## phase transition

Today, our youngest daughter (aka PD2 on this blog) gave birth to a little boy, Gust.

I’m in transition, trying to adjust to this new phase in our lives.

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## Kasha-eating dragons

This semester I’m teaching a first course in representation theory. On campus, IRL! It’s a bit strange, using a big lecture room for a handful of students, everyone wearing masks, keeping distances, etc.

So far, this is their only course on campus, so it has primarily a social function. The breaks in between are infinitely more important than the lectures themselves. I’d guess breaks take up more than one third of the four hours scheduled.

At first, I hoped to make groups and their representations relevant by connecting to the crisis at hand, whence the the symmetries of Covid-19 post, and the Geometry of Viruses series of posts.

Not a great idea. I guess most of us are by now over-saturated with Corona-related news, and if students are allowed to come to campus just one afternoon per week, the last thing they want to hear about is, right, Covid.

So I need to change tactics. By now we’ve reached the computation of character tables, and googling around I found this MathOverflow-topic: Fun applications of representations of finite groups.

The highest rated answer, by Vladimir Dotsenko, suggests a problem attributed to Kirillov:

An example from Kirillov’s book on representation theory: write numbers 1,2,3,4,5,6 on the faces of a cube, and keep replacing (simultaneously) each number by the average of its neighbours. Describe (approximately) the numbers on the faces after many iterations.

A bit further down the list, the Lecture notes on representation theory by Vera Serganova are mentioned. They start off with a variation of Kirillov’s question (and an extension of it to the dodecahedron):

Hungry knights. There are n hungry knights at a round table. Each of them has a plate with certain amount of food. Instead of eating every minute each knight takes one half of his neighbors servings. They start at 10 in the evening. What can you tell about food distribution in the morning?

Breakfast at Mars. It is well known that marsians have four arms, a standard family has 6 persons and a breakfast table has a form of a cube with each person occupying a face on a cube. Do the analog of round table problem for the family of marsians.

Supper at Venus. They have five arms there, 12 persons in a family and sit on the faces of a dodecahedron (a regular polyhedron whose faces are pentagons).

Perhaps the nicest exposition of the problem (and its solution!) is in the paper Dragons eating kasha by Tanya Khovanova.

Suppose a four-armed dragon is sitting on every face of a cube. Each dragon has a bowl of kasha in front of him. Dragons are very greedy, so instead of eating their own kasha, they try to steal kasha from their neighbors. Every minute every dragon extends four arms to the four neighboring faces on the cube and tries to get the kasha from the bowls there. As four arms are fighting for every bowl of
kasha, each arm manages to steal one-fourth of what is in the bowl. Thus each
dragon steals one-fourth of the kasha of each of his neighbors, while at the same
time all of his own kasha is stolen. Given the initial amounts of kasha in every
bowl, what is the asymptotic behavior of the amounts of kasha?

I can give them quick hints to reach the solution:

• the amounts of kasha on each face gives a vector in $\mathbb{R}^6$, which is an $S_4$-representation,
• calculate the character of this kasha-representation,
• use the character table of $S_4$ to decompose the representation into irreducibles,
• identify each of the irreducible factors as instances in the kasha-representation,
• check that the food-grabbing operation is an $S_4$-morphism,
• remember Schur’s lemma, and compute the scaling factors on each irreducible component,
• conclude!

But, I can never explain it better than Khovanova’s treatment of the kasha-eating dragons problem.

## this one’s for M.

and for everyone else feeling a bit lost in these tiring times.

Photo credit: @itoldmythe

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## taking stock

The one thing harder than to start blogging after a long period of silence is to stop when you think you’re still in the flow.

(image credit Putnam Consulting)

The Januari 1st post a math(arty) 2018 was an accident. I only wanted to share this picture, of a garage-door with an uncommon definition of prime numbers, i saw the night before.

I had been working on a better understanding of Conway’s Big Picture so I had material for a few follow-up posts.

It was never my intention to start blogging on a daily basis.

I had other writing plans for 2018.

For years I’m trying to write a math-book for a larger audience, or at least to give it an honest try.

My pet peeve with such books is that most of them are either devoid of proper mathematical content, or focus too much on the personal lives of the mathematicians involved.

An inspiring counter-example is ‘Closing the gap’ by Vicky Neal.

From the excellent review by Colin Beveridge on the Aperiodical Blog:

“Here’s a clever way to structure a maths book (I have taken copious notes): follow the development of a difficult idea or discovery chronologically, but intersperse the action with background that puts the discovery in context. That’s not a new structure – but it’s tricky to pull off: you have to keep the difficult idea from getting too difficult, and keep the background at a level where an interested reader can follow along and either say “yes, that’s plausible” or better “wait, let me get a pen!”. This is where Closing The Gap excels.”

So it is possible to publish a math-book worth writing. Or at least, some people can pull it off.

Problem was I needed to kick myself into writing mode. Feeling forced to post something daily wouldn’t hurt.

Anyway, I was sure this would have to stop soon. I had plans to disappear for 10 days into the French mountains. Our place there suffers from frequent power- and cellphone-cuts, which can last for days.

Thank you Orange.fr for upgrading your network to the remotest of places. At times, it felt like I was working from home.

I kept on blogging.

Even now, there’s material lying around.

I’d love to understand the claim that non-commutative geometry may offer some help in explaining moonshine. There was an interesting question on an older post on nimber-arithmetic I feel I should be following up. I’ve given a couple of talks recently on $\mathbb{F}_1$-material, parts of which may be postable. And so on.

Problem is, I would stick to the same (rather dense) writing style.

Perhaps it would make more sense to aim for a weekly (or even monthly) post over at Medium.

Medium offers no MathJax support forcing me to write differently about maths, and for a broader potential audience.

I may continue to blog here (or not), stick to the current style (or try something differently). I have not the foggiest idea right now.