Happy 3*11*61

It’s common for the Putnam exam to include problems that involve the number of the current year in some way. (There weren’t any such problems on the past exam, but in 2011 there were two, A1 and B4.) Usually these problems don’t rely on any specific property of the number, but they simply use the number as a stand-in for “n”, so as to make the problem a little bit more concrete; for example, for either of the 2011 problems, you could replace the number 2011 with any other number without changing the nature of the problem.

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On 2012 Putnam problem B2

This post continues my effort to write up solutions (aimed at students) to the problems on this year’s Putnam exam. Here is problem B2:

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On 2012 Putnam Problem B1

This post continues my effort to write solutions, aimed at students, to the problems on this year’s Putnam exam. Here is problem B1:

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On 2012 Putnam Problem A3

This post continues my effort to write detailed solutions to the problems on this year’s Putnam exam (previous solutions are here and here). More than the actual solutions, I hope that students will find my commentary valuable. When writing these, I try to emphasize the thought processes and the strategies that were used to solve the problem, so that a reader might feel empowered to solve similar problems. I feel like some of the solutions in the MAA’s Putnam Directory are too slick and might reinforce a student’s feeling that they never would have come up with a solution on their own. These solutions attempt to counter that feeling.

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On 2012 Putnam Problem A2

This post continues my effort to write solutions to the problems on this year’s Putnam competition. I previously posted a solution to problem A1. Problem A2 seems like it should have been reasonable for students familiar with abstract algebra, but it could have been tricky for students without much exposure to axiomatic theories.

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On 2012 Putnam problem A1

The Putnam competition was last weekend, on December 1. I was coaching Smith’s team this year, so I’d like to go through solutions to each of the problems (or as many as I can solve), with my students being the intended audience. But I’ll post the solutions on this blog, since I’m sure they’ll be of interest to a wider audience.

I’ll start with the first one, Problem A1. It’s often the case that Putnam problems can be solved in several different ways, but sometimes you discover a solution that is clearly the most elegant of all possible approaches. That’s the case with this one, although I will include greater-than-necessary detail for pedagogical purposes.

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Imagist poetry and mathematics

This post is about something I’ve been thinking about since my undergraduate days–the parallels between mathematics and the Imagist movement in poetry.

The key idea of Imagism was that a poem should describe its subject as it “truly is”, without unnecessary words, and without introducing the feelings and personal perspectives of the poet. In its unattainable ideal, an Imagist poem is poet-independent, in that one would hope that two Imagist poets writing about the same subject would write the same poem. And that poem would be the perfect poem on that subject, using exactly the right words in the most elegant possible way.

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Homological interpretation of the supertrace

Let \mathcal{V} = V_0 \oplus V_1 be a \mathbb{Z}_2-graded vector space, where both V_0 and V_1 are finite-dimensional. Suppose that \phi is a degree-preserving endomorphism of \mathcal{V} (so it consists of endomorphisms \phi_i for both V_i). Then the supertrace of \phi is defined as the difference of the two traces:

\mathrm{str}\phi = \mathrm{tr}\phi_0 - \mathrm{tr}\phi_1.

Obviously, the definition of the supertrace is very simple, but it begs for a justification. For example, why is this better than just adding the two traces? I’ve seen two facts offered in defense of the supertrace (see the Wikipedia article): 1) The supertrace is invariant when applied to endomorphisms of modules over commutative superalgebras, and 2) The supertrace of a supercommutator is 0.

I feel like neither of these justifications is particularly satisfying; the first requires a much more complicated setup than what I described above, and the second passes the burden of proof onto the notion of supercommutator. So I’d like to offer another way of thinking about the supertrace, which should be reasonably convincing to anyone who knows a little bit of homological algebra. I feel like this is one of those ideas that the practitioners in the field all know, but nobody bothers to write down.

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MathIM for chatting with LaTeX

I often talk with my collaborators over Skype. Since math equations can be difficult to verbalize, we inevitably end up typing pseudo-latex code to each other through the chat window. Today I discovered MathIM. It’s not the prettiest interface, but it allows multiple people to chat with each other using LaTeX, which is pretty cool.

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Superman’s mathematics

I recently came across this comic panel in a series of “Rubbish superpowers”. I have to say I’m a little bit ambivalent about it. On the one hand, it’s awesome that Superman used the term “super-mathematics”. On the other hand, the math he uses is just multiplication.

When we refer to mathematics as a superpower, we suggest that it’s powerful and cool. But superpowers also have the connotation of being arcane or completely inaccessible to mere mortals. It’s important to emphasize that everyone can learn math, and everyone should have at least basic math skills.

Of course, there’s no doubt that math is more useful than X-ray vision.

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