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Note:These pages make extensive use of the latest XHTML and CSS Standards. They ought to look great in any standards-compliant modern browser. Unfortunately, they will probably look horrible in older browsers, like Netscape 4.x and IE 4.x. Moreover, many posts use MathML, which is, currently only supported in Mozilla. My best suggestion (and you will thank me when surfing an ever-increasing number of sites on the web which have been crafted to use the new standards) is to upgrade to the latest version of your browser. If that's not possible, consider moving to the Standards-compliant and open-source Mozilla browser.

October 25, 2022

Booleans, Natural Numbers, Young Diagrams, Schur Functors

Posted by John Baez

There’s an adjunction between commutative monoids and pointed sets, which gives a comonad. Then:

Take the booleans, apply the comonad and get the natural numbers.

Take the natural numbers, apply the comonad and get Young diagrams.

Take the Young diagrams, apply the comonad and get Schur functors.

Let me explain how this works!

Posted at 3:07 PM UTC | Permalink | Followups (15)

October 16, 2022

Partition Function as Cardinality

Posted by John Baez

In classical statistical mechanics we often think about sets where each point has a number called its ‘energy’. Then the ‘partition function’ counts the set’s points — but points with large energy count for less! And the amount each point gets counted depends on the temperature.

So, the partition function is a generalization of the cardinality |X||X| that works for sets XX equipped with a function E:XE\colon X \to \mathbb{R}. I’ve been talking with Tom Leinster about this lately, so let me say a bit more about how it works.

Posted at 4:12 PM UTC | Permalink | Followups (26)

October 11, 2022

Two Talks on Measuring Diversity

Posted by Tom Leinster

Later this month, I’ll give a pair of lectures at Riken, the major research institute in Japan, at the kind invitation of Ryosuke Iratani.

Both lectures will be on measuring diversity, and the aim is to touch on some biological and information theory aspects as well as the mathematics. Titles, abstracts and dates follow. You can attend online (and I’ll be speaking online too), but Ryo asks that you fill in the registration form.

Posted at 9:20 AM UTC | Permalink | Followups (2)

October 7, 2022

The Eventual Image, Eventually

Posted by Tom Leinster

More than ten years ago, I wrote a series of posts (1, 2, 3) about what happens when you iterate a process for an infinite amount of time. I know just how that feels: it’s taken me until now to finish writing it up. But finally, it’s done:

Tom Leinster, The eventual image. arXiv:2210.00302, 2022.

What is the eventual image? I’ll tell you in nine ways.

Posted at 6:03 PM UTC | Permalink | Followups (5)

October 5, 2022

CWRU Is Hiring

Posted by Tom Leinster

Guest post by Nick Gurski and Mark Meckes

The Department of Mathematics, Applied Mathematics, and Statistics at Case Western Reserve University is hiring a tenure-track assistant professor in pure math, to start in fall 2023. Preference will be given to candidates in algebra, topology, and related areas, although excellent candidates in all areas of pure mathematics will be considered.

At CWRU, patrons of the nn-Category Café may be familiar with not only our names, Nick Gurski and Mark Meckes, but also our colleagues Juan Orendain (Math, Applied Math and Statistics) and Colin McLarty (Philosophy).

Applications are through MathJobs, where you can see the full ad. Questions can be emailed to Nick (, who is chairing the search committee.

Posted at 9:51 PM UTC | Permalink | Post a Comment