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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!
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 that works for sets equipped with a function . I’ve been talking with Tom Leinster about this lately, so let me say a bit more about how it works.
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.
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.
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 -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 (nick.gurski@case.edu), who is chairing the search committee.