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February 28, 2022

Hardy, Ramanujan and Taxi No. 1729

Posted by John Baez

In his book Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, G. H. Hardy tells this famous story:

He could remember the idiosyncracies of numbers in an almost uncanny way. It was Littlewood who said every positive integer was one of Ramanujan’s personal friends. I remember once going to see him when he was lying ill at Putney. I had ridden in taxi-cab No. 1729, and remarked that the number seemed to be rather a dull one, and that I hoped it was not an unfavourable omen. “No,” he replied, “it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways.”

Namely,

10 3+9 3=1000+729=1729=1728+1=12 3+1 3 10^3 + 9^3 = 1000 + 729 = 1729 = 1728 + 1 = 12^3 + 1^3

But there’s more to this story than meets the eye.

Posted at 12:55 AM UTC | Permalink | Followups (12)

February 25, 2022

Applied Category Theory 2022

Posted by John Baez

The Fifth International Conference on Applied Category Theory, ACT2022, will take place at the University of Strathclyde from 18 to 22 July 2022, preceded by the Adjoint School 2022 from 11 to 15 July. This conference follows previous events at Cambridge (UK), Cambridge (MA), Oxford and Leiden.

Applied category theory is important to a growing community of researchers who study computer science, logic, engineering, physics, biology, chemistry, social science, linguistics and other subjects using category-theoretic tools. The background and experience of our members is as varied as the systems being studied. The goal of the Applied Category Theory conference series is to bring researchers together, strengthen the applied category theory community, disseminate the latest results, and facilitate further development of the field.

Posted at 5:09 PM UTC | Permalink | Followups (14)

February 15, 2022

Questions About the Néron–Severi Group

Posted by John Baez

A friend of mine with good intuitions sometimes says things without proof, and sometimes I want to know why — or even whether — these things are true.

Here are some examples from algebraic geometry.

Posted at 8:16 PM UTC | Permalink | Followups (6)

February 12, 2022

Ergodic Set Theory is Trivial

Posted by Tom Leinster

On the one hand, an ultrafilter on a set can be seen as a primitive sort of probability measure, in which every subset is assigned a probability of either 0 or 1 and the measure only has to be finitely additive.

On the other, ergodic theory studies measure-preserving endomorphisms of probability spaces.

What happens if you put the two together? That is, given an ultrafilter 𝒰\mathcal{U} on a set XX, what can be said about the measure-preserving endomorphisms of (X,𝒰)(X, \mathcal{U})?

The answer is: very little. The situation is trivial. Essentially the only measure-preserving endomorphism of (X,𝒰)(X, \mathcal{U}) is the identity. A bit more exactly, such an endomorphism must be the identity almost everywhere.

Here’s why.

Posted at 2:30 PM UTC | Permalink | Followups (11)

February 10, 2022

Submission to arXiv

Posted by John Baez

Philip Helbig is an astrophysicist who wrote a paper called The flatness problem and the age of the Universe. It’s a good review of some very important problems, but the arXiv refused to accept it in the category where it belongs, astro-ph. Instead they tried to shunt it off to gen-ph. This is their usual strategy for dealing with bad papers, designed to keep the more important categories clean. Helbig protested, and this blog article is the story of what happened next.

Briefly: by now his paper has been published in the Monthly Notices of the Royal Astronomical Society, but the arXiv still refuses to accept his paper in astro-ph. Recently Stein Sigurðsson, the Scientific Director of the arXiv, gave this cryptic response:

As I have noted, we do not discuss decisions on individual papers with third parties. When people express concerns, we discuss process and broader issues that affect the processes. The SCOAP3 agreement constrains how appeals for published submissions are handled.

and

SCOAP3 impacts all appeals to arXiv based on the submission having been published in a journal. arXiv gets sued, our policies and processes are constrained by advice of counsel and rulings in those cases. I believe in all case arXiv prevailed, but judge rulings provide guidance.

I don’t understand this. Is he claiming that the arXiv might get sued if they put Helbig’s paper on the arXiv now… because it’s been published?

Posted at 4:37 PM UTC | Permalink | Followups (9)

February 8, 2022

Concepts and Profunctor Nuclei

Posted by Simon Willerton

Guest post by Matthew Di Meglio and Owen Lewis

This article is part of the ongoing work at the 2021 Adjoint School. Thanks to Sophie Libkind, David Jaz Myers, Simon Willerton, Tai-Danae Bradley, Callum Reader, and participants of the 2021 Adjoint School for helpful discussions and feedback.

Previously on this blog, Simon Willerton introduced the notion of the nucleus of an enriched profunctor, and alluded to the fact that the formal concept lattice of a relation is a particular instance of this general categorical construction. Continuing from Simon’s posts, we will present, through the lens of enriched category theory, a generalization of Formal Concept Analysis called Quantitative Concept Analysis (QCA). The material in this blog post is based on Dusko Pavlovic’s Quantitative Concept Analysis and Jonathan Elliot’s On the Fuzzy Concept Complex.

Posted at 4:42 PM UTC | Permalink | Followups (1)

February 7, 2022

Compositional Thermostatics (Part 2)

Posted by John Baez

guest post by Owen Lynch

In Part 1, John talked about a paper that we wrote recently:

and he gave an overview of what a ‘thermostatic system’ is.

In this post, I want to talk about how to compose thermostatic systems. We will not yet use category theory, saving that for another post; instead we will give a ‘nuts-and-bolts’ approach, based on examples.

Posted at 9:54 PM UTC | Permalink | Post a Comment