The “Hot spots conjecture” proposal has taken off, with 42 comments as of this time of writing. As such, it is time to take the proposal to the next level, by starting a discussion thread (this one) to hold all the meta-mathematical discussion about the proposal (e.g. organisational issues, feedback, etc.), and also starting a wiki page to hold the various facts, strategies, and bibliography around the polymath project (which now is “officially” the Polymath7 project).
I’ve seeded the wiki with the links and references culled from the original discussion, but it was a bit of a rush job and any editing would be greatly appreciated. From past polymath experience, these projects can get difficult to follow from the research threads alone once the discussion takes off, so the wiki becomes a crucial component of the project as it can be used to collate all the progress made so far and make it easier for people to catch up. (If the wiki page gets more complicated, we can start shunting off some stuff into sub-pages, but I think it is at a reasonable size for now.)
One thing I see is that not everybody who has participated knows how to make latex formatting such as appear in their comments. The instructions for that (as well as a “sandbox” to try out the code) are at this link.
Once the research thread gets long enough, we usually start off a new thread (with some summaries of the preceding discussion) to make it easier to keep the discussion at a manageable level of complexity; traditionally we do this at about the 100-comment mark, but of course we can alter this depending on how people are able to keep up with the thread.
Chris Evans has proposed a new polymath project, namely to attack the “Hot Spots conjecture” for acute-angled triangles. The details and motivation of this project can be found at the above link, but this blog post can serve as a place to discuss the problem (and, if the discussion takes off, to start organising a more formal polymath project around it).
This post marks the official opening of the mini-polymath3 project to solve a problem from the 2011 IMO. I have decided to use Q2, in part to see how the polymath format would cope with a more geometrically themed problem.
be a finite set of at least two points in the plane. Assume that no three points of
are collinear. A windmill
is a process that starts with a line
going through a single point
. The line rotates clockwise about the pivot
until the first time that the line meets some other point
. This point
takes over as the new pivot, and the line now rotates clockwise about
, until it next meets a point of
. This process continues indefinitely.
Show that we can choose a point
and a line
such that the resulting windmill uses each point of
as a pivot infinitely many times.
The comments to this post shall serve as the research thread for the project, in which participants are encouraged to post their thoughts and comments on the problem, even if (or especially if) they are only partially conclusive. Participants are also encouraged to visit the discussion thread
for this project, and also to visit and work on the wiki page
to organise the progress made so far.
- All are welcome. Everyone (regardless of mathematical level) is welcome to participate. Even very simple or “obvious” comments, or comments that help clarify a previous observation, can be valuable.
- No spoilers! It is inevitable that solutions to this problem will become available on the internet very shortly. If you are intending to participate in this project, I ask that you refrain from looking up these solutions, and that those of you have already seen a solution to the problem refrain from giving out spoilers, until at least one solution has already been obtained organically from the project.
- Not a race. This is not intended to be a race between individuals; the purpose of the polymath experiment is to solve problems collaboratively rather than individually, by proceeding via a multitude of small observations and steps shared between all participants. If you find yourself tempted to work out the entire problem by yourself in isolation, I would request that you refrain from revealing any solutions you obtain in this manner until after the main project has reached at least one solution on its own.
- Update the wiki. Once the number of comments here becomes too large to easily digest at once, participants are encouraged to work on the wiki page to summarise the progress made so far, to help others get up to speed on the status of the project.
- Metacomments go in the discussion thread. Any non-research discussions regarding the project (e.g. organisational suggestions, or commentary on the current progress) should be made at the discussion thread.
- Be polite and constructive, and make your comments as easy to understand as possible. Bear in mind that the mathematical level and background of participants may vary widely.
Richard Lipton has just proposed on his blog to discuss the following conjecture of Erdos as a polymath project: that there are no natural number solutions to the equation
with . Previous progress on this problem (including, in particular, a proof that any solution to this equation must have an extremely large value of , and specifically that ) can be found here.
In October 2010 there was a discussion about polymath projects at an event organized by the I.A.S in NYC. Tim Gowers described the endeavor and some prospects, and hopes, and Peter Sarnak responded with some concerns. An interesting discussion followed. Some of the discussion is described in the IAS Institute Letter for fall 2010 .
An update on the status of the Polymath4 paper on finding primes. I’ve received a referee report from Mathematics of Computation on the submission, which can be found here. The referee liked the result but wanted a fair number of expository changes before he or she was willing to recommend acceptance, so the editor has asked for a revision. I will be happy to make the relevant changes, but if there are any other changes that other participants would like to make, now would be a good time to suggest them. (The most recent version of the paper can be found at the Subversion repository or at this link; see also the arXiv version.)
One change requested is to add a list of participants to the project. In analogy with what we did for Polymath1, I therefore started a “signup sheet” on the wiki at
for people to self-report their participation, contact information, and grant information for the project. There is the usual problem of trying to decide who is a “main participant” of the project, and who is a “contributor” (though I think I can safely add Ernie, Harald, and myself as participants); as with Polymath1, I will leave it to each of you to self-report what level of participation you feel is appropriate.
I’ve been feeling slightly guilty over the last few days because I’ve been thinking privately about the problem of improving the Roth bounds. However, the kinds of things I was thinking about felt somehow easier to do on my own, and my plan was always to go public if I had any idea that was a recognisable advance on the problem.
I’m sorry to say that the converse is false: I am going public, but as far as I know I haven’t made any sort of advance. Nevertheless, my musings have thrown up some questions that other people might like to comment on or think about.
Two more quick remarks before I get on to any mathematics. The first is that I still think it is important to have as complete a record of our thought processes as is reasonable. So I typed mine into a file as I was having them, and the file is available here to anyone who might be interested. The rest of this post will be a sort of digest of the contents of that file. The second remark is that I am writing this as a post rather than a comment because it feels to me as though it is the beginning of a strand of discussion rather than the continuation of one, though it grows out of some of the comments made on the last post. Note that since we are operating on the Polymath blog, anybody else is free to write a post too (if you are likely to be one of the main contributors, haven’t got moderator status and want it, get in touch and I can organize it).
The starting point for this line of thought is that the main difficulty we face seems to be that Bourgain’s Bohr-sets approach to Roth is in a sense the obvious translation of Meshulam’s argument, but because we have to make a width sacrifice at each iteration it gives a type bound rather than a type bound. Sanders’s argument gives a type bound, but if we use that then it is no longer clear how to import the new ideas of Bateman and Katz. Therefore, peculiar as it might seem to jettison one of the two papers that made this project seem like a good one in the first place, it is surely worth thinking about whether the width sacrifice that Bourgain makes (and that is also made in subsequent refinements of Bourgain’s method, due to Bourgain and Sanders) is fundamentally necessary or merely hard to avoid. (more…)
For the time being this is an almost empty post, the main purpose of which is to provide a space for mathematical comments connected with the project of assessing whether it is possible to use the recent ideas of Sanders and of Bateman and Katz to break the barrier in Roth’s theorem. (In a few hours’ time I plan to write a brief explanation of what one of the main difficulties seems to be.)
Added later. Tom Sanders made the following remarks as a comment. It seems to me to make more sense to have them as a post, since they are a good starting point for a discussion. So I have taken the liberty of upgrading the comment. Thus, the remainder of this post is written by Tom.
This will hopefully be an informal post on one aspect of what we might need to do to translate the Bateman-Katz work into the setting.
One of the first steps in the Bateman-Katz argument is to note that if is a cap-set (meaning it is free of three-term progressions) of density then we can assume that there are no large Fourier coefficients, meaning
They use this to develop structural information about the large spectrum, , which consequently has size between and . This structural information is then carefully analysed in the `beef’ of the paper. (more…)