Calculus without limits
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 … The post Crouzeix-Conjecture has fallen appeared first on Quantum Calculus .
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 … The post Euler Characteristic for metric spaces appeared first on Quantum Calculus .
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 … The post Axiomatics of natural groups appeared first on Quantum Calculus .
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 … The post Erwin Engeler 1930-2026 appeared first on Quantum Calculus .
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 … The post Manifold variational problems appeared first on Quantum Calculus .
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: … The post A Green Function Example appeared first on Quantum Calculus .
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 … The post Summing over all Submanifolds appeared first on Quantum Calculus .
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 … The post More on random manifolds appeared first on Quantum Calculus .
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 … The post Jacobian conjecture solution appeared first on Quantum Calculus .
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 … The post Monte Carlo for random manifolds appeared first on Quantum Calculus .
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 post Random Manifold Theorem appeared first on Quantum Calculus .
some random codimension 2 surfaces in a 4 manifold The post World Sheet Problem appeared first on Quantum Calculus .
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 … The post Riemann Hurwitz appeared first on Quantum Calculus .
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 post Epistemology appeared first on Quantum Calculus .
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 … The post De Rham for quadratic cohomology? appeared first on Quantum…
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 … The post Shifts in the Mood appeared first on Quantum Calculus .
[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 … The post Moebius Kantor Graph appeared first on Quantum Calculus .
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 post Der Zauberlehrling appeared first on Quantum Calculus .
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 … The post Cylinder and Moebius strip again appeared first on Quantum…
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 … The post Game adventures appeared first on Quantum Calculus .
This spring, the red-blue button dilemma went viral. Every human has to make a decision to push a red or blue button. If more than 50 percent push blue, all humanity continues to live. Otherwise, only the ones who have pushed red survive. It went viral as it is considered … The post Devastating Nash equilibria appeared first on Quantum Calculus .
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 post God numbers appeared first on Quantum Calculus .
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) … The post Graphs Groups and Games appeared first on Quantum Calculus .
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 … The post Clover Construction appeared first on Quantum Calculus .
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 … The post Spider Lemma appeared first on Quantum Calculus .
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 … The post Discrete Grassmannian appeared first on Quantum Calculus .
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 … The post Exponential Patch appeared first on Quantum Calculus .
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 … The post R2-D2 Curvature Formula appeared first on Quantum Calculus .
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 … The post Calculus on Wave fronts Wrap up appeared first on Quantum Calculus .
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 … The post A Sturm Liouville Problem appeared first on Quantum Calculus .
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 post Black Scholes appeared first on Quantum Calculus .
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 … The post Polarization Identity appeared first on Quantum Calculus .
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 … The post Quantized Distances appeared first on Quantum Calculus .
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 post Konrad Osterwalder (1942-2025) appeared first on Quantum Calculus .
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 … The post 3D Dirac operator appeared first on Quantum Calculus .
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 … The post 2D wave calculus appeared first on Quantum Calculus .
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 … The post Single variable Wave front Calculus appeared first on Quantum Calculus .
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 post Spherical Averages appeared first on Quantum Calculus .
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 … The post Huygens Principle appeared first on Quantum Calculus .
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 … The post Density of Wave Front on the Cube appeared first on Quantum Calculus .
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 … The post Quantum Multivariable Calculus Flavors appeared first on Quantum Calculus .
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 … The post Wave density conjectures appeared first on Quantum Calculus .
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 … The post Code for QR inflation appeared first on Quantum Calculus .
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 … The post Nonlinear Waves appeared first on Quantum Calculus .
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 … The post Discrete Time Wave Dynamics appeared first on Quantum Calculus .
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 … The post Quantum Wave Animation appeared first on Quantum Calculus .
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 … The post Sudan’s talk about P-NP appeared first on Quantum Calculus .
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 … The post Wave interpolation appeared first on Quantum Calculus .
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 post Discrete Calculus appeared first on Quantum Calculus .
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 post Quaternion Integers appeared first on Quantum Calculus .