# Random three dimensional partitions

Back in graduate school, I read a beautiful paper of Kenyon, Okounkov and Sheffield. It started with the following physical story.

This is the corner of a crystal of salt, as seen under an electron microscope. (I took the image from here, unfortunately I couldn’t find better information about the sourcing.) As you can see, the corner is a bit rounded, where some of the molecules have rubbed away. They ask the question: “What is the shape of that rounded corner?”

# 3 continuing positions at the ANU, in statistics, probability, stochastic analysis, mathematical finance, biomathematics.

We’ve just posted an ad for up to 3 continuing positions at the Mathematical Sciences Institute, at the Australian National University, in Canberra. (Where I work!)

It’s up on mathjobs, but applicants will need to apply through the university website. Here’s the pitch:

The Mathematical Sciences Institute at the Australian National University is seeking to invigorate its research and teaching profile in the areas of statistics, probability, stochastic analysis, mathematical finance and/or biomathematics/biostatistics. We wish to fill several continuing positions at the Academic Level B and/or Level C (which equates to the position of Associate Professor within the United States of America). Up to 3 full time positions may be awarded.

You will be joining an internationally recognised leading team of academics with a focus on achieving excellence in research and teaching. The Institute comprises of approximately fifty academics, within seven mathematical research programs. Applicants are expected to have an outstanding record in research, teaching and administration. All positions will involve some teaching, in the specialised areas advertised and/or standard mathematics undergraduate courses, but this may be at a reduced level for several years.

It’s a great place to work, excellent opportunities for research grant funding, and a really nice place to live. Feel free to contact me if you have any questions about the job or living in Canberra.

Please pass this on to friends with relevant interests!

(Oh, and don’t forget those two postdocs I’m hiring in quantum algebra, higher category theory, subfactors, representation theory, etc.)

# The shape of a random partition

In this post we will give a heuristic derivation of a result of Vershik, describing the shape of a random partition of a large integer $N$. (Vershik’s Russian original is available here; English translation is pay-walled.)

By a partition of $N$, we mean positive integers $\lambda_1 \geq \lambda_2 \geq \cdots \geq \lambda_r > 0$ with $\sum \lambda_i = N$. We draw a partition as a collection of boxes: For example, this is $4+2+1+1+1$:

Suppose we let $N \to \infty$, select partitions of $N$ uniformly at random and rescale the size of the boxes by $1/\sqrt{N}$, so that the diagram of the partition always has area $1$. What is the shape of the most likely diagram?

# Legendre duality and statistical mechanics

I’d like to make another attempt at a topic I handled badly before: How Legendre duality shows up in statistical mechanics (or, at least, toy models thereof).

# Short story recommendation

It is a minor spoiler to say why mathematicians will enjoy this story by Scott Alexander but I predict many of you will.

# Postdoc position at ANU

Update — there are now not one, but two, positions available! The application has been extended to the end of November.

We’ve just put up an ad for a new 2 year postdoctoral position at the ANU, to work with myself and Tony Licata. We’re looking for someone who’s interested in operator algebras, quantum topology, and/or representation theory, to collaborate with us on Australian Research Council funded projects.

The ad hasn’t yet been crossposted to MathJobs, but hopefully it will eventually appear there! In any case, applications need to be made through the ANU website. You need to submit a CV, 3 references, and a document addressing the selection criteria. Let me know if you have any questions about the application process, the job, or Canberra!

# Michigan Math In Action

Those of you who are interested in college math instruction may be interested in a no-longer-so-new blog “Michigan Math In Action”, which a number of our faculty started last year. (I was involved in the sense of telling people “blogs are fun!”, but haven’t written anything for them yet.) It mostly features thoughtful pieces on teaching calculus and similar courses.

Recently, Gavin Larose put up a lengthy footnoted post on the effort that goes into running our “Gateway testing” center, and the benefits we get from it. This is a room designed for proctoring computerized tests of basic skills, and we use it for things like routine differentiation or putting matrices into reduced row echelon form, which we want every student to know but which are a waste of class time. Check it out!

# Register soon for ALGECOM Fall 2015!

Just a quick reminder that, if you are looking for graduate support to attend ALGECOM at the University of Michigan on Saturday October 24, or to register for the poster session, you should please send an e-mail to speyer@umich.edu by Tuesday Sept 15. (Yes, after sunset but before midnight is fine, I won’t be online during Rosh Hoshanah either.)
Even if you are not requesting support, I’d appreciate knowing that you are coming.

Our speakers are Jonah Blasiak (Drexel), Laura Escobar (UIUC), Joel Kamnitzer (Toronto) and Tri Lai (IMA and Minnesota). Please see our website for more information.

# A counting argument for Frobenius’ theorem

Let $G$ be a finite group, and let $n$ be a positive integer dividing $|G|$. Then the number of solutions to $g^n=1$ in $G$ is divisible by $n$.

This is a 1907 theorem of Frobenius. Along with the Sylow theorems, it is one of the few nontrivial elementary results about a completely general finite group. And it has some nice applications, which you can read about on Mathoverflow. But it has never made it into the standard basic group theory syllabus the way the Sylow theorems have. I wanted to give it as a challenging problem last time I taught group theory, but I didn’t find a proof that I liked enough.

The last few days, I’ve been thinking about the problem again, and I found what I think is a decent counting proof. I have the feeling there is a really slick proof in here waiting to get out. Let me know if you can find it!

# van Ekeren, Möller, Scheithauer on holomorphic orbifolds

There aren’t many blog posts about vertex operator algebras, so I thought I’d help fill this gap by mentioning a substantial advance by Jethro van Ekeren, Sven Möller, and Nils Scheithauer that appeared on the ArXiv last month. The most important feature is that this paper resolves several folklore conjectures that have been around since near the beginning of vertex operator algebra theory. This was good for me, since I was able to use some of these results to prove the Generalized Moonshine Conjecture much more quickly than I had expected. I won’t say much about moonshine here, as I think it deserves its own post.