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!

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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.

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Announcing ALGECOM Fall 2015

The University of Michigan at Ann Arbor is proud to be hosting
ALGECOM, the twice annual midwestern conference on algebra, geometry
and combinatorics on Saturday, October 24. We will feature four
speakers, namely,

Jonah Blasiak (Drexel University)
Laura Escobar (University of Illinois at Urbana-Champaign)
Joel Kamnitzer (University of Toronto)
Tri Lai (IMA and University of Minnesota)

as well as a poster session. If you would like to submit a poster, please e-mail (David Speyer) with a quick summary of your work by September 15.

A block of rooms has been reserved at the (Lamp Post Inn) under the name of ALGECOM.

This conference is supported by a conference grant form the NSF. Limited funds are available for graduate student travel to the conference. Please contact (David Speyer) to request support, and include a note from your adviser.

More information will be added to our website as it becomes available.

We hope to see you there!

The dishonest stopping paradox

A number of blogs I read are arguing about a paradox, posed by tumblr blogger perversesheaf. Here is my attempt to explain what the paradox says.

Suppose that a drug company wishes to create evidence that a drug is beneficial, when in fact its effect is completely random. To be concrete, we’ll say that the drug has either positive or negative effect for each patient, each with probability 1/2. The drug company commits in advance that they will state exactly what their procedure will be, including their procedure for when to stop tasks, and that they will release all of their data. Nonetheless, they can guarantee that a Bayesian analyst with a somewhat reasonable prior will come to hold a strong belief that the drug does some good. Below the fold, I’ll explain how they do this, and think about whether I care.

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ANU hiring postdocs

Amnon Neeman has just put up an ad for two postdoctoral positions at the ANU. He says: “The successful applicants should have strong research interests and activities in or related to one of the following fields: Algebraic Geometry, Commutative Algebra, Representation Theory, Algebraic Topology, Algebraic K-Theory. Skills at applying the techniques of triangulated categories to these areas would be a plus.”

These are excellent positions — available for up to 3 years, with no teaching requirements, and salaries in the AUD81-89k range.

Applications close at the end of January, and I hear Amnon is keen to hire as soon as possible.

Editorial board of “Journal of K-Theory” on strike, demanding Tony Bak hands over the journal to the K-Theory foundation.

Text of the announcement below:

Dear Colleagues,

We the undersigned announce that, as of today 15 September 2014, we’re starting an indefinite strike. We will decline all papers submitted to us at the Journal of K-Theory.

Our demand is that, as promised in 2007-08, Bak’s family company (ISOPP) hand over the ownership of the journal to the K-Theory Foundation (KTF). The handover must be unconditional, free of charge and cover all the back issues.

The remaining editors are cordially invited to join us.

 Yours Sincerely,
Paul Balmer, Spencer Bloch, Gunnar Carlsson, Guillermo Cortinas, Eric Friedlander, Max Karoubi, Gennadi Kasparov, Alexander Merkurjev, Amnon Neeman, Jonathan Rosenberg, Marco Schlichting, Andrei Suslin, Vladimir Voevodsky, Charles Weibel, Guoliang Yu
More details to follow!

What makes the Monster Lie Algebra special?

This is a post I’d been meaning to write for several years, but I was finally prompted to action after talking to some confused physicists. The Monster Lie Algebra, as a Lie algebra, has very little structure – it (or rather, its positive subalgebra) is quite close to being free on countably infinitely many generators. In addition to its Lie algebra structure, it has a faithful action of the monster simple group by Lie algebra automorphisms. However, the bare fact that the monster acts faithfully on the Lie algebra by diagram automorphisms is not very interesting: the almost-freeness means that the diagram automorphism group is more or less the direct product of a sequence of general linear groups of unbounded rank, and the monster embeds in any such group very easily.

The first interesting property of the Monster Lie Algebra has nothing to do with the monster simple group. Instead, the particular arrangement of generators illustrates a remarkable property of the modular J-function.

The more impressive property is a *particular* action of the monster that arises functorially from a string-theoretic construction of the Lie algebra. This action is useful in Borcherds’s proof of the Monstrous Moonshine conjecture, as I mentioned near the end of a previous post, and this usefulness is because the action satisfies a strong compatibility condition that relates the module structures of different root spaces.

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