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In the last two years, I ran a “mini-polymath” project to solve one of the problems of that year’s International Mathematical Olympiad (IMO).  This year, the IMO is being held in the Netherlands, with the problems being released on July 18 and 19, and I am planning to once again select a question (most likely the last question Q6, but I’ll exercise my discretion on which problem to select once I see all of them).

The format of the last year’s mini-polymath project seemed to work well, so I am inclined to simply repeat that format without much modification this time around, in order to collect a consistent set of data about these projects.  Thus, unles the plan changes, the project will start at a pre-arranged time and date, with plenty of advance notice, and be run simultaneously on three different sites: a “research thread” over at the polymath blog for the problem solving process, a “discussion thread” over at this blog for any meta-discussion about the project, and a wiki page at the polymath wiki to record the progress already made at the research thread.  (Incidentally, there is a current discussion at the wiki about the logo for that site; please feel free to chip in your opinion on the various proposed icons.)  The project will follow the usual polymath rules (as summarised for instance in the 2010 mini-polymath thread).

There are some kinks with our format that still need to be worked out, unfortunately; the two main ones that keep recurring in previous feedback are (a) there is no way to edit or preview comments without the intervention of one of the blog maintainers, and (b) even with comment threading, it is difficult to keep track of all the multiple discussions going on at once.  It is conceivable that we could use a different forum than the WordPress-based blogs we have been using for previous projects for this mini-polymath to experiment with other software that may help ameliorate (a) and (b) (though any alternative site should definitely have the ability to support some sort of TeX, and should be easily accessible by polymath participants, without the need for a cumbersome registration process); if there are any suggestions for such alternatives, I would be happy to hear about them in the comments to this post.  (Of course, any other comments germane to the polymath or mini-polymath projects would also be appropriate for the comment thread.)

The other thing to do at this early stage is set up a poll for the start time for the project (and also to gauge interest in participation).  For ease of comparison I am going to use the same four-hour time slots as for the 2010 poll.  All times are in Coordinated Universal Time (UTC), which is essentially the same as GMT; conversions between UTC and local time zones can for instance be found on this web site.   For instance, the Netherlands are at UTC+2, and so July 19 4m UTC (say) would be July 19 6pm in Netherlands local time.  (I myself will be at UTC-7.)

In a few weeks (and more precisely, starting Friday, September 24), I will begin teaching Math 245A, which is an introductory first year graduate course in real analysis. (A few years ago, I taught the followup courses to this course, 245B and 245C.)  The material will focus primarily on the foundations of measure theory and integration theory, which are used throughout analysis.  In particular, we will cover

1. Abstract theory of $\sigma$-algebras, measure spaces, measures, and integrals;
2. Construction of Lebesgue measure and the Lebesgue integral, and connections with the classical Riemann integral;
3. The fundamental convergence theorems of the Lebesgue integral (which are a large part of the reason why we bother moving from the Riemann integral to the Lebesgue integral in the first place): Fatou’s lemma, monotone convergence theorem, and the dominated convergence theorem;
4. Product measures and the Fubini-Tonelli theorem;
5. The Lebesgue differentiation theoremabsolute continuity, and the fundamental theorem of calculus for the Lebesgue integral.  (The closely related topic of the Lebesgue-Radon-Nikodym theorem is likely to be deferred to the next quarter.)

See also this preliminary 245B post for a summary of the material to be covered in 245A.

Some of this material will overlap with that seen in an advanced undergraduate real analysis class, and indeed we will be revisiting some of this undergraduate material in this class.  However, the emphasis in this graduate-level class will not only be on the rigorous proofs and on the mathematical intuition, but also on the bigger picture.  For instance, measure theory is not only a suitable foundation for rigorously quantifying concepts such as the area of a two-dimensional body, or the volume of a three-dimensional one, but also for defining the probability of an event, or the portion of a manifold (or even a fractal) that is occupied by a subset, the amount of mass contained inside a domain, and so forth.  Also, there will be more emphasis on the subtleties involved when dealing with such objects as unbounded sets or functions, discontinuities, or sequences of functions that converge in one sense but not another.  Being able to handle these sorts of subtleties correctly is important in many applications of analysis, for instance to partial differential equations in which the functions one is working with are not always a priori guaranteed to be “nice”.

I’ll be taking a break from blogging and other work for a few weeks.

I’ve just opened the Mini-polymath2 project over at the polymath blog.  I decided to use Q5 from the 2010 IMO in the end, rather than Q6, as it seems to be a little bit more challenging and interesting.

This post will serve as the discussion thread of the project, intended to focus all the non-research aspects of the project such as organisational matters or commentary on the progress of the project.     (Is it possible for one blog thread to “live-blog” another?) The third component of the project is the wiki page, which is intended to summarise the progress made so far on the problem.

As with mini-polymath1, I myself will be serving primarily as a moderator, and hope other participants will take the lead in the research and in keeping the wiki up-to-date.   (Ideally, once we get all the kinks in the format ironed out, it should be possible for anyone with a blog and an interested pool of participants to start their own mini-polymath project, without being overly dependent on one key contributor.)

In view of feedback, I have decided to start the mini-polymath2 project at 16:00 July 8 UTC, at which time I will select one of the 2010 IMO questions to solve.  The actual project will be run on the polymath blog; this blog will host the discussion threads and post-experiment analysis.

It’s been a while since I’ve added to my career advice and writing pages on this blog, but I recently took the time to write up another such page on a topic I had not previously covered, entitled “Write in your own voice“.  The main point here is that while every piece of mathematical research inevitably builds upon the previous literature, one should not mimic the style and text of that literature slavishly, but instead develop one’s own individual style, while also updating and adapting the results and insights of previous authors.

Starting on Monday, March 29, I will begin my graduate class for the winter quarter, entitled “Higher order Fourier analysis“.  While classical Fourier analysis is concerned with correlations with linear phases such as $x \mapsto e(\alpha x)$ (where $e(x) := e^{2\pi i x}$), quadratic and higher order Fourier analysis is concerned with quadratic and higher order phases such as $x \mapsto e(\alpha x^2)$, $x \mapsto e(\alpha x^3)$, etc.

In recent years, it has become clear that certain problems in additive combinatorics are naturally associated with a certain order of Fourier analysis.  For instance, problems involving arithmetic progressions of length three are connected with classical Fourier analysis; problems involving progressions of length four are connected with quadratic Fourier analysis; problems involving progressions of length five are connected with cubic Fourier analysis; and so forth.  The reasons for this will be discussed later in the course, but we will just give one indication of the connection here: linear phases $x \mapsto e(\alpha x)$ and arithmetic progressions $n, n+r, n+2r$ of length three are connected by the identity

$e(\alpha n) e(\alpha(n+r))^{-2} e(\alpha(n+2r)) = 1,$

while quadratic phases $x \mapsto e(\alpha x^2)$ and arithmetic progressions $n, n+r, n+2r, n+3r$ of length four are connected by the identity

$e(\alpha n^2) e(\alpha(n+r)^2)^{-3} e(\alpha(n+2r)^2)^3 e(\alpha(n+3r)^2)^{-1} = 1,$

and so forth.

It turns out that in order to get a complete theory of higher order Fourier analysis, the simple polynomial phases of the type given above do not suffice.  One must also consider more exotic objects such as locally polynomial phases, bracket polynomial phases (such as $n \mapsto e( \lfloor \alpha n \rfloor \beta n )$, and/or nilsequences (sequences arising from an orbit in a nilmanifold $G/\Gamma$).  These (closely related) families of objects will be introduced later in the course.

Classical Fourier analysis revolves around the Fourier transform and the inversion formula.  Unfortunately, we have not yet been able to locate similar identities in the higher order setting, but one can establish weaker results, such as higher order structure theorems and arithmetic regularity lemmas, which are sufficient for many purposes, such as proving Szemeredi’s theorem on arithmetic progressions, or my theorem with Ben Green that the primes contain arbitrarily long arithmetic progressions.  These results are powered by the inverse conjecture for the Gowers norms, which is now extremely close to being fully resolved.

Our focus here will primarily be on the finitary approach to the subject, but there is also an important infinitary aspect to the theory, originally coming from ergodic theory but more recently from nonstandard analysis (or more precisely, ultralimit analysis) as well; we will touch upon these perspectives in the course, though they will not be the primary focus.  If time permits, we will also present the number-theoretic applications of this machinery to counting arithmetic progressions and other linear patterns in the primes.

As an experiment, I’ve recently started using Google Buzz as an outlet for various things I wanted to say or share, but which were too insubstantial to merit a mention on this blog. (In turn, one of the reasons of starting this blog was to share various bits of mathematics which were too insubstantial for a published paper.  Presumably the process becomes degenerate if iterated any further…)    I don’t know how frequently I will be updating, though.

In view of the sustained interest in new polymath projects, Tim Gowers, Gil Kalai, Michael Nielsen, and I have set up a new blog to propose, plan, and run these projects.  This blog is not intended to hold a “monopoly” on the polymath enterprise, but to merely be a convenient central location for discussing and running such projects should one choose.

We have started the ball rolling on this blog with some proposed rules for running a polymath, a mock-up of what a research thread and a discussion thread for a project would look like, two new proposals for the next polymath project (deterministic location of primes, and a problem of Michael Boshernitzan), and a thread on how one should select the next project (which we intend to do in a few months, with a tentative plan to actually start the next project at around October or so).  Please come give the blog a visit, and contribute your thoughts and suggestions, though it should be noted that we are not planning to start a new polymath project right away, but merely to plan for the next one for now.

[Update, July 28: Actually, this may change.  There has already been enough progress on the "deterministic location of primes" project that a discussion thread and wiki page has been created, and this polymath project may in fact formally launch much sooner than anticipated, perhaps in a matter of weeks.  However, much work still needs to be done in laying the groundwork of this project, in particular developing preparatory materials in the wiki and elsewhere to allow participants to get up to speed.]

On a somewhat related note, now that we have a dedicated blog for these sorts of polymath projects, I am thinking of revisiting the ratings system for comments that I recently turned on here.   I guess this would be a good question to poll the readers on:

Note that the ratings system is somewhat customisable: see this wordpress page for details.

The mini-polymath project to find solutions to Problem 6 of the 2009 IMO is still ongoing, but I thought that, while the memories of the experience are still fresh, it would be a good time to open a parallel thread to collect the impressions that participants and observers had of how the project was conducted, how successful it was, and how it (or future projects) could be made to run more smoothly.

Just to get the ball rolling, here are some impressions I got as a (rather passive) moderator:

1. There is no shortage of potential interest in polymath projects. I was impressed by how the project could round up a dozen interested and qualified participants in a matter of hours; this is one particular strength of the polymath paradigm.  Of course, it helped that this particular project was elementary, and was guaranteed to have an elementary (and relatively short) solution.  Nevertheless, the availability of volunteers does bode well for future projects of this type.
2. A wiki needs to be set up as soon as possible. The wiki for polymath1 was an enormously valuable resource, once it was set up.  I had naively thought that the mini-polymath1 project would be short enough that a wiki was not necessary, but now I see that it would have come in handy for organising and storing the arguments, strategies, insights, and ideas that arose through the linear blog thread format, but which was difficult to summarise in that format.  (I have belatedly set a wiki for this project up here.)  For the next polymath project (I have none planned yet, but can imagine that one would eventually arise), I will try to ensure a wiki is available early on.
3. There is an increasing temptation to work offline as the project develops. In the rules of the polymath projects to date, the idea is for participants to avoid working “offline” for too long, instead reporting all partial progress and thoughts on the blog and/or the wiki as it occurs.  This ideal seems to be adhered to well in the first phases of the project, when the “easy” but essential observations are being made, and the various “low-hanging fruits” are harvested, but at some point it seems that one needs to do more non-trivial amounts of computation and thought, which is still much easier to do offline than online.  It is possible that future technological advances (e.g. the concurrent editing capabilities of platforms such as Google Wave) may change this, though; also a culture and etiquette of collaborative thinking might also evolve over time, much as how mathematical research has already adapted to happily absorb new modes of communication, such as email.  In the meantime, though, I think one has accommodate both online and offline modes of thinking to make a polymath project as successful as possible, avoiding degeneration into a mass of low-quality observations on one hand, and a fracturing into isolated research efforts on the other.
4. Without leadership or organisation, the big picture can be obscured by chaos. As I was distracted by other tasks (for instance, flying from Bremen back to Los Angeles), and had already known of a solution to the problem, I adopted a laissez faire attitude to task of moderating the project.  This worked to some extent, and there was certainly no shortage of ideas being tossed back and forth, arguments being checked and repaired, etc., but I think that with more active moderation, one could have had a bit more focus on longer-term strategy and vision than there was.  Perhaps in future projects one could be more explicit in the rules about encouraging this sort of perspective (for instance, in encouraging periodic summaries of the situation either on the blog or on the wiki).
5. Polymath projects tend to generate multiple solutions to a problem, rather than a single solution. A single researcher will tend to focus on only one idea at a time, and is thus generally led to just a single solution (if that idea ends up being successful); but a polymath project is more capable of pursuing several independent lines of attack simultaneously, and so often when the breakthrough comes, one gets multiple solutions as a result.  This makes it harder to do direct comparison of success between polymath projects and individual efforts; from the (limited) data points available, I tentatively hypothesise that polymath projects tend to be slower, but obtain broader and deeper results, than what a dedicated individual effort would accomplish.
6. Polymath progress is both very fast and very slow. I’ve noticed something paradoxical about these projects.  On the one hand, progress can be very fast in the sense that ideas get tossed out there at a rapid rate; also, with all the proofreaders, errors in arguments get picked up much quicker than when only one mathematician is involved.  On the other hand, it can take a while for an idea or insight obtained by one participant to be fully absorbed by the others, and sometimes the key observation can be drowned out by a large number of less important observations.  The process seems somewhat analogous to that of evolution and natural selection in biology; consider for instance how the meme of “try using induction”, which was the ultimately successful approach, had to first fight among competing memes such as “try using contradiction”, “try counting arguments”, “try topological arguments on the cube”, etc., before gaining widespread acceptance.  In contrast, an individual might through luck (or experience) hit upon the right approach (in this case, induction) very early on and end up with a solution far quicker than a polymath effort; conversely, he or she may select the wrong approach and end up wasting far more time than a polymath would.
7. The wordpress blog format is adequate, but far from ideal. Technical problems (most notably, the spam filter, the inability to preview or edit comments [except by myself], and the (temporary) lack of nesting and automatic comment numbering) made things more frustrating and clunky than they should be.  Adding the wiki helps some of the problems, but definitely not all, especially since there is no integration between the blog and the wiki.  But the LaTeX support included in the WordPress blog is valuable, even if it does act up sometimes. Hopefully future technologies will provide better platforms for this sort of thing.  (As a temporary fix, one might set up some dedicated blog (or other forum) for polymath projects with customised code, rather than relying on hosts.)
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