Quantum Calculus

Abstract: the upper envelope of the Lyapunov exponent functional on SL(2,R) cocycle A over an automorphism T of a probability space can be estimated below by the Herman functional H(A) which is the expectation of log||A||/2 where ||A|| is the Hilbert Schm…

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”.

In my recent Saturday nostalgia presentation on the entropy problem, I mentioned Alex Honold and Dean Potter.

Math 2.0 as Terry Tao has called it, gives at first a rather depressive outlook onto the future of mathematics.

A rather general positive entropy problem is that Baire generically, a volume preserving diffeomorphism T on a 2-dimensional manifold M has positive entropy.

October 6 is now often called Mathocalypse event.

I got Hans Bethe much in my youtube feed recently, probably due to my current interests in group theory.

The September 8 event in mathematics was the solution to the Navier Stokes Millenium problem.

The structure theorem is now fully proven thanks to Alex Sutherland.

Every abelian group A can be either ‘dihadrated’ or “dicycled”.

When we were teenagers, we were huge fans of Ruediger Nehberg.

During our high school class reunion I took the following picture (bottom of the page).

The current crisis in mathematics is unparalleled.

In 2-divisible groups, we can divide by 2.Are all 2-divisible Abelian groups natural?

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.

The September 8 event is comparable with the sputnik moment in the late 50ies.

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.”

[Update 9/7/2026: as things change so fast, I decided to put up the most recent note on natural groups here.

This Sunday morning, I listened to the german podcast of the Spektrum der Wissenschaft about Hermann Weyl.

The first problem is whether every additive group of every field is natural.

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.

While updating “Graphs, Groups and Geometry” I also updated other things.

As the new semester starts, I had to wrap up updating the revision of “graphs, groups and geometry”.

[ Update August 23, 2026: the updated paper looks better now.

It happened again.

Euler characteristic super sums energy of space.

Jakub Gismatullin informed me about an issue with the definition of natural groups, I gave here in 2022.

I just learned that Erwin Engeler died on July 28, 2026.

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.

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.

A write-up on the random manifold theorem is now available as a PDF.

Update: July 26, 2026: A write-up.

There had been a lot o buzz about Levent Alpoge’s counter example of the Jacobian conjecture.

I love short code.

Define the curvature K(G) of a simplicial complex as .

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.

Epistemology, the study of knowledge starts with a leap of faith.

The de Rham theorem equates simplicial cohomology with de Rham cohomology.

Last week we have seen some insensitive administrators praise AI to graduates during commencement speeches.

[Update: May 28: I have an expository paper on this uploaded now.

Johann Wolfgang von Goethe wrote in 1797 the “Sourcerer’s apprentice” (der Zauberlehrling).

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.

In the video below, I tell a few more personal stories related to games.

This spring, the red-blue button dilemma went viral.

The axiomatic set-up of game theory has some similarity with the setup in statistical mechanics.

The work of John von Neumann and Oskar Morgensten from 1944 is usually considered the start of mathematical game theory.

We still have not yet implemented the construction of geodesic manifolds in the computer.

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).

A good discrete differential geometry should mirror the continuum as closely as possible.