Calculus without limits

## Updates on QR and Curvature

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 …

## Moving along the Symmetry of a Geometry

Here is the proof that is solved by with , the QR decomposition. Proof. We show that is solved . From by $Q_t$ with , we get . Look at the differential equations for Q,R with initial condition . The first equation gives the conjugating orthogonal transformation. Then, . But …

## McKean’s Themes

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 of his work and met some of this students. Eugene Trubowitz was my Analysis 3 (complex analysis) and Analysis 4 …

## Dark Matter Lemma

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, …

## QR deformation

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. …

## Kublanovskaya-Francis Transform of Dirac matrix

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 …

## Curvatures for 2-manifolds with boundary

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 …

## Form Curvatures

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 …

## More Curvatures

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 …

## Stability of the Vacuum

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 …

## Geometry of Delta Sets

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 …

## Fusion Inequality for Quadratic Cohomology

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 …

## The most general finite geometric structure

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 …

## Delta sets from Quivers

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 …

## Kruskal-Katona

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 …

## Atoms of Space

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 …

## Geometry of Delta Sets

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 …

## Discrete Hopf Theme

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 …

## Gauss-Bonnet for Delta Sets

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 …

## Sard for delta sets

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 …

## Arnold’s Theme

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 …

## Last geometric theorem

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 Topologies

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 …

## Wu Betti Conjecture

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 …

## Isospectral Simplicial Complexes

One can not hear a complex! After some hope that some kind of algebraic miracle allows to recover the complex from the spectrum (for example by looking for the minimal polynomial which an eigenvalue has and expecting that the factorization reflects some order structure in the abstract simplicial complex), I …

## Wenjun Wu, 1919-2017

According to Wikipedia, the mathematician Wen-Tsun Wu passed away earlier this year. I encountered some mathematics developed by Wu when working on Wu characteristic. See the Slides and the paper on multi-linear valuations. There is an other paper on this in preparation, especially dealing with the cohomology belonging to Wu …

## Hearing the shape of a simplicial complex

A finite abstract simplicial complex has a natural connection Laplacian which is unimodular. The energy of the complex is the sum of the Green function entries. We see that the energy is also the number of positive eigenvalues minus the number of negative eigenvalues. One can therefore hear the Euler characteristic. Does the spectrum determine the complex?

## Symmetry via Ergodic Theory

One of the attempts to quantize space without losing too much symmetry is ergodic theory. Much of my thesis belongs to this program. It is a flavor of quantum calculus, as “no limits” are involved. The story is closely related to Jacob Feldman, one of my heroes of my graduate …

## What is geometry?

In the context of quantum calculus one is interested in discrete structures like graphs or finite abstract simplicial complexes studied primarily in combinatorics or combinatorial topology. Are they geometry? Are they calculus? What is geometry? In MathE320 I try to use the following definition: Geometry is the science of shape, …

## Jones Calculus

The mathematics of evolving fields with two complex components is known already in Jones calculus.

## A quaternion valued elliptic complex

This blog entry delivers an other example of an elliptic complex which can be used in discrete Atiyah-Singer or Atiyah-Bott type setups as examples. We had seen that when deforming an elliptic complex with an integrable Lax deformation, we get complex elliptic complexes. We had wondered in that blog entry …

## Discrete Atiyah-Singer and Atiyah-Bott

As a follow-up note to the strong ring note, I tried between summer and fall semester to formulate a discrete Atiyah-Singer and Atiyah-Bott result for simplicial complexes. The classical theorems from the sixties are heavy, as they involve virtually every field of mathematics. By searching for analogues in the discrete, …

## Strong Ring of Simplicial Complexes

The strong ring is a category of geometric objects G which are disjoint unions of products of
simplicial complexes. Each has a Dirac operator D and a connection operator L. Both are related in
various ways to topology.

## The Dirac operator of Products

Implementing the Dirac operator D for products of simplicial complexes without going to the Barycentric refined simplicial complex has numerical advantages. If G is a finite abstract simplicial complex with n elements and H is a finite abstract simplicial complex with m elements, then is a strong ring element with …

## Do Geometry and Calculus have to die?

In the book ‘This Idea Must Die: Scientific Theories That Are Blocking Progress’, there are two entries which caught my eye because they both belong to interests of mine: geometry and calculus. The two articles are provided below. [I believe it is “fair use” as a reprint of these two …

## The Two Operators

The strong ring The strong ring generated by simplicial complexes produces a category of geometric objects which carries a ring structure. Each element in the strong ring is a “geometric space” carrying cohomology (simplicial, and more general interaction cohomologies) and has nice spectral properties (like McKean Singer) and a “counting …

## Space and Particles

Elements in the strong ring within the Stanley-Reisner ring still can be seen as geometric objects for which mathematical theorems known in topology hold. But there is also arithemetic. We remark that the multiplicative primes in the ring are the simplicial complexes. The Sabidussi theorem imlies that additive primes (particles) have a unique prime factorization (into elementary particles).

## Graph limits with Mass Gap

The graph limit We can prove now that the graph limit of the connection graph of Ln x Ln which is the strong product of Ln‘ with itself has a mass gap in the limit n to infinity. The picture below shows this product graph for n=13, and to the …

## One ring to rule them all

Arithmetic with networks The paper “On the arithmetic of graphs” is posted. (An updated PDF). The paper is far from polished, the document already started to become more convoluted as more and more results were coming in. There had been some disappointment early June when realizing that the Zykov multiplication …

## Three Kepler Problems

Depending on scale, there are three different Kepler problems: the Hydrogen atom, the Newtonian Kepler problem as well as the binary Blackhole problem. The question whether there is a unifying model which covers all of them is part of the quest of finding a quantum theory of gravity.

## More about the ring of networks

The dual multiplication of the ring of networks is topological interesting as Kuenneth holds for this multiplication and Euler characteristic is a ring homomorphism from this dual ring to the ring of integers.

## Unique prime factorization for Zykov addition

We give two proofs that the additive Zykov monoid on the category of finite simple graphs has unique prime factorization. We can determine quickly whether a graph is prime and also produce its prime factorization.

## Hardy-Littlewood Prime Race

The Hardy-Littlewood race has been running now for more than a year on my machine. The Pari code is so short that it is even tweetable. Here are some slides which also mention Gaussian Goldbach: What do primes have to do with quantum calculus? First of all, analytic number theory …

## The Hydrogen trace of a complex

Motivated by the Hamiltonian of the Hydrogen atom, we can look at an anlogue operator for finite geometries and study the spectrum. There is an open conjecture about the trace of this operator.

## The quantum plane

Update of May 27, 2017: I dug out some older unpublished slides authored in 2015 and early 2016. I added something about the quantum gap and something on the quantum plane at the very end. Here is the presentation, just spoken now. The quantum line In one dimension, there is …