Here is the link to a mathoverflow question asking for polymath proposals. There are some very interesting proposals. I am quite curious to see some proposals in applied mathematics, and various areas of geometry, algebra, analysis and logic.

Here is the link to a mathoverflow question asking for polymath proposals. There are some very interesting proposals. I am quite curious to see some proposals in applied mathematics, and various areas of geometry, algebra, analysis and logic.

**Dinesh Thakur David Speyer**

A beautiful polymath proposal by Dinesh Thakur was posted by Terry Tao on the this blog. The task was to explain some remarkable, numerically observed, identities involving the irreducible polynomials in the polynomial ring over the finite field of characteristic two. David Speyer managed to prove Thakur’s observed identities! Here is the draft of the paper. Congratulations to Dinesh and David!

The MIT PRIMES program and the Art of Problem Solving are planning to run a “Crowdmath” project for high school students with advanced mathematical backgrounds, based on the polymath approach to mathematical research. The project, which officially starts on March 1, will be devoted to original research on a mathematics problem to be specified at the time of the project (but judging from the reference material provided, it will probably involve the combinatorics of 0-1 matrices). Participation is open to all high school students (though they will need an Art of Problem Solving account).

I am posting this proposal on behalf of Dinesh Thakur.

Let be the ring of polynomials over the finite field of two elements, and let

be the set of irreducible polynomials in this ring. Then infinite series such as

and

can be expanded as formal infinite power series in the variable .

It was numerically observed in http://arxiv.org/abs/1512.02685 that one appears to have the remarkable cancellation

and

For instance, one has

and all other terms in are of order or higher, so this shows that has -valuation at least 3. Similarly, if one expands the first sum for all primes of degree (in ) up to 37, one obtains (the calculation took about a month on one computer), implying that the -valuation of the infinite sum is at least 38; in fact a bit of theory can improve this to 42. (But we do not know whether this 42 is the answer to everything!).

For the second sum, calculation for degrees up to 28 shows that the difference between the two sides has -valuation at least 88.

The polymath proposal is to investigate this phenomenon further (perhaps by more extensive numerical calculations) and supply a theoretical explanation for it.

Background links:

- The paper http://arxiv.org/abs/1512.02685 where these (and many more guesses of this type) are given with some background on zeta deformation etc, and
- http://www.math.rochester.edu/people/faculty/dthakur2/primesymmetryrev.pdf where the updated version is and will be maintained.

Below the fold is some more technical information regarding the above calculations.

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.

Polymath5 was devoted to the Erdős discrepancy problem. It ran in 2010 and there were a few additional posts in 2012, without reaching a solution. The problem has now been solved by Terry Tao using some observations from the polymath project combined with important recent developments in analytic number theory. See this blog post from Tao’s blog and this concluding blog post from Gowers’s blog.

This post is meant to propose and discuss a polymath project and a sort of polymath project.

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.)

____________________

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.

The main objectives of the polymath8 project, initiated by Terry Tao back in June, were “to understand the recent breakthrough paper of Yitang Zhang establishing an infinite number of prime gaps bounded by a fixed constant , and then to lower that value of as much as possible.”

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.)

This post is the new research thread for the Polymath7 project to solve the hot spots conjecture for acute-angled triangles, superseding the previous thread; this project had experienced a period of low activity for many months, but has recently picked up again, due both to renewed discussion of the numerical approach to the problem, and also some theoretical advances due to Miyamoto and Siudeja.

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.