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It’s been a busy semester, and I haven’t done a great job of updating this blog lately. I have a couple of posts in preparation, but in the meantime:
• I gave a talk on the nilpotence and periodicity theorems in stable homotopy theory at the pre-Talbot seminar (a.k.a. Juvitop) at MIT. All the talks this semester were videotaped; the video of mine is here. The results are really beautiful, showing that the “global” picture of stable homotopy theory exactly parallels the geometry of the moduli stack of formal groups.
• I’ve been taking notes from a course of Joe Harris on the representation theory of Lie groups. Unfortunately, I’m unable to include the many pictures that were drawn in lectures, and the notes are somewhat incomplete.
• I’m spending the summer at the REU program at Emory, and I’ll be thinking about problems in moduli of curves. It should be interesting to get a little experience with algebraic geometry. In particular, I’m going to try to focus this blog in that direction over the next couple of months.

I’m sorry for the lack of posts over the past few weeks! I have a few things that I’d like to blog about during the winter break. But to start with, I’d like to use this post to advertise a student organization with which I’m involved — the Harvard College Math Review. The HCMR is a yearly student magazine, and it publishes expository articles by mathematics students (from various universities and occasionally high schools) in all areas of mathematics. We’re looking for submissions for the 2013 issue.

General submissions are around 10-15 pages in length, and there are several “features” slots: Mathematical Minutiae, Statistics Corner, Applied Mathematics Corner, and My Favorite Problem. The HCMR also publishes a list of problems and solutions contributed by readers. Final papers written for courses make a good fit!

Zach Abel, a graduate student in the math department at MIT, has started a blog intended at explaining a wide variety of mathematical topics. Go visit it!

I’m not an expert on tech policy (or policy in general), but as far as I can tell the proposed “Stop Online Piracy Act” is really atrocious in its (possibly unconstitutional?) over-reach, and would probably be very bad for bloggers, mathematics, and the internet. Following Tim Gowers, Wikipedia, and others, I’ll be blocking this blog tomorrow. I couldn’t find out a good way to do this (those with a custom stylesheet can directly edit things to do so), so like Gowers, I’ll make this blog private for a day.

Edit: It actually seems that WordPress has provided a means to do this directly. Let’s see how that works.

Cathy O’Neill’s new blog Mathbabe seems to be fairly active already, but I thought I would promote it here in case some of this blog’s readers have not seen it.  I don’t think I would be able to summarize this extremely interesting and varied blog about quantitative issues, politics, and mathematics, so you should read it instead of listening to me recommend it!

Sorry I have not been updating this blog lately. BBD turns out to be a fairly difficult paper, and I have been reading other things, like trying to understand the Fourier-Deligne transform, which I might blog about soon. In addition, I have been reading some very nice notes on cohomological descent, which I recommend.

I’ve been taking the year-long algebraic topology sequence, which has been really great — I’ve started dipping my toes into areas that I had no prior familarity with, like the theory of model categories and simplicial sets. Michael Hopkins, who is teaching the course this year, is making the second half much more algebraic than a typical course (covering, say, characteristic classes) than usual; the goal, it seems, is to prove a Quillen equivalence of the rational homotopy category with an algebraic model category (I think the category of differential graded algebras).

Eva Belmont and I have been TeXing notes from the course; she has posted them here.

One of the things you’ll notice that was reflected in the notes is that Hopkins’s style is to sketch big ideas without necessarily delving into details. Part of this is the nature of the subject. The rudiments of homotopical algebra have a similar feel to the basic proofs in homological algebra — diagram-chasing left and right, lots of abstract nonsense, proofs that may be hard digest but which are equally hard to forget. In short, the details of the foundations seem to be one of those things that one checks in the privacy of one’s room rather than in public. Just as people don’t write out full proofs of the five-lemma in class, we didn’t formally verify the construction of the homotopy category of a model category, or even prove formally the details of the model structure on simplicial sets — had we, there wouldn’t have been enough time for the fun stuff.

But, as a student, getting the details of the foundations straight in one’s mind is important. And, especially if we want to make these notes more suitable for public consumption, including full Bourbaki-esque proofs for each result would be helpful — not least because the modern, post-Whitehead style of homotopy theory where “space” means “simplicial set” seems to have rather few textbooks. There is no EGA for algebraic topology!

So, we are planning eventually to turn these notes into an open-source and collaborative project, currently named the PATH project. The pressures of the academic semester makes it unlikely that this will really get off the ground before the summer, but when it does, it should be a useful experience for anyone interested in working through the nuts and bolts of the foundations. (The plan is to have a significant amount of overlap with the CRing project. After all, you need a certain amount of algebra to do algebraic topology, and homotopical methods figure in commutative algebra in, for instance, the theory of the cotangent complex.)

So MaBloWriMo was kind of a failure for me. I did make the fifteen posts I planned for, but I said much less than I had hoped to, and dropped out near the end of the month. I also did not keep my promise of a series of posts on local cohomology. Those may happen someday, but I do not think I will have the time to do that in the near future.

Nonetheless, it turned out that my commutative algebra course covered most of the material that I was planning to write about (e.g. regular local rings, homological theory, etc.). Since I have taken detailed notes for that class, I still ended up writing about the material in some way. I have created a website where I have posted these notes and many more things.

For the winter, my priority is to study stable homotopy theory. There are other topics that I also want to read about (mostly in algebraic geometry), but I am planning to take a topics course in topology next semester that will require this, so it probably will be the focus on this blog. My knowledge of homotopy theory in general right now is very limited, though. So I hope to begin talking about spectra soon, and with the additional free time that I now have, posts may actually be somewhat frequent.

This is the lemma we shall use in the proof of the reciprocity law, to reduce the cyclic case to the cyclotomic case:

Lemma 4 (Artin) Let ${L/k}$ be a cyclic extension of degree ${n}$ and ${\mathfrak{p}}$ a prime of ${k}$ unramified in ${L}$. Then we can find a field ${E}$, a subextension of ${L(\zeta_m)}$, with ${E \cap L = k}$ such that in the lattice of fields

we have:

1. ${\mathfrak{p}}$ splits completely in ${E/k}$

2. ${E(\zeta_m) = L(\zeta_m)}$, so that ${LE/E}$ is cyclotomic

3. ${\mathfrak{p}}$ is unramified in ${LE/k}$

Moreover, we can choose ${m}$ such that it is divisible only by arbitrarily large primes.

The proof of this will use the previous number-theory lemmas and the basic tools of Galois theory.

So, first of all, we know that ${E}$ is a subextension of some ${L(\zeta_m)}$. We don’t know what ${m}$ is, but pretend we do, and will start carrying out the proof. As we do so, we will learn more and more about what ${m}$ has to be like, and eventually choose it.
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The following topic came up in the context of my project, which has been expanding to include new areas of mathematics that I did not initially anticipate. Consequently, I have had to learn about several new areas of mathematics; this is, of course, a common experience at RSI. For me, the representation theory of Lie algebras has been one of those areas, and I will post here about it to help myself understand it. Right here, I’ll aim to cover the groundwork necessary to get to the actual representation theory in a future post.

Lie Algebras

Throughout, we work over ${{\mathbb C}}$, or even an algebraically closed field of characteristic zero.

Definition 1 A Lie algebra is a finite-dimensional vector space ${L}$ with a Lie bracket ${[\cdot, \cdot]: L \times L \rightarrow L}$ satisfying:

• The bracket ${[\cdot, \cdot]: L \times L \rightarrow L}$ is ${{\mathbb C}}$-bilinear in both variables.
• ${[A,B] = -[B,A]}$ for any ${A,B \in L}$.
• ${[A, [B,C]] + [B, [C,A]] + [C, [A,B]] = 0}$. This is the Jacobi identity.

To elucidate the meaning of the conditions, let’s look at a few examples. (more…)