Following Emily’s advice, I recently signed up to be mentor in the AWM Mentor Network. It’s been pretty good thus far (I recommend it to any of you who would like to do some menting), but I got a request from my mentee that I thought some of our audience might have better ideas about than me.

What math books would you suggest for relatively casual summer reading for an undergraduate math major finishing their third year? This is not the sort of thing I think about a lot, but I know a reasonable number of readers have a lot more experience with young mathematicians than I do.

The past few weeks there has been a summer school and conference on geometric representation theory and extended affine Lie algebras at University of Ottawa. As part of this event, I gave a week long lecture series entitled “three geometric constructions of the irreducible representations of “. Specifically I discussed the Borel-Weil theorem, Ginzburg’s construction using Springer fibres, and the geometric Satake correspondence. I focused on to keep the root system combinatorics and the geometry as elementary as possible.

The typed lecture notes from my talk are now available. If you do read them, please let me know if you have any comments/corrections. (You can also find videos of the talks.)

The other lectures at the summer school were given by Neher, Kang, Wang, Savage, and Chari. I recommend reading their notes/watching their videos if you want to learn more about geometric representation theory, crystals, and affine Lie algebras.

I recently read a charming little paper: Quasi-elliptic integrals and periodic continued fractions, by van der Poorten and Tran. Most of us who have taken a number theory course of some kind learned how to solve Pell’s equation: where is a nonsquare positive integer. The usual method is to compute the continued fraction
.
One then defines the convergents of by


etcetera.

Then tends to be very small and, if you compute long enough, for some you will have .

What van der Poorten and Tran do is to ask what happens if is not an integer, but a polynomial . Before I get into details, I want to tell you about something gorgeous that I won’t explain at all. Using the methods in their paper, van der Poorten and Trap can discover identities like

Isn’t that pretty?

It turns out that the continued fraction algorithm for is actually much prettier than for integers. Everything should be understood in terms of the curve cut out by . This is a curve of genus , with two points at infinity. (One of these points is the limit of and the other is the limit of .) I’ll call these two points and . The theory is controlled by the line bundles . In particular, there are nontrivial solutions to if and only if the continued fraction is periodic, if and only if for some .

Below the fold, I’ll explain what is meant by the continued fraction algorithm for an algebraic function, and tell you some of the other nice results from the paper.

Given any power series in , we define the continued fraction of .

Define . Set and define by . Then set and . Continuing in this way, we get a sequence of polynomials, a sequence of power series, and a continued fraction
.
We can also define the convergents as before; they do converge to in the sense that each ratio agrees with to a higher order than the ratio does.

In particular, suppose that is a polynomial of the form .
Then is a power series in :

So we can define the continued fraction of .
We keep the notations , , and from above.

I’ll explain just one key idea from the paper. Let’s think about the zeroes and poles of . Since is a polynomial in , its only poles are at , and it has a pole of the same order at both ’s. So, other than , the function has the same poles as . Then has zeroes at the poles of .

That’s what happens away from . Suppose that has a pole of order at , and a zero of order at . Then has a pole of order at both ’s. The difference has a zero of order at and a pole of order at . So has a pole of order at and a zero of order at .

Summing up the last two paragraphs, let the poles of be and let the zeroes be . Then the poles of are , for some and some and the zeroes are . (Here , and are supported away from .) In other words, there is a sequence of positive integers and a sequence of divisors such that the poles of are while the zeroes are .

Note that is the degree of . Note also that in the Picard group, so
.

It’s not too hard to work out what happens if the coefficients of are chosen generically. The first has degree and all the other are . A bit of effort checks that has degree (exercise!), so all of have degree and, in fact, in the Picard group. You may remember that a generic divisor of degree has a unique effective representative in PIcard; is that unique representative. So, we have just found an explicit way to write down an arithmetic progression in , where is a hyperelliptic curve.

Of course, the fun comes in the nongeneric case. In that case, the can skip around. It’s really fun when is torsion in the Picard group or, in other words, when there is a unit in the coordinate ring of . Then, eventually, the sequence in Picard will repeat. It turns out, when this happens, the corresponding approximation gives your unit!

There are plenty of other ideas in the paper. What is the analogue of the result that the are the best approximations to ? The are all of the form : how do we relate the polynomials and to the divisors ? And how did I come up with that integral above? All this and more, so read the paper!

I’ve just uploaded a paper to the arXiv, Man and machine thinking about the smooth 4-dimensional Poincaré conjecture, joint with Michael Freedman, Robert Gompf, and Kevin Walker.

The smooth 4-dimensional Poincaré conjecture (SPC4) is the “last man standing in geometric topology”: the last open problem immediately recognizable to a topologist from the 1950s. It says, of course:

A smooth four dimensional manifold homeomorphic to the 4-sphere is actually diffeomorphic to it, .

We try to have it both ways in this paper, hoping to both prove and disprove the conjecture! Unsuprisingly we’re not particularly successful in either direction, but we think there are some interesting things to say regardless. When I say we “hope to prove the conjecture”, really I mean that we suggest a conjecture equivalent to SPC4, but perhaps friendlier looking to 3-manifold topologists. When I say we “hope to disprove the conjecture”, really I mean that we explain an potential computable obstruction, which might suffice to establish a counterexample. We also get to draw some amazingly complicated links:

Our new formulation of SPC4 might be thought of as a generalization of Gabai’s Property R [euclid.jdg/1214441488, Corollary 8.3]:

If surgery on gives then is the unknot.

We point out that SPC4 is equivalent to the following (Conjecture 3.3 in the paper)

Let be a link in with a dotted p-component unlink and a framed link of p+q components. Suppose that normally generates , and that surgery on (with dotted components 0-framed) is diffeomorphic to . Then there is a sequence of moves

transforming to the empty diagram.

(you’ll have to read the paper if you want to know how “dotted” and “framed” come into the picture; essentially there are restrictions on move (3) depending on this data, which you’ll already know about if you understand link presentations for 4-manifolds) This is a generalization in the sense that Property R says that for p = 0 and q = 1, a single move suffices. The proof that this is equivalent to SPC4 is essentially trivial, and makes use of the Kirby calculus theorem. We discuss some variations of this conjecture, both weaker and stronger, that might be more approachable.

In the later half of the paper, we discuss some potential counterexamples to SPC4, the Cappell-Shaneson spheres, and explain how Rasmussen’s s-invariant (related to Khovanov homology) might give a computable obstruction preventing these homotopy spheres from being diffeomorphic to the standard sphere.

The basic idea is quite simple. The Cappell-Shaneson spheres have a handle presentation, due to Bob Gompf, that have no 3-handles.

Pull off the top 4-handle, and think of this handle presentation as a presentation of boundary , which is certainly standard. Inside this , we can see a certain link, the meridian links of the loops along which we glue the 2-handles. This link is obviously slice in the homotopy sphere (because it’s just the meridian links, we can see the slice disks immediately!), but once we do the long sequence of Kirby calculus moves converting this presentation of into the standard (empty) presentation, this link has become tremendously complicated (see the diagram near the top of the post!) and it’s not at all obvious that it’s slice in $B^4$. Moreover, if it isn’t, then our homotopy ball can’t have been diffeomorphic to the standard ball!

Our problem now is to find computable obstructions to sliceness. Until recently, not many were available, and those that were (for example the invariant from knot Floer homology) were suspected to be unable to detect slice genus in homotopy balls. Rasmussen’s s-invariant, defined using a deformation of Khovanov homology, fits our requirements however! Unfortunately, computing the s-invariant, although in principal combinatorial, suffers from terrible scaling as the complexity of the link (especially its girth) increases. It seems that of the many examples coming from different Cappell-Shaneson spheres, very few cases are computable with modern hardware and the current best algorithms for computing s, and even then we need to cheat a little, and look only at various knots obtained by band connect summing the link components together. We’ve done two cases (the and Cappell-Shaneson spheres; the is known to be standard), and got both times — after more than a week of computing time on a huge machine!

If anyone out there thinks up a new way to compute the s-invariant, and can prove (perhaps via a gauge theoretic interpretation of Khovanov homology?) that the slice genus bound from the s-invariant can’t tell the difference between homotopy spheres, please let us know!

The European Math Society Publishing House (a non-profit publishing company which also publishes the Journal of the EMS, CMH, and half a dozen other journals) just announced a new journal: Quantum Topology. I think this is very exciting as it fills a nice hole in the existing journal options. The list of main topics include knot polynomials, TQFT, fusion categories, categorification, and subfactors. So there should be lots of material of interest to people here.

Early in my graduate student career, I was told by several people that I should go to conferences and talk to professors. If you work in mathematics, you’ve probably heard this piece of advice before, and it’s hard to see how you could damage your career by following it (given reasonable assumptions on your behavior). I encountered two problems:

1. What sort of talking am I supposed to do with a professor if I don’t know anything?
2. How do I make my way into one of those small circles of people that inevitably form between talks?

I’ve heard that some advisors actually go to conferences with their students and introduce them to colleagues, and this pretty much solves both problems, but I’d like to focus on the case that this doesn’t happen, since I imagine it will be the norm for a while. This isn’t meant to be a definitive guide, and I’d really appreciate further suggestions and anecdotes.

My basic solution to the first problem was to talk about something other than math, e.g., after some introductory banter ask for funny stories about other mathematicians, or solicit general grad school advice. Unfortunately, this didn’t help with my presumed goal, which was to communicate an impression like, “Scott is a promising student in this field,” so I still felt pressure to perform in a way that was, in retrospect, impossible. I’ve talked to some other mathematicians about this, and I’ve concluded that a pure career focus can be an unhealthy mindset when attending a conference. If you don’t feel comfortable talking to professors, then I don’t think there’s a problem with approaching grad students and just hanging out (you can pick out the grad students, because they’re the ones that look afraid to talk to professors). If you have a good time, then maybe you’ll form a positive mental association to conferences, and if you make friends with grad students in your field, then later conferences will be like reunions. As you get older, some of the grad students might even become professors.

The second problem was not a big deal when I was friends with someone at the conference, since I felt more comfortable barging into a conversation, and I’d often find a circle forming around me. Therefore, one solution is prevent the problem in the first place by getting other people from your school to go to the same conference. This is much easier if you go to a big research school – in some number theory conferences, I felt like I already knew half of the people there, because a large fraction of them had overlapped with me at Berkeley, or given seminars-with-dinner at some time. However, I’ve been in situations where I didn’t know anyone, and without a social group at a conference, the experience could be rather awkward and lonely. I think one way to remedy this is to seek out “conference friends” quickly, before people get into a routine. A conference friend doesn’t need have a whole lot in common with you, but you shouldn’t be afraid of each other, so you should look for people who seem relatively approachable and willing to talk to you. One tactic I’ve heard is to find people who dress like you, e.g., at a similar level of formality. If the conference has a breakfast spread, this is a good time to find a circle, since the groups are often smaller, and the people often aren’t alert enough to engage in intimidating mathematics. I’m told there are some artificial ways of starting a conversation, like awkwardly commenting about the food, then transitioning to an introduction. I’m afraid I haven’t mastered this art yet.

I think I should add a few more tips on talking to professors. If you ask them about their work, you shouldn’t feel bummed if you don’t understand an explanation. In particular, don’t worry too much about looking stupid for interrupting to ask an elementary question. If you wear a name tag, you can avoid the situations where people recognize you but don’t remember your name while they feel like they should. A name tag also provides another opening for conversation about your school or the town it’s in. (If you happen to be a super-famous professor, your name tag might frighten people, so you may want to leave it off.) You should go to informal events, like meals and bars, even if you’re really tired, because that’s where a lot memorable events happen. If you’re far enough along your grad career to have a problem you can talk about, the rules change a bit. You can say things like, “I like your work on xyz. Can I talk to you about my work?” This gives you an opportunity to feel kind of smart, because even experts will probably take some time to process what you say. In this case it really helps to have a big picture view of your project, so you can describe why one would want to do what you’re doing. This is good for multiple reasons, as it is often the answer to the first question people ask, failure to give a clear explanation can be a conversation-killer, and having a coherent view of a research program can help you find your way through your dissertation. If you don’t understand your project well enough to say how it fits into the rest of mathematics, you should probably bug your advisor a bit for answers.

Finally, I’d like to say something about social rules that can make conferences more welcoming places for grad students. Conferences can be punishing experiences for everybody, but I think a little effort can go a long way. The first thing is that informal conversation circles shouldn’t be completely closed off. I think people should be welcomed into a conversation, and even so, I’ve unintentionally engendered cliquish atmospheres in the past. If you’re going out for lunch or dinner, it’s nice to invite people along, even if they’re annoying, and if you end up with a big train of awkward grad students, it’s not the end of the world. I want to emphasize that ditching people is pretty bad form (Prof. H, I haven’t forgotten after all these years…), and so is ignoring students from less famous schools. It’s not necessary to talk to every grad student for a half hour, but making some effort to include everyone in a conversation helps to spread the happiness around. When talking to grad students about their projects, it is often nice to ask if their work is influenced by their advisors’ work. They tend to be relatively familiar with that, and it can be a basis for further conversation.

Konni Rietsch writes to me

The Department of Mathematics at King’s College London is advertising a permanent Lectureship in the area of geometry with application deadline the 27th of July.  

Further information can be found at:

http://www.kcl.ac.uk/depsta/pertra/vacancy/external/pers_detail.php?jobindex=7951

King’s College London, part of the University of London, is based in central London with its mathematics department located in the Strand campus, next to the river Thames and surrounded by theatres and art galleries.

More information about King’s College London can be found on http://www.kcl.ac.uk/about/

Information on the department and its research groups can be found on http://www.kcl.ac.uk/schools/pse/maths/

For those not familiar with British academic ranks, my understanding is that a permanent lectureship is the first stage leading to a professorship; and that the term “professor” at a British University is far more prestigious than at an American one.

I haven’t been to King’s College myself, but I am a frequent user and admirer of Konni’s work on total positivity. Looking through their department, I see a lot of a lot of geometry and representation theory, including classical Lie Theory, mapping class groups, Langlands, quantum groups and higher category theory. I also see a lot of number theory/arithmetic geometry, including Diamond, and a lot of mathematical physics. If you like the Secret Blogging Seminar, you might fit in very well there.

Welcome to the next installation of my series on local systems. In this post I’ll be talking about connections. This post should require less sophistication than the last few — no schemes, no functors — I’ll almost be coming at the subject afresh. There will be another post later, explaining how you might get to connections if you started out thinking about the infinitesimal site.

To start out with, let’s talk about derivatives; ordinary, single variable calculus derivatives. We have a function of a variable . Then the derivative of is the function . There are two directions in which we might want to generalize this idea. The first is to work with functions on a manifold, on a space which has no inherent coordinate system. This is the subject of your standard Calculus on Manifolds course, and I am going to assume that my readers are at least vaguely familiar with it. The second is to work, not with functions, but with sections of vector bundles. That’s our subject in this post.

So, let’s think about a vector bundle on the line , and let be a section of . If we want to define , we need to subtract and , two vectors which live in different fibers. To think of it another way, we need to distinguish between , the point in the fiber over , and , the constant function which assigns the same value at every point. Suppose that, for any , we had a local section of with ; we think of as a constant function. Then we could define .

A local system gives us the constant functions . (Indeed, in definitions A.2 and B.6, we took a local system to be the constant functions, along with the data of certain maps between them.) Today, we will take the fundamental object to be the operation of derivation, and see how to build everything else from it.

We start with a special case, and then build up to the whole. Let be a vector bundle on . We’ll write for the coordinate on . Of course, can be trivialized, but for the moment we don’t want to trivialize it. A connection on is a map from sections of to sections of , such that

(1) For any two sections and of , we have .

(2) For any section of , and any scalar-valued function on , we have .

If we do trivialize , then the operation of taking the derivative with respect to obeys these axioms. The point is that these are axioms which hold without talking about any choice of a trivialization. If you’re an algebraist, the following might help you: a derivation is a map from an algebra (to something); a connection is a map from a module.

A section of is locally constant if . So this is how to go from connections to the more sheafy perspectives which are focused on locally constant sections.

I said early on that a local system is a vector bundle with isomorphisms between different fibers. How, in the setting of connections, do we build an isomorphism between one fiber of and another? Let and be two points of and let be a point in the fiber . Solve the differential equation , with initial condition . The isomorphism between and will send to . So, this is how to go from a connection to the path groupoid approach, when our as space is . Later, when we work on more complicated spaces, we’ll have to keep track of the path along which we solve the differential equation, but everything else will look the same.

Let’s see this differential equation in coordinates. Choosing an arbitrary trivialization of in order to write things down, we have where is an matrix, varying with . (Exercise!) So we have to solve the differential equation . If , this has the solution ; for larger , one usually cannot give a closed form solution.

I have now presented all the main ideas, in the case where our space is the real line. We will now move to the case of an arbitrary manifold . This will introduce two difficulties: we won’t have a natural coordinate system on , and there is a genuinely new phenomenon that happens when has dimension larger than one — the possibility of curvature.

Let’s start by addressing the lack of coordinates; this will just be a matter of careful notation. Let be a vector field on ; we will think of this as a derivation on the scalar-valued functions on . Then we want a way, , of differentiating with respect to . This should obey

(1′) , where and are real constants and

(2′) .

We also need a condition on how this depends on :

(0) , where and are scalar valued functions.

You now have reached the definition of a connection. I always found it hard to remember the difference between (1′) and (0).
Why are the coefficients and just constants, while and get to be functions? Unfortunately, wordpress won’t let me do a hidden section inside a hidden section, so click on this asterisk to read the explanation*.

We won’t be interested in all connections, we will only be interested in the integrable ones. (Also known as flat, or zero-curvature.) This will be easiest to explain in coordinates. Let , , …, be coordinates on near . Then we expect to equal . After all, if we were doing honest differentiation of functions, we would have .
A connection is said to be integrable if we have for some (equivalently any) set of coordinates. An equivalent condition is that for any vector fields and .

It turns out that, if is an integrable connection, then the differential equation is uniquely solvable for any initial condition . These solutions give a local trivialization of our vector bundle, just like in the one dimensional case we discussed above.

If is not integrable, then we can still solve $\latex \nabla (\sigma)=0$ along any path. But making even topologically trivial changes to the path may change the result. In other words, you get holonomy, not just monodromy.

Next up: some other ways to think about integrability, such as -modules, curvature and the deRham complex. Then, the relationship between connections and the infinitesimal perspective on local systems.

* Suppose we have a vector field , and a section , and we want to compute the value of at a point . Then it is enough to know at . However it is not enough to know at ; we have to know the first order variation of .

This is part of a more general distinction that everyone should learn at some point. Suppose that and are two vector bundles on , and and are the sheaves of sections of . Suppose we have a map . If for and constants, then is called a “map of sheaves” and the stalk of will depend on the stalk of . But if , then is called a “map of -modules”, and the fiber of will depend only on the fiber of . In other words, will be induced by a map of vector bundles . So $\nabla$ has the first kind of linearity in and the second (more local) kind in .

As another application of the above ideas, if and are two connection on the same vector bundle , then is a map of -modules from to itself. Hey, I just solved on of your exercises for you! (New exercise: which one!)

I’m obsessing over the aftermath of the Iranian election today, and thought our readership might be interested in this analysis by Walter Mebane, from UMich, of Iranian election results.

He’s taken the Ministry of Interior’s posted results (or a copy in Google docs), and done a quick check against Benford’s law, which predicts the statistical distribution of initial digits of numbers. With the available data (returns for ~350 districts), he reports that nothing looks particularly wrong, but would love to have more detailed (polling station level) data, as you shouldn’t expect to see much anyway.

Nate Silver also has a critique up of some earlier statistical complaints made about the announced election results. If anyone sees something else along this lines, I’d love to hear.

Whenever I’m finishing off a paper, at some point I have to sit down and clean up all the references, which generally look something like \cite{Popa?} or \cite{that paper by Marco and co}. Wouldn’t it be nice if someone else could do the rest?

If you don’t already know about it, one great resource is mathscinet, which will produce nicely formatted BIBTEX entries for you (example). If you want to be even more efficient, you can wander around mathscinet, saving articles to your “clipboard”, and then ask mathscinet to give you the BIBTEX entries for everything at once. (After you have articles on the clipboard, follow the “clipboard” link in the top right of the page, then select BIBTEX from the drop-down box and click “SaveClip”.)

If you’re even lazier, you could use the two command-line scripts that I use (download find-missing-bibitems and get-mathscinet-bibtex and put them on your path; you’ll need linux/OSX/cygwin to run). Now, when you cite items in LaTeX, cite them via their mathscinet identifiers, e.g. \cite{MR1278111} instead of \cite{Popa?}. Now, if you usually type latex article to compile, and bibtex article to generate the bibliography, you can also type find-missing-bibitems article, and all the missing BIBTEX entries will appear! For example, after adding \cite{MR1278111} somewhere in my text, the output of find-missing-bibitems article is

@article {MR1278111,
    AUTHOR = {Popa, Sorin},
     TITLE = {Classification of amenable subfactors of type {II}},
   JOURNAL = {Acta Math.},
  FJOURNAL = {Acta Mathematica},
    VOLUME = {172},
      YEAR = {1994},
    NUMBER = {2},
     PAGES = {163--255},
      ISSN = {0001-5962},
     CODEN = {ACMAA8},
   MRCLASS = {46L37 (46L10 46L40)},
  MRNUMBER = {MR1278111 (95f:46105)},
MRREVIEWER = {V. S. Sunder},
}

If you’re brave, you could run something like

find-missing-bibitems article >> bibliography.bib

to automatically append any missing entries to your BIBTEX file. The really enthusiastic could incorporate this script into the standard latex-latex-bibtex-latex cycle.

Really, I like to have more in my BIBTEX file: I generally use the note field to include a link to the mathscinet review, and a link to the DOI for the paper on the publisher’s webpage. If available, I want a link to the arxiv version of the paper too, for people without institutional access to the published version. Currently, the scripts can’t do this automatically, but it’s might not be much more work. Maybe next time.