Hydrogen operator (leaving out constants) essentially explains the periodic system of elements and so the starting point of chemistry. The eigenvalue difference explain spectral lines (like Lyman (UV light) , Balmer …
This is a continuation. We look more closely at the 2 dimensional case. The transition from 1 to 2 dimensions is crucial in the context of discretization. In the one dimensional case there is little choice on how to discretize calculus because there is only one possible discretization of the …
We discuss the exterior derivative on a compact Riemannian manifold (M,g) in dimension 1. It is a bounded operator with the property that for a k-form f, the (k+1)-form only depends on f located on the wave front . This notion emerged when searching for a natural multi-variable calculus in …
Here is a bit of calculus in dimension q. I use the letter q and not d for the dimension as we will use the exterior derivative d later. Multivariable calculus is very similar to single variable if one uses multi-index notation like or or or for partial derivatives or …
The dispute of 1689 between Newton and Huygens was won by Newton (he was by far the more important person) but in reality, Newton was wrong (*). With all the information available at that time, the “guess” that light is made of corpuscular objects should not have been tenable at …
If P is a light source on a cube we can look at , the wave front. Every point of this wave front can be assigned a group element, the rotation which is needed to rotate the initial Frenet frame at the start to the frame at time t. Whenever …
This winter, I started to revive some seeds which have been placed in the winter of 2010. It is a calculus in a general compact Riemannian manifold in which the exterior derivative is bounded. It builds on the usual exterior derivative d defined on a manifold (M,g) and builds for …
We currently make some more experiments for the density of wave fronts project. We conjecture there for example that for any polygon wave fronts become dense. In cases like eqilateral triangles or right angle triangles this can be proven because doubling produces a parallelogram that tessellates the plane. This is …
Here is again some Mathematica code. What is new here, that we do not bother to solve the Lax differential equations but directly compute the deformation using QR which is exact. We build a small 3 sphere as a simplicial complex G, then build the Dirac matrix B=D (we chose …
A geometry is a space on which one has a derivative and notion of integration. This can be expressed more elegantly as a space with a cohomology as cohomology are kernels of matrices defined by the derivative and when looking at operators one needs a Hilbert space which intrinsically defines …
Causality Principle One of the most exciting principles in physics is the Huygens principle relating the geodesic motion with the wave dynamics. If we are in a Riemannian manifold, there are two type of “wave fronts”. We can look at the solution of the wave equation , where is the …
This is a bit a blast from the past. While thinking about waves, I thought of this little C program, I wrote in April 2000 (we moved in June 2000 to Boston which was quite an adventure, our car broke down, and I drove from Austin to Arlington twice). As …
For slides and first remarks about the P-NP problem, see this page. I mentioned there my own personal predictions about the Millenium problems. See also this talk from 8 months ago about the perfect number problem in which also all Hilbert, Millenium and Landau problems are mentioned. So, here is …
Having talked a bit about GR (general relativity) and the SM (standard model), lets talk a bit about QM (quantum mechanics). Without any doubt, GR,SM and QM are three extremely successful pillars of modern fundamental physics. Their track record with experiments is monumental (*). We have played with simple games …
In 1960, Regge formulated a discrete calculus as a numerical scheme for relativity. We can formulate it as a theory on a weighted graph . where is a distance which is a discrete q-manifold. The graph defines a q-dimensional pure simplicial complex generated by the maximal complete subgraphs as facets. …
The number theory of quaternion integers has connections to many beautiful parts of mathematics. The quaternions themselves are selected out from all the structures in mathematics as the only associative real non-commutative division algebra and so play a rather unique role in the entire landscape of mathematics. Its unit sphere …
The topic is a combination of the “particles and primes” and geodesics in discrete manifolds” topics, I had worked on before. The first one is an elementary connection between primes in division algebras and particle phenomenology, the second is a completely deterministic dynamical system in combinatorics that has many properties …
AXIOMS: Most basic mathematical structures are heavenly in the sense that they would be discussed in any part of the universe. In algebra these would be groups or rings. In topology these would be topological spaces or metric spaces, in probability theory it would be sigma algebras and stochastic processes. …
We have seen last time that describing many particles moving on a manifold is best done by keeping on each cell a tag telling how many particles there are there. In other words, we do not evolve a sequence of maps but a divisor . This is more effective if …
On October 15, 2025, Sourav Chatterjiee gave the second of the Millenium prize lectures. I have started a page on this here where the slides are included. A bit of nostalgia Some of my own course assistants (when I was an undergrad) or grad student colleagues (when I was a …
This is a bit about getting back to the story of geodesics. See “Geodesics for Discrete Manifolds” and “interacting geodesics on discrete manifolds”, all from earlier this year. I wanted to get next week on how to evolve Gaussian integer and quaternion integer particles. The quaternion primes show a nice …
Manifolds are in the following two dimensional, compact, connected and without boundary. Their classification of these geometric objects is a classical piece of mathematics, completed in a satisfactory way in 1907 by Dehn and Heegaard: the structure is that there are two monoids. The monoid of orientable manifolds that is …
If K(t) is a function in a Lie algebra then the Frenet equations Q'(t)=K(t) Q(t), Q(0)=1 defines a path in the corresponding Lie group. We can now use K(t) to define the Frenet path using the equation R'(t)=Q(t). This defines r(t), where r'(t) is the first row of R(t). The …
The arrow of time is probably one of the most written fundamental topics in physics. I myself have written about it once. To the right are some writings and movies related to time and these are only the movies or books I have seen. My favorite math book pair about …
A periodic Frenet curve in is up to isometry uniquely defined by a periodic curve in where the later describes the motion of the Frenet frame. Differentiation then defines a periodic curve in the Lie algebra . We can now reverse the story and start with a periodic curve in …
Maybe it is a good idea to add current “discovery code” also with the Dirac part. It can be helpful to post code also to be able to see in the future how one was thinking at the time. In the talk of Freedman, there was a slide with a …
Update October 7, 2025: Kitsch is already produced by professional mathematicians . No problem if it is declared as such. I myself like some kitsch. It just has to be declared as fake. It feels a bit like climbing “action direct” with the help of the rope. No problem, if …
Let (G,T) be a finite dynamical system. This means that G is a finite abstract simplicial complex and T is an invertible simplicial map (a permutation of G coming from a permutation of the vertex set V or alternatively, a homeomorphism of the topological space which origins from a permutation …
Yesterday was 9/16/25. You can easily check that it is the only date that is a Pythagorean triple MM/DD/YY. I tweeted And made a brief short on youtube about Elkies’s puzzle explaining visually that . If one is asked about the minimal number of puzzle pieces which can explain this …
Here is my “discovery code” used over the weekend to detect first experimentally that there are dynamical connection and green function matrices which are inverse of each other. I might talk about the proof next time. But first to the experiments. I take here a concrete example of a finite …
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 …
When I had been studying mathematics at ETHZ, taking two semesters of calculus and at the same time taking two semesters of linear algebra had been mandatory. My Calc 1 (1a/1b here) and Calc 2 (21a here) teacher from the first year was Hans Laeuchli. His advisor Ernst Specker during …
When trying to translate everything that has been done in mathematics to the finite world it is good first to look at the most important theorems. Some of the theorems like the pigeon hole principle or the law of the product or the fundamental theorem of arithmetic do not need …
For a finite abstract simplicial complex, we can look at the Dirac Zeta function, the Connection Zeta function and the Lefschetz zeta function. My work on this in the last couple of years
Here is an other “poem” that has been added to the “math poetry page”. Given a simplical complex G, it computes the automorphism group Aut (its group of symmetries, meaning invertible simplicial mapss) and for each T computes both the Lefschetz number and the sum of the indices of the …
During a nice 23 K run on Friday, I was thinking a bit about Lefschetz. This is one of the first stories, I tackled. See the paper. It is one of my 10 favorite theorem in graph theory. I originally formulated it in graph theory in 2012 generalizing the one …
A conference earlier this year had the effect that some main stream media started to report about it. I saw it here in an Austrian journal. But the article does not report on any new discoveries. The 2025 Ultrafinitism Conference was also a cover story for NewScientist Magazine. I thought …
Given a finite abstract simplicial complex G, there are two amazing matrices which are two sides of a coin. Both matrices are symmetric n x n matrices if G has n elements. The Dirac matrix D =d+d* is defined by the exterior derivative d(x,y) = sign(x,y) if y is incident …
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 …
Theorem: if a graph G=(V,E) with n=|V| vertices and m=|E| edges and maximal vertex degree d satisfies , then G satisfies the Brouwer conjecture . This adds confidence that the conjecture is true. As mentioned in the video (and now having worked on it for 3 weeks and wrapping up), …
Let G=(V,E) be a graph with n vertices and m edges. So far all experiments indicate that also the signless Kirchhoff Laplacian |K| =D+A satisfies the Brouwer bound for all $late 1 \leq k \leq n$, where is the sum of the largest eigenvalues of |K| and . The signless …
If K is the Kirchhoff matrix of a finite simple graph with n vertices and m edges and eigenvalues and edge degrees . Define and and and . This is . Then the following general inequalities are known (the first 4 inequalities in the following list) or conjectured (the last …
One of the most amazing formulas in linear algebra is the Hadamard first variation formula which tells how an eigenvalue changes if the matrix entries are changed. Take a symmetric matrix K and perturb it as K+t E, where E is an other symmetric matrix. How do the eigenvalues change? …
John Walker (1949-2024) was an amazing programmer. His website “Fourmilab” was a page of inspiration for me since 1994, when the web started to get serious with Mosaic allowing to brouwse the internet. I still remember in early spring 1994, when I was busy finishing my PhD, that near our …
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 …
While thinking about some fundamental parts in the story of Dehn-Sommerville, I decided to make a bit of an exercise in imagining how Euclid would have done finite geometry. The result is given in the movie below. Euclid’s elements is undoubtedly one of the milestones in the whole of mathematics …
Next week, I will be back in my office. As they are constructing labs just near my temporay office, I made this “talk to myself session” in a seminar room of the department. I try to finish these days a review about Dehn-Sommerville, a rather unpopular topic historically speaking (not …
Goldbach has been hugely popular recently. It appeared in movies, and books. on some major youtube math sites like numberphile or veritasium recently. That show featured also Harald Helfgott, who had shown the ternary conjecture more than a decade ago. The conjecture is probably the most popular math problem of …
Dehn-Sommerville identities are symmetries for manifolds. First detected by Max Dehn in small dimensions, they were generalized by Duncan Sommerville in 1927. The relations were classically studied for simplicial polytopes which means q-spheres. It gives relations for the f-vector of a simplicial complex that is a q-sphere. The original work …
In a recent interview with Lex Freedman, Terrence Tao brought up the allegory of foxes and hedg-hogs in mathematics. Foxes know lots of things, hedgehogs know one thing well. As for a reference, I found this allegory. There are other analogies: Freeman Dyson indicated how Francis Bacon and Rene Descartes …
While walking through the ETH library in Zuerich on Monday, I saw a new book from 2025 by Brian Hopkins called “Hands-On Combinatorics: Building Colorful Trains to Manifest Pascal’s Triangle, Fibonacci Numbers, and Much More”. The ETH library is very nice. I used to work there myself often as a …
Summary We look at 5 results of Peter Lax (1926-2025) in a youtube short (1 minute clip) 1) 1956: Lax Pairs for integrable system2) 1990: Pedal Map in geometry: a chaotic system3) 1971: Approximation theorem of volume preserving continuous maps4) 1954: Lax Milgram theorem generalizing the Riesz representation theorm5) 1956: …
As promised in the talk, here are the 12 lines of code. The 13th line is an example and take the smallest 3 dimensional manifold G, the 16 cell which is a small example of a 3-sphere. The fiber bundle P has 384 elements in this case. Every single particle …
Here is the start of the novel “Schild’s ladder” by Greg Egan from 2004. It has sometimes been called the hardest SciFi Novel. Indeed, you can already be stuck in the first paragraph. The text describes the diamond lattice (there is a Wolfram demonstration file here by Sandor Kabai from …
One of the problems which has bothered me in the last couple of months is the fact that in a locally finite geometry G, it is improbable to get a notion of geodesics which satisfies the properties: 1. any two points can be joined by a geodesics, 2. there is …
After talking about this on Saturday, I tried some other things (this is a perfect topic to think about before getting to sleep or even contemplate about while sleeping). First of all, we can extend how to evolve the geodesic flow given a triangle t=(a,b,c) in a q-simplex x of …
[Update June 17, 2025: The Senior thesis: Spectra and Similarity by Jessie Pitisillides and the preprint on the ArXiv: (Segre Characteristic Equivalence) of Jessie Pitsillides.] Every linear algebra course battles the concept of similarity. We learn that trace, determinant, rank or eigenvalues allow to check whether two matrices are similar …
While teaching probability this semester, I had naturally to think again at the central limit theorem and took this as an opportunity to warm up a bit something I abandoned in 2022 after learning that what I was following has been understood already by mathematicians like Levy or Gnedenko. Still, …
A closed geodesic in a q-manifold is a q-manifold C with boundary dC. This boundary dC is a circle bundle. It can be for example. But it can also be a non-trivial bundle. Note that everything is purely combinatorial and pretty small. For the Moebius strip for example C is …
Given a q-manifold, we have a geodesic dynamics T:P -> P, where P is the oriented frame bundle. Most geodesics are not simple but we can look for the number of simple closed geodesics and so investigate questions close to the Ljusternik-Schnirelmann theme in the classical setup. For any 2 …
One can for discrete billiards ask questions which are classical in the continuum. One of them is Guillemin’s conjugacy problem, which is an inverse problem. It is problem 6 in my personal favorite list of open problems in Hamiltonian dynamics from 25 years ago. This is a problem which in …
[Update March 23: a paper draft is up. The ArXiv version contains also more code.] Spring break is a good time for some programming and cleanup. I reorganized our home office, took everything apart and rewired the half a dozen computers new, got rid of about 30 old harddrives, mostly …
This spring, scientific american was asking around in departments about problems mathematicians are thinking about. I immediately wrote back mentioning the “odd perfect number problem“, a problem which has for strange reasons been snubbed by problem collections. The article in SCIAM has now appeared. As I had once taken the …
Recently, monopoles was mentioned in a youtube episode of Hossenfelder’s show. For, me, it had always been more natural that monopoles do not exist. Let me explain. Electromagnetism is defined if one has a geometry with an exterior derivative d. This is very general and works for delta sets. The …
Riemannian geometry is related to general relativity, probability theory is related to quantum mechanics. Length enters in GR in the form of geodesics, paths of mass points and curvature, the deformation of space through mass. Probability enters in quantum mechanics by seeing solutions of wave or Schrödinger equations in terms …
Shortly after working on Gauss-Bonnet-Chern for graphs, I wrote about Poincare-Hopf for graphs. It took a larger part of the winter break 2011/2012 to come up with the formula where is part of the unit sphere where the function value is smaller than the function value at v. The function …
A discrete 2-manifold [PDF] is a finite simple graph for which all unit spheres have d(v)=4 or more vertices. The smallest example is which is the octahedron graph and where every unit sphere is a circular graph with 4 elements. The Eberhard curvature is very rigid: the Mickey Mouse theorem …
Curvature is a local notion in a finite geometry that adds up to a topological invariant. This is Gauss-Bonnet. I’m only really interested in curvatures that satisfy this exactly. This does not exclude sectional curvature, the latest venture because sectional curvature integrated over a geodesic sheet is the Euler characteristic …
We continue to look at examples of a-manifolds. Besides level sets we can also do connected sum constructions. In the talk, I glue together two manifolds along a q-simplex. An other possibility is to glue along a wall, a (q-1) simplex, removing two simplices attached at a hypersimplex and glue …
We have to adapt the geodesic flow in two situations: for manifolds with boundary or for manifolds obtained as level sets in other manifolds where we want to define the geodesic flow directly in the open set and not first pass to the Barycentric refinement. In both cases, we sort …
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 …
If G is a q-manifold, we have defined a geodesic dynamics T on the frame bundle P, a principle fiber bundle with structure group . As we have seen last time, the geodesic update step , where is the dual sphere of the -simplex . Everything is finite. G is …
The Shashibo puzzle is a game in the category of geometrically realized simplicial complexes of dimension q=3. One can also play it combinatorially, meaning to disregard geometric realization and just look at finite set of finite sets. One can see it as a playing with 3-dimensional complexes generated by twelve …
Here are two fundamental problems for discrete q-manifolds (a notion which can be defined for the trinity of geometric structures, that is for finite simple graphs as well as for simplicial complexes or delta sets). As for 1, one could ask that every embedded wheel graph has positive curvature. This …
Herbert Groetzsch and Jan Mycielski worked in the 50ies on the chromatology of triangle free graphs. Groetzsch’s 3 color theorem assures that planar triangle free graphs have chromatic number 3 or less. Mycielski defined an operation on graphs which preserves triangle free graphs and increases the chromatic number exactly by …
Coloring manifolds is a wonderful theme because it is not that well studied and so has many low hanging fruits. Here is an other one. For d-manifolds, the chromatic number is between d+1 and 2d+2 and a growth rate (3d+1)/2 is observed and reasonably conjectured upper bound. I started to …
The Regge approach to discrete relativity is also related to graph coloring problems of d-manifolds. While one can look at the length of the dual sphere of a codimension-2 simplex as a notion of curvature, the distinction of whether this circle has even or odd length is relevant when wanting …
The definition of soft Barycentric refinement needed adaptation so that it preserves manifolds with boundary. If G is a finite abstract simplicial complex. The Barycentric refinement is the Whitney complex of the graph in which G are the vertices and where two points are connected, if one is contained in …
A finite abstract simplicial complex G, a finite set of non-empty sets closed under the operation of taking non-empty subsets, has not only a wonderful topology in which the stars form a basis and the cores are closed, but also a hyperbolic structure as the unit sphere is the join …
General relativity plays on a pseudo Riemannian manifold (M,g). The Einstein equations describe how matter defines the space time and the geodesic equations describe how mass moves in space time. It is maybe the most beautiful theories that exist as it describes a relatively complicated frame work as part of …
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 …
What is the reason for the structure of the Standard Model? For a mathematician, it better has to be something inevitable. The model is the periodic system of elements is all based on the geometry of the space G one considers. If G has an exterior derivative, one has a …
There are exactly 3 associative real normed division algebras as the Frobenius theorem from 1879 tells. Each of them produce natural Lie groups . Each of them produces natural dense sphere packings . Each of them produces natural rings , the ring of integers, the Eisenstein ring and the ring …
The 2 dimensional plane can be characterized as the only simply-connected two dimensional flat manifold. In the discrete there is a similar uniqueness. The discrete hexagonal plane is the only 2-manifold that is flat and simply connected. Compact flat 2-manifolds like the torus or Klein bottle are not simply connected …
There is a soft Barycentric refinement of graphs or simplicial complexes which can be seen as the usual Barycentric refinement in which the second highest dimensional simplices are collapsed. It is a triangulation of the dual complex. The soft Barycentric refinement of a d-dimensional cross polytop for example is a …
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 …
Here is the proof that is solved by with , the QR decomposition. Proof. We show that is solved . From by $Q_t$ with $latex Q’=QB, Q^*=- BQ^*$, we get . Look at the differential equations for Q,R with initial condition . The first equation gives the conjugating orthogonal transformation. …
Update of November 4th 2024: There is a memorial in honor of Henry McKean at NYU on November 15th 2024. Last week, when looking up Henry McKean, I saw that he passed away on April 20, 2024. I don’t recall having met him in person but I have seen some …
A real symmetric matrix is called a Dirac matrix, if it is a block Jacobi matrix in which the side diagonal entries are nilpotent, meaning . For such a matrix, the square is called the Laplacian. It is block diagonal . If is a continuous function such that is invertible, …
A discrete geometry does not have a lot of symmetry as the automorphism group is in general empty. The isospectral set of the Laplacian or Dirac matrix is large enough however. Note that when dealing with a specific class of operators like Dirac matrices, then not all isospectral matrices qualify. …
Since finding the isospectral deformation of the exterior derivative (see “An integrable evolution equation in geometry” from June 1, 2013 and “Isospectral Deformations of the Dirac operator” from June 24, 2013), I tried to find discrete time integrable evolutions of the Dirac operator. Last Sunday, while experimenting in a coffee …
A 2-manifold with boundary is a finite simple graph for which every unit sphere is a circular graph with 4 or more nodes or a path graph with 3 more nodes. The boundary is the set of vertices for which the unit sphere is a path graph, the interior is …
An abstract delta set (G,D,R) is a finite set G with n elements, a selfadjoint Dirac matrix with and a dimension vector defining a partition and Hilbert spaces called the spaces of k-forms. The exterior derivative maps to . The Hodge Laplacian is a block diagonal matrix defining the Hodge …
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 …
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 …
I looked at a quadratic cohomology example. For theoretical backgroun, see the ArXiv paper “Fusion inequality for quadratic cohomology”. It is the case when U is a union of two disjoint smallest open sets in a 2-sphere for which I take the Icosahedron, one of the Platonic solids and a …
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 …
Delta sets are very general. They include simplicial complexes, open sets in simplicial complexes, quotients of simplicial complexes, quivers and so multi-graphs or simply hypergraphs, sets of sets. For the later, the geometry is not that interesting in general. As for quivers, the associated delta set is one dimensional only …
Quivers are graph in which multiple connections and loops are allowed. Since there is a Dirac operator d+d* with exterior derivative for them, they define a one-dimensional delta set (G,D,r), where G is the union of vertices and edges (loops count as edges) and r is the dimension function which …
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 …
The smallest open sets in a finite topological space form the atom of space. It was almost 100 years ago, when one has turned away from non-Hausdorff topological spaces and decided they are less relevant (Hausdorff seems have convinced Alexandrov and Hopf to focus on Hausdorff property). This is unfortunate …
Historically, geometry started in Euclidean spaces. There was no concept of coordinate when Euclid wrote the “elements”. Using “points” and “lines” as building blocks and some axioms, the reader there is lead to quantitative concepts like “length”, “angle” or “area” and many propositions and theorem. Only with Descartes, the concept …
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 …
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 …
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 …
Again a little bit of a flashback to my earliest steps in doing mathematical research. The Pentagonal number theorem is one of the most beautiful theorems in number theory. It uses the Pentagonal numbers to get a recursion for the partition function . This can be written as ( typo …
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 …
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 …
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.
We look at four different features or properties which can be invoked when looking at a mathematical problem. Why is something beautiful, when is it difficult, how hard is it or are there any surprises?
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 aim to write down a short proof of the statement that a planar graph has arboricity 3 or less.
The Acyclic Chromatic Number is bounded above by the arboricity. We can improve this by one if the Acyclic Chromatic number is even.
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.
