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Small research update
#1
Hi to everyone. I was very quiet since June. I wanted to make a brief update for myself in the future and for who is interested.
James' contribute and amount of new insight paralyzed my brain, literally. I was burned out. I used the following 3-4 months connecting some of the dots (few of the many, many dots).
I'm delighted of the aesthetic of the new tetration plots that are being produced: I love them. Even if I can't follow the computational implementation issues, it seems to me that lot of progress were made thanks to James, Sheldon and Ember Edision teaming up on the algorithms+Tommy's insights, correct me if I'm wrong. If someone is willing to make a very brief recap of the latest episodes, I will be grateful.



On my side In June I was working mostly on three research fronts:

-algebraic superfunction tricks: an algebraic rewriting of the classical kneser's one to the more involved James' trick.
-categorical description of James's compositional calculus. in order to understand better the limit tricks and apply them to more general time monoids, see [Composition].
-general foundations of the theory of superfunctions (complete spaces): this effort filled my time since August 2020, when I finally found the key language in the category theory of monoid actions linking iteration to general recursion. This aims to provide a universal language to describe all the possible problems of iteration, i.e. tetration, hyperoperations, fractional-hyperoperations, classifications of iteration methods, see [SupSpaces].

What crashed my brain's cpu was the realization of some facts, mainly brought up by James:

1-Reading James' posts in [Congruent integral] that compositional integral assigning values to "composition paths" and that seems to require, for a deep unification and categorical rewriting, to consider the general theory of differential forms. The Jabotinsky iteration logarithm connection , that involved infinitesimal generators of fields, convinced me that I needed to study seriously 1-parameter groups, differential forms and fields, to vector bundles, parallel transport and gauge-groups and monodromy/holonomy actions and way up to fundamental groups, Grothendieck-Galois theory and cohomology.

2-During that period I had a sequence of Eureka moment while trying to approach String theory and Quantum Field theory. I started to understand, for the first time, what bundles and parallel transport are and that it is possible to use their language to understand deeply the structure of class of solutions of super-functions: the 1-million dollar question of how to chose THE solution. In fact A METHOD can be formally defined to be a section of the solution bundle, see [Adjoints], i.e. a field configuration over the space of solutions.

3- That discussions [Subfunction is functorial], [FracCalcApproach] on the link between categories and Hilbert spaces were the cherry on the pie.

Needles to say that this made me halt. I went back to basics and tried to study more, narrowing down my efforts. So I paused on the Jabotinsky/differential forms/differential geometry front because it is the most broad and scary one.
But this summer, thanks to the sun and the sea, I had a further eureka moment. I connected all the dots concerning James' trick and super-function spaces with my old research on fractional hyperoperations. I finally found it. The theory of ranks, of Goodstein hyperoperations and of \(\diamond\)-operations(I call them Nixon algebras/diamond algebras) can be grounded on the theory of super-function spaces and Kneser and Nixon tricks can be applied to this:I found a closed limit formula for general hyperoperations not only with real/complex ranks but over arbitrary ranks!

Current fronts of research:
A-So atm I'm working on giving the foundations category-theoretical theory of super-functions. The road is long because that is classical literature that goes under the name of topos theory (topoi of sheaves on monoids/topoi of monoid actions) and was treated deeply by Lawvere. I just need to make it accessible and apply it to our needs in the tetration/hyperoperations field. This amounts to turning this draft into a full book. The interesting point is that I guess I found a purely-algebraic characterization of rational (\(t\in\mathbb Q\)) iteration that, basically, is an extension of the classical recursion theorem over the natural numbers and generalize the folklore of defining real iteration piecewise over the [0,1) interval , i.e. the circle.

B- I'm currently writing at full speed a paper on the general theory of Goodstein hyperoperations. Sadly I was not able to describe it category-theoretically yet but only set-theoretically: the impasse that blocked me for years, since 2016, was caused by my stubborness in wanting a fully category theoretic description when it was too early. The set theoretic formulation do its job well. I have already a bunch of interesting and non-trivial results. For example over finite sets there cannot be goodstein hyperoperations and that there exists at least one non-trivial (non-constant) goodstein sequence over some infinite group. Another result is that a goodstein sequence can be realized, over groups, a solution to a fixed point problem, hence we can use kneser's trick to obtain it if the structure admits a compatible convergence notion, in particular if we have topological completeness over the function space. Another gem I found for free is that the set of hyperoperations over a given group  has a partition into pieces that makes it into a bundle over the connected components: section of this bundle, field configurations, are in bijections with the class of possible Nixon Algerbras over that fixed group.

C-Studying the possibility of starting a discord server, a blog and hyperoperations' wiki.

Regards

MathStackExchange account:MphLee

Fundamental Law
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#2
Wooo! Sounds like you've had a productive summer! I spent mine teaching myself pari!

I'm excited to see what comes of this! I'm a little wary of the term Nixon-algebra; as I have no idea what it is and it has my name on it, LOL! I assume it's the hyper-operation algebra equipped with the diamond operator \(\diamond \uparrow^s = \uparrow^{s+1}\).

Fascinated to see what your paper will look like!

Regards, James
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#3
(10/25/2021, 08:40 PM)JmsNxn Wrote: I'm a little wary of the term Nixon-algebra; as I have no idea what it is and it has my name on it, LOL!

The term Hilbert space comes with a funny anecdote on its origin:
Saunders Mac Lane, one of the founding fathers of category theory, in [1989] writes

[Image: image.png]
Translation: "Dr. von Neumann, I would like to know what is a Hilbert space?" (tr. by \(n\)Lab)

[1989] S. Mac Lane, 'Concepts and categories in perspective: Addendum', pp. 323–365 in A century of mathematics in America, part I. Edited by P. Duren, R. A. Askey, and U. C. Merzbach. History of Mathematics 1. American Mathematical Society (Providence, RI), 1988. An addendum to this article was published in History of Mathematics, part III (1989).


ps: A \(\diamond\)-algebra/Nixon Algebra is something much bolder than what you are thinking then, yet,  originated from you. Let \((J,{}^+)\) be a set equipped with an function \({}^+:J\to J\), i.e. a discrete dynamical system maaping \(\sigma \mapsto \sigma^+\) that we call the system of ranks, define a Nixon algebra over \(J\) - with ranks in \(J\) - to be a pair \((M,\diamond)\) where
  • \(M\) is a monoid \((M,\cdot, 1_M)\);
  • equipped with an operation \(\diamond:J\times M\to M\) such that for every \(\sigma\in J\) and \(f\in M\) we have
$$(\sigma^+\diamond f)\cdot f=(\sigma\diamond f)\cdot(\sigma^+\diamond f)$$

MathStackExchange account:MphLee

Fundamental Law
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