This is the second in a sequence of posts where I look at extensions of the rational numbers in a very hands on, low tech, way. This time, we’re going to start checking the Kronecker-Weber theorem. The Kronecker-Weber Theorem states:

The Kronecker-Weber Theorem: If is an abelian Galois extension of , then is contained in for some integer .

So, when I said before that the only abelian extensions I was familiar with were the quadratic and the cyclotomic extensions, I had basically already listed them all! In the last post, I already listed all of the extensions of ; they were indexed by order three subgroups of . Today, we’ll try to fit these inside cyclotomic fields.

By the way, I just discovered the very nice article Kronecker-Weber via Stickelberger by Franz Lemmermeyer; if you are enjoying this series of posts, you might want to work your way through that.

(more…)

After what seemed like roughly the longest two weeks of my life (which included roughly 20 hours of driving, 5 hours on trains, a conference, an apartment search, two seminar talks, taking a friend to an urgent medical care center and one frantic packing of everything I own), I’ve finally managed to leave Princeton permanently (well, at least for the foreseeable future), and take up residence in Cambridge, MA.

As soon as I recover, I have some exciting math posts to write (but only after I get some furniture). Keep your eyes open.

The triumph of early twentieth century number theory is the formulation and proof of the main results of class field theory. In technical terms, class field theory classifies the abelian extensions of a number field and describes how primes split in such extensions. In more elementary (but less accurate) terms, class field theory answers two questions.

First, what are the polynomials such that is a Galois extension of with abelian Galois group? Remember that, if is a generic polynomial of degree , then the splitting field of has Galois group , and the field is fixed by . So the polynomials which give Galois extensions are very rare, in that the Galois group is much smaller than the generic case.

Second, if is such a polynomial, how does factor modulo various primes ? I already wrote a post explaining what this has to do with Galois theory, so for now I’ll just refer you to there. Let me point out, though, that the description of that post is pretty impractical, if you actually want to answer a question like “for which primes does the polynomial have three roots modulo ?” That’s because these questions are really hard! The amazing thing about class field theory is that it will give you a completely explicit answer to the question “for which primes does the polynomial have three roots modulo ?”.

The first special case of these questions is the case where is a quadratic polynomial. In this case, the Galois group is always abelian, because it is contained in the abelian group . The field is isomorphic to for some unique square-free integer . The question of whether has roots modulo comes down to computing the Legendre symbol which, by quadratic reciprocity, comes down to evaluating the class of modulo .

In this post and the sequel, I will explain how elementary methods are good enough to get you through the next case, where has degree . In this post, we will solely be concerned with the first question — what are the cubic extensions of with Galois group ?

One of the reasons I like this problem it that it shows you how far a little Galois theory can get you. Another reason is that it shows you that the main results of Class Field Theory don’t come out of nowhere. When I learned the subject, I knew very few examples of abelian Galois extensions, basically just the quadratic fields and the cycloctomic fields, and so the big classification results were baffling to me. Later, I realized that the number theorists who formulated these results knew tons of examples and could were very comfortable playing with them. In this post, I hope to get you comfortable playing with the extensions of .

(more…)

So, suppose you have a Subfactor category. Remember that this is a bi-oidal category with four different kinds of objects (A-A, A-B, B-A, and B-B), with tensor products and , with a good theory of duals, and a chosen generating object X. How does one study categories with tensor products? By drawing pictures!

(more…)

You heard me right. Fractal cookies. Why aren’t they being served at the Institute? They really should put their bakers on the case.

Of course, those aren’t the only physical realization of fractals on the web, but they may well be the most delicious.

Does anyone know where the following useful facts were first proved? A lot of papers just say, "It is known that…" and I’d like to give proper attribution in some future work.

Let A be an abelian group, and let denote the monoidal category of A-graded complex vector spaces. Then:

1. Equivalence classes of braided structures on are classified by elements of .
2. also classifies -valued quadratic forms on A.
3. , where the right side is "Eilenberg-MacLane abelian group cohomology" (defined in MacLane’s 1950 ICM address).

There is an additional neat interpretation involving double loop maps and multiplicative torsors on A, but I don’t need that level of sophistication for the near future.

Last week I talked about the Temperley-Lieb algebra - the algebra of boxes with n top points connected in a non-crossing way to n bottom points, with multiplication as stacking boxes.  Some of you may have noticed (but weren’t picky enough to point out) that I didn’t specify whether AB meant A on top of B, or B on top of A.  Of course, it doesn’t really matter, but we should pick one, right?

But wait … why are these two stackings the only candidates for multiplication?  Why shouldn’t we multiply by connecting the right side of a box to the left side of the next box?

or by connecting some top points and some bottom points of each box?  

The observation that there are lots of different multiplications on Temperley-Lieb might lead you to wonder about other operators on Temperley-Lieb.  For instance, we can map to by connecting any point of the first box to a point of the second, and the rest of the points to the boundary:

Everything I’ve drawn above is an example of a “planar tangle” - and the trace we used last week is also a planar tangle, which takes to :

In general, a planar tangle is a diagram where the strings of k input boxes and an output box are connected among themselves in non-crossing ways.  Here’s another example - which is a fine planar tangle, although it’s not clear that it should have any particular meaning if we let it act on Temperley-Lieb inputs.

Planar tangles can sometimes be composed with each other:  we can connect the output of one tangle to the input of another tangle, if both have the same number of strings.  Here’s an unnecessarily complicated example:

Notice that in the LHS, we have labels 1, 2 and 3 in the boxes — this is just so we know what order to do the compositions in.  In the MHS, we’ve stuck the tangles in the boxes they go into; and on the RHS, we’ve discarded the information of the old boundaries and isotoped the strings to make a nicer picture.

The set of planar tangles, with the operation of composition, is an operad.  (I’m not going to tell you what an operad is in general, but if you’re curious  http://homepages.ulb.ac.be/~fschlenk/Maths/What/operad.pdf is a nice introduction.)  A planar algebra is, basically, a representation of the planar operad:  a family of vector spaces with an action by planar tangles which is compatible with composition.  

Temperley-Lieb is not just the simplest and most natural example of a planar algebra; it’s also one of the most important ones.  Coming soon:  Some other examples!

Sorry for the delay, Scott’s been in town and so I’ve been too busy doing actual research to get much blogging done. This post was also a little delayed because I didn’t understand this example as well as I’d hoped. I still don’t fully grok it so maybe you all can help me out. If you don’t follow this example, don’t worry, we’ll be moving on to pictures in the next post.

Take a finite group G. Let C[G] denote the group algebra and let F(G) denote the algebra of functions on G with pointwise product (that is F(G) has a basis of elements of the form and is 0 unless in which case it is ). Recall that for both C[G] and F(G) there is a notion of tensor product for modules (for the former g acts on a tensor product via , whereas for the latter acts on a tensor product via

We define a bioidal category called the “group Subfactor” as follows

• The A-A objects are C[G]-modules
• The A-B objects are vector spaces (thought of as representations of the trivial group)
• The B-A objects are vector spaces (thought of as representations of the trivial group)
• The B-B objects are F(G)-modules

(more…)

Isabel Lugo had a nice post recently on advice for prospective graduate students. Although not all of her advice rang true to me (the first year of graduate school was pretty fun for most people I know), she makes an excellent point in comment 2 that I wanted to respond to.

Your mathematical interests will change during the first year in graduate school, because a lot of subjects “feel” different at the undergraduate level than at the graduate level, and there are some things you just don’t see as an undergraduate at all.

Personally, my favorite subjects in mathematics are:

1. The representation theory of finite groups from Frobenius through Brauer
2. Algebraic and Analytic Number Theory from Gauss through sometime in the late 1800s
3. Quantum Topology from the Jones polynomial through the present

On the one hand you can tell from this that I like algebra more than I like geometry or analysis. This was something I was quite aware of as an undergraduate and beginning graduate student. However, all of these have something else important in common: they are/were all young subjects. With the exception of Euler’s prescient work on zeta functions, there’s not a whole lot of precursors prior to the beginnings I’ve stated above. You don’t see on my list anything like modern homotopy theory, ell-adic cohomology, the classification of simple groups project, or 20th century number theory. I went into graduate school thinking I wanted to do number theory because I love number theory up until about 1920. The lesson I should have taken from that is that I like younger topics in algebra, not that I want to do number theory. Older subjects feel different from younger subjects. You have more tools, bigger machines, but the most natural questions have already been answered and the field has moved on to harder things. But there was no way for me to know this as an undergraduate, because undergraduates don’t know enough to have been exposed to any material in an older subject.

There are lots of other key differences between fields: Do people do more theory building or problem solving? Is the topic dominated by a few giants’ research programs with other people following their lead or is it more each researcher having their own smaller programs? Is it on the intersection of multiple fields or more isolated from other fields of mathematics?

Obviously a first-year graduate student isn’t going to know the answers to these questions, but these are the sorts of questions that you need to ask yourself rather than just “which subject did I like as an undergraduate.”

So, I felt I should try to explain a bit about this hypertoric stuff. I think the first order of business should be explaining what Gale duality is. This is all “just” linear algebra, in principle very simple, though not exactly the usual perspective on these things. To slightly paraphrase some notes from Daniel Kleitman’s website:

The act of writing down explicit dual equations of a given hyperplane arrangement is not complicated but it is quite unnatural to human beings. You can expect to do it hesitantly and incorrectly the first three times you try to do it.

(more…)