Quantum Calculus

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.

John Walker (1949-2024) was an amazing programmer.

The elements of Euclid of Byrne (internet archive) are a nice example also in how to illustrate mathematics.

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.

Next week, I will be back in my office.

Goldbach has been hugely popular recently.

Dehn-Sommerville identities are symmetries for manifolds.

In a recent interview with Lex Freedman, Terrence Tao brought up the allegory of foxes and hedg-hogs in mathematics.

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”.

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

As promised in the talk, here are the 12 lines of code.

Here is the start of the novel “Schild’s ladder” by Greg Egan from 2004.

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…

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).

[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.]

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 ma…

A closed geodesic in a q-manifold is a q-manifold C with boundary dC.

Given a q-manifold, we have a geodesic dynamics T:P -> P, where P is the oriented frame bundle.

One can for discrete billiards ask questions which are classical in the continuum.

[Update March 23: a paper draft is up.

This spring, scientific american was asking around in departments about problems mathematicians are thinking about.

Recently, monopoles was mentioned in a youtube episode of Hossenfelder’s show.

Riemannian geometry is related to general relativity, probability theory is related to quantum mechanics.

Shortly after working on Gauss-Bonnet-Chern for graphs, I wrote about Poincare-Hopf for graphs.

A discrete 2-manifold [PDF] is a finite simple graph for which all unit spheres have d(v)=4 or more vertices.

Curvature is a local notion in a finite geometry that adds up to a topological invariant.

We continue to look at examples of a-manifolds.

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 refinemen…

We continue the quest to define a sectional curvature for q-manifolds.

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 .

The Shashibo puzzle is a game in the category of geometrically realized simplicial complexes of dimension q=3.

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).

Herbert Groetzsch and Jan Mycielski worked in the 50ies on the chromatology of triangle free graphs.

Coloring manifolds is a wonderful theme because it is not that well studied and so has many low hanging fruits.

The Regge approach to discrete relativity is also related to graph coloring problems of d-manifolds.

The definition of soft Barycentric refinement needed adaptation so that it preserves manifolds with boundary.

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 t…

General relativity plays on a pseudo Riemannian manifold (M,g).

Gauss-Bonnet tells that integrating curvature K over the geometry G is Euler characteristic X.

What is the reason for the structure of the Standard Model?

There are exactly 3 associative real normed division algebras as the Frobenius theorem from 1879 tells.

The 2 dimensional plane can be characterized as the only simply-connected two dimensional flat manifold.

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.

This is just an update on two topics looked over during the summer.

Here is the proof that is solved by with , the QR decomposition.

Update of November 4th 2024: There is a memorial in honor of Henry McKean at NYU on November 15th 2024.

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 .

A discrete geometry does not have a lot of symmetry as the automorphism group is in general empty.

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 e…

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.