Two great Peter Frankls

Tim Gowers launched polymath 11 aimed at Peter Frankl’s conjecture asserting that for every union-closed family there is an element that belongs to at lease half the sets in the family. Here are links to Post number 0 and Post number 1. (Meanwhile polymath10 continues to run on “Combinatorics and More.”)

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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.

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**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!

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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.

To show how complicated the cancellations are we record triples with degree , then -power for the first sum for degree primes followed by the -valuation

for the first sum for degrees up to .

Since the computation was done to precision , we only know that for degree the sum had -valuation at least , but do not know precise value. (I would like to know!).

Remarks: It might be best for the first sum to calculate to precision about (ignoring and more) and second about . (The valuations are even, so e.g., or will be waste.). Rather than making a full list (huge) and then summing, it is better to calculate both sums term by term (simultaneously) as the P’s are produced.

Probably different range of polynomials can be checked for irreducibility and sum calculated for it on different computers and then combined at the end. Simplest calculation will be degree 38 calculation for the first sum to precision (since the earlier calculation had this precision). (To go beyond , I suggest , as you cannot improve it later). There are many more sums for different ‘s and powers ‘s. I have written up some answers in the links above, any of which can be checked independently.

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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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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.

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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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