Polymath10 has started on my blog. The aim is to prove the Erdos-Rado sunflower conjecture (also known as the delta-system conjecture). Here is the wikipage.

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Filed under: polymath5 ]]>

One of the interesting questions regarding the polymath endeavor was:

Can polymath be used to develop a theory/new area?

My idea is to have a project devoted to develop a theory of “convex hulls of real algebraic varieties”. The case where the varieties are simply a finite set of points is a well-developed area of mathematics – the theory of convex polytopes, but the general case was not studied much. I suppose that for such a project the first discussions will be devoted to raise questions/research directions. (And mention some works already done.)

In general (but perhaps more so for an open-ended project), I would like to see also polymath projects which are on longer time scale than existing ones but perhaps less intensive, and that people can “get in” or “spin-off” at will in various times.

The Riemann hypothesis is arguably the most famous open question in mathematics. My view is that it is premature to try to attack the RH by a polymath project (but I am not an expert and, in any case, a project of this kind is better conducted with some specific program in mind). I propose something different. In a sort of polymath spirit the project I propose invite participants, especially professional mathematicians who thought about the RH over the years, to share their thoughts about RH.

Ideally each comment will be

1) One or a few paragraphs long

2) Well-thought, focused and rather polished

A few comments by the same contributors are also welcome.

To make it clear, the thread I propose is **not** going to be a research thread and also **not** a place for further discussions beyond some clarifying questions. Rather it is going to be a platform for interested mathematician to make statements and expressed polished thoughts about RH. (Also, if adopted, maybe we will need a special name for such a thing.)

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This thread is **not launching** any of the two suggested projects, but rather a place to discuss further these proposals. For the second project, it will be better still if the person who runs it will be an expert in the area, and certainly not an ignorant. For the first project, maybe there are better ideas for areas/theories appropriate for polymathing.

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Tim Gowers Proposed and launched a new polymath proposal aimed at a certain approach he has for proving that .

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Polymath8 was a **remarkable success!** Within two months the best value of *H* that was 70,000,000 in Zhang’s proof was reduced to 5,414. Moreover, the polymath setting looked advantageous for this project, compared to traditional ways of doing mathematics. (I have written a post with some more details and thoughts about it, looked from a distance.)

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On the numerical side, we have decided to focus first on the problem of obtaining validated upper and lower bounds for the second Neumann eigenvalue of a triangle . Good upper bounds are relatively easy to obtain, simply by computing the Rayleigh quotient of numerically obtained approximate eigenfunctions, but lower bounds are trickier. This paper of Liu and Oshii has some promising approaches.

After we get good bounds on the eigenvalue, the next step is to get good control on the eigenfunction; some approaches are summarised in this note of Lior Silberman, mainly based on gluing together exact solutions to the eigenfunction equation in various sectors or disks. Some recent papers of Kwasnicki-Kulczycki, Melenk-Babuska, and Driscoll employ similar methods and may be worth studying further. However, in view of the theoretical advances, the precise control on the eigenfunction that we need may be different from what we had previously been contemplating.

These two papers of Miyamoto introduced a promising new method to theoretically control the behaviour of the second Neumann eigenfunction , by taking linear combinations of that eigenfunction with other, more explicit, solutions to the eigenfunction equation , restricting that combination to nodal domains, and then computing the Dirichlet energy on each domain. Among other things, these methods can be used to exclude critical points occurring anywhere in the interior or on the edges of the triangle except for those points that are close to one of the vertices; and in this recent preprint of Siudeja, two further partial results on the hot spots conjecture are obtained by a variant of the method:

- The hot spots conjecture is established unconditionally for any acute-angled triangle which has one angle less than or equal to (actually a slightly larger region than this is obtained). In particular, the case of very narrow triangles have been resolved (the dark green region in the area below).
- The hot spots conjecture is also established for any acute-angled triangle with the property that the second eigenfunction has no critical points on two of the three edges (excluding vertices).

So if we can develop more techniques to rule out critical points occuring on edges (i.e. to keep eigenfunctions monotone on the edges on which they change sign), we may be able to establish the hot spots conjecture for a further range of triangles. In particular, some hybrid of the Miyamoto method and the numerical techniques we are beginning to discuss may be a promising approach to fully resolve the conjecture. (For instance, the Miyamoto method relies on upper bounds on , and these can be obtained numerically.)

The arguments of Miyamoto also allow one to rule out critical points occuring for most of the interior points of a given triangle; it is only the points that are very close to one of the three vertices which we cannot yet rule out by Miyamoto’s methods. (But perhaps they can be ruled out by the numerical methods we are also developing, thus giving a hybrid solution to the conjecture.)

Below the fold I’ll describe some of the theoretical tools used in the above arguments.

Let be an acute-angled triangle that is not equilateral, and let be the second Neumann eigenvalue; as discussed in previous posts, we know that this eigenvalue is simple. The method of Miyamoto allows one to control the structure of the second eigenfunction through an analysis of the quadratic form

for (we restrict attention here to real-valued functions). From the spectral theorem, we know that this quadratic form is non-negative when has mean zero, with equality if and only if is a multiple of . This leads to the following consequence:

Lemma 1Let have disjoint supports. Then is non-negative for all but at most one of the . If and none of the vanish identically, we may upgrade “non-negative” in the previous assertion to “strictly positive”.

\bein{proof} Suppose for contradiction that and are negative for some distinct . If we take to be a linear non-trivial combination of which has mean zero, then we see from the disjoint supports of that is also negative, contradicting the non-negativity of on mean-zero functions.

Now suppose that and are merely non-positive instead of non-negative. Then the above argument shows that there is a non-trivial linear combination of that is a non-zero multiple of . On the other hand, if and none of the vanishing identically, then this linear combination of will be zero on a set of positive measure, which is impossible for a non-zero multiple of the eigenfunction (which is real analytic).

We have a further non-negativity property of :

Lemma 2Let vanish on two of the three sides of . Then , with equality occuring if only if solves the eigenfunction equation and obeys Neumann conditions on the remaining side of .

*Proof:* Write . If vanishes on and with , we reflect across and obtain a function on the kite formed by reflecting across , with

and so the first Dirichlet eigenvalue of is less than . But by a result of Friedlander, the first Dirichlet eigenvalue of the convex planar domain is at least as large as the third Neumann eigenvalue of that domain. Hence, the symmetric reflection of across cannot be the second or third Neumann eigenfunction for , and so these functions must both be anti-symmetric instead of symmetric across . But at least one of these anti-symmetric eigenfunctions must change sign on (as they are orthogonal to each other), and will then have at least four nodal domains, contradicting the Courant nodal line theorem. The second claim of the lemma follows by similar arguments and is omitted.

This lemma turns out to be particularly useful when applied to the nodal components of a solution to the eigenfunction equation :

Corollary 3Let be a solution to the eigenfunction equation , not necessarily obeying the Neumann boundary condition. Let be the nodal domains of (i.e. the connected components of in ). Thenfor all but at most one , where is the derivative in the outward normal direction. If , then we can make the inequality (1) strict. Finally, (1) holds (with strict inequality) whenever is contained in one of the three sides of

*Proof:* We apply the previous lemmas with , and observe from integration by parts that

As worked out in previous polymath7 threads, applying this corollary to the Neumann eigenfunction yields that the nodal curve is simple and connects two distinct sides of the triangle . However, the new advances of Miyamoto and Siudeja have come from applying this corollary to other solutions to the eigenfunction equation. For instance:

Corollary 4Let be a non-trivial solution to the eigenfunction equation , not necessarily obeying the Neumann boundary condition. Then the nodal curve does not contain any loops.

This leads to a variant of the maximum principle:

Corollary 5Let be a solution to the eigenfunction equation , not necessarily obeying the Neumann boundary condition. If on , then on .

One particularly nice solution to use in the above corollary is a directional derivative of , yielding the following result of Siudeja:

Corollary 6Suppose that has no critical points on the interior of two of the three sides of the triangle . Then has no critical points in the interior of either. In particular, the hot spots conjecture is true for this triangle.

*Proof:* Apply the previous corollary to the derivative of in the direction normal to the third side, to conclude that that derivative does not change sign in the interior of the triangle. But this is incompatible with a critical point in the interior (as can be seen for instance by a Bessel expansion around that point).

Another fruitful solution to use is some linear combination of and another solution designed to create a degenerate critical point . This gives the following criterion of Miyamoto for excluding critical points at certain locations:

Corollary 7Let be an interior point of , and let be a solution to the eigenfunction equation with , , and for all (excluding vertices), but is not identically zero on . Then does not have a critical point at at .

*Proof:* Suppose for contradiction that . We first eliminate a degenerate case when . In this case the nodal curve of crosses itself at , which by Corollary 4 creates at least four nodal domains, contradicting the Courant nodal line theorem. Thus we may assume without loss of generality that . If we subtract a suitable multiple of from we then get another solution to the eigenfunction equation with , and on . Again, from Corollary 4 has at least four nodal domains, including at least two in which is negative. But this contradicts Corollary 3.

Miyamoto uses this corollary with being a radial solution to the eigenfunction equation centered at to establish the hot spots conjecture for sufficiently round domains, but perhaps one can adapt the method to other solutions to also cover many cases of critical points inside various triangles.

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It seems that this naturally suggests a Polymath project with two interrelated goals:

- Further improving the numerical upper bound on gaps between primes; and
- Understanding and clarifying Zhang’s argument (and other related literature, e.g. the work of Bombieri, Fouvry, Friedlander, and Iwaniec on variants of the Elliott-Halberstam conjecture).

Part 1 of this project splits off into somewhat independent sub-projects:

- Finding narrow
~~prime~~admissible tuples of a given cardinality (or, dually, finding large~~prime~~admissible tuples in a given interval). This part of the project would be relatively elementary in nature, relying on combinatorics, elementary number theory, computer search, and perhaps some clever algorithm design. (Scott Morrison has already been hosting a de facto project of this form at this page, and is happy to continue doing so). - Solving a calculus of variations problem associated with the Goldston-Yildirim-Pintz argument (discussed at this blog post, or in this older survey of Soundararajan) [in particular, this could lead to an improvement of a certain key parameter , currently at 341,640, even without any improvement in the parameter mentioned in part 3. below.]
- Delving through the “hard” part of Zhang’s paper in order to improve the value of a certain key parameter (which Zhang sets at 1/1168, but is likely to be enlargeable).

Part 2 of this project could be run as an online reading seminar, similar to the online reading seminar of the Furstenberg-Katznelson paper that was part of the Polymath1 project. It would likely focus on the second half of Zhang’s paper and would fit well with part 1.3. I could run this on my blog, and this existing blog post of mine could be used for part 1.2.

As with other polymath projects, it is conceivable that enough results are obtained to justify publishing one or more articles (which, traditionally, we would publish under the D.H.J. Polymath pseudonym). But it is perhaps premature to discuss this possibility at this early stage of the process.

Anyway, I would be interested to gauge the level of interest and likely participation in these projects, together with any suggestions for improving the proposal or other feedback.

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From Holroyd’s sorting networks picture gallery

A celebrated theorem of Ajtai, Komlos and Szemeredi describes a sorting network for $n$ numbers of depth $O(log N)$. rounds where in each runs $n/2$. Tim Gowers proposes to find collectively a randomized sorting with the same properties.

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Let us have a little discussion about it.

We may also discuss both general and specific open research mathematical projects which are of different flavor/rules.

Proposals for polymath projects appeared on this blog, in this post on Gowers’s blog, and in several other places.

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Activity has now focused on a numerical strategy to solve the hot spots conjecture for all acute angle triangles ABC. In broad terms, the strategy (also outlined in this document) is as follows. (I’ll focus here on the problem of estimating the eigenfunction; one also needs to simultaneously obtain control on the eigenvalue, but this seems to be to be a somewhat more tractable problem.)

- First, observe that as the conjecture is scale invariant, the only relevant parameters for the triangle ABC are the angles , which of course lie between 0 and and add up to . We can also order , giving a parameter space which is a triangle between the values .
- The triangles that are too close to the degenerate isosceles triangle or the equilateral triangle need to be handled by analytic arguments. (Preliminary versions of these arguments can be found here and Section 6 of these notes respectively, but the constants need to be made explicit (and as strong as possible)).
- For the remaining parameter space, we will use a sufficiently fine discrete mesh of angles ; the optimal spacing of this mesh is yet to be determined.
- For each triplet of angles in this mesh, we partition the triangle ABC (possibly after rescaling it to a reference triangle , such as the unit right-angled triangle) into smaller subtriangles, and approximate the second eigenfunction (or the rescaled triangle ) by the eigenfunction (or ) for a finite element restriction of the eigenvalue problem, in which the function is continuous and piecewise polynomial of low degree (probably linear or quadratic) in each subtriangle; see Section 2.2 of these notes. With respect to a suitable basis, can be represented by a finite vector .
- Using numerical linear algebra methods (such as Lanczos iteration) with interval arithmetic, obtain an approximation to , with rigorous bounds on the error between the two. This gives an approximation to or with rigorous error bounds (initially of L^2 type, but presumably upgradable).
- After (somehow) obtaining a rigorous error bound between and (or and ), conclude that stays far from its extremum when one is sufficiently far away from the vertices A,B,C of the triangle.
- Using stability theory of eigenfunctions (see Section 5 of these notes), conclude that stays far from its extremum even when is not at a mesh point. Thus, the hot spots conjecture is not violated away from the vertices. (This argument should also handle the vertex that is neither the maximum nor minimum value for the eigenfunction, leaving only the neighbourhoods of the two extremising vertices to deal with.)
- Finally, use an analytic argument (perhaps based on these calculations) to show that the hot spots conjecture is also not violated near an extremising vertex.

This all looks like it should work in principle, but it is a substantial amount of effort; there is probably still some scope to try to simplify the scheme before we really push for implementing it.

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