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.

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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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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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The many principles of conservation of number

In algebraic geometry, we like to make statements like: “two conics meet at {4} points”, “a degree four plane curve has {28} bitangents”, “given four lines in three space, there are {2} lines that meet all of them”. In each of these, we are saying that, as some parameter (the conics, the degree four curve, the lines) changes, the number of solutions to some equation stays constant. The “principle of conservation of number” refers to various theorems which make this precise.

In my experience, students in algebraic geometry tend to pick up the rough idea but remain hazy on the details, most likely because there are many different ways to make these details precise. I decided to try and write down all the basic results I could think of along these lines.

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Mathematical Research Community on Cluster Algebras in Utah this summer

This June 8 to 14, there will be a week long gathering in Snowbird, Utah for young mathematicians working on cluster algebras. The target audience here are either current graduate students, or people with Ph. D. in the last 3 or so years, who would be ready to start working on problems in cluster algebras. The hope is to spend a lot of time getting collaborations and projects going during the week. The organizers are Michael Gekhtman, Mark Gross, Gregg Musiker, Gordana Todorov and me.

We still have room for a number more applicants, so we would like to encourage more of you to apply. Please note that the application deadline of March 1 is firm.

The Hoffman-Singleton graph and groupoids

The Hoffman-Singleton graph is the unique graph on 50 vertices with the following property: Every vertex is of degree 7 and, between any two vertices, there is either an edge or a path of length two, but not both. The Hoffman-Singleton graph has a large symmetry group — order 252,000 — and there are many ways to describe it that emphasize different symmetry properties. Various constructions describe it in terms of the geometry of the affine plane \mathbb{F}_5^2, the projective space \mathbb{P}^3(\mathbb{F}_2) or just pure combinatorics. Here is one more that I noticed the other day when reading through the original Hoffman-Singleton paper. While turning it into a blogpost, I noticed that the same observation was made by Markus Junker in 2005.

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New open access journal in algebraic geometry

I just received an e-mail announcing that Compositio has launched an Open Access journal entitled Algebraic Geometry. Their website is live and promises “Open access implies here that the electronic version of the journal is freely accessible and that there are no article processing charges for authors whatsoever. The printed version of the journal will be available at the end of the calendar year against printing costs.”

The editorial board looks great, including L. Caporaso, J. Ellenberg, D. Maulik and R. Pandharipande. They will definitely get my next algebraic geometry paper.

This is really good news. It’s seemed clear from the debates on journals of the last year that what is needed is for people and institutions of high reputation to commit to running open journals. Compositio, and the editors they have found, are top of the line. From a selfish perspective, what makes me really happy is that I didn’t wind up on the editorial board.

Good work, and good luck, to Algebraic Geometry.