# The polymath blog

## February 7, 2016

### Polymath 11 is Now Open

Filed under: news — Gil Kalai @ 3:45 am

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.”)

### Polymath Proposals on Math Overflow

Filed under: news,polymath proposals — Gil Kalai @ 3:21 am

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.

### Explaining Polynomials Identities – Success!

Filed under: news — Gil Kalai @ 3:03 am

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 $P$ in the polynomial ring ${\bf F}_2[t]$ 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!

## January 2, 2016

### “Crowdmath” project for high school students opens on March 1

Filed under: polymath proposals — Terence Tao @ 4:25 pm
Tags: ,

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).

## December 28, 2015

### Polymath proposal: explaining identities for irreducible polynomials

Filed under: planning,polymath proposals — Terence Tao @ 7:05 pm

I am posting this proposal on behalf of Dinesh Thakur.

Let $F_2[t]$ be the ring of polynomials over the finite field $F_2$ of two elements, and let

$\displaystyle {\mathcal P} = \{t, t+1, t^2+t+1, \dots \}$

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

$\displaystyle \sum_{P \in {\mathcal P}} \frac{1}{1+P} = \frac{1}{t+1} + \frac{1}{t} + \frac{1}{t^2+t} + \dots$

and

$\displaystyle \sum_{P \in {\mathcal P}} \frac{1}{1+P^3} = \frac{1}{t^3+1} + \frac{1}{(t+1)^3+1} + \frac{1}{(t^2+t+1)^3+1} + \dots$

can be expanded as formal infinite power series in the variable $t = 1/u$.

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

$\displaystyle{}\sum_{P \in {\mathcal P}} \frac{1}{1+P} = 0$

and

$\displaystyle{}\sum_{P\in{\mathcal P}} \frac{1}{1+P^3}=\frac{1}{t^4+t^2}$

$\displaystyle = u^4 + u^6 + u^8 + \dots.$

For instance, one has

$\displaystyle \frac{1}{t+1} = u + u^2 + u^3 + \dots$

$\displaystyle \frac{1}{t} = u$

$\displaystyle \frac{1}{t^2+t} = u^2 + u^3 + \dots$

and all other terms in $\sum_{P \in {\mathcal P}} \frac{1}{1+P}$ are of order $u^3$ or higher, so this shows that ${}\sum_{P \in {\mathcal P}} \frac{1}{1+P}$ has $u$-valuation at least 3.  Similarly, if one expands the first sum for all primes of degree (in $t$) up to 37, one obtains ${}u^{38}+u^{39}+u^{44}+u^{45}+\dots$ (the calculation took about a month on one computer), implying that the $u$-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 $u$-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.

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

## November 6, 2015

### Polymath10 is now open

Filed under: news — Gil Kalai @ 12:42 pm
Tags:

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.

## September 22, 2015

### The Erdős discrepancy problem has been solved by Terence Tao

Filed under: polymath5 — Gil Kalai @ 12:41 pm
Tags: ,

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.

## January 20, 2014

### Two polymath (of a sort) proposed projects

Filed under: discussion,polymath proposals — Gil Kalai @ 5:20 pm
Tags: , ,

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

# I. A polymath proposal: Convex hulls of real algebraic varieties.

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.

# II. A polymath-of-a-sort proposal: Statements about the Riemann Hypothesis

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.

## November 4, 2013

### Polymath9: P=NP? (The Discretized Borel Determinacy Approach)

Filed under: polymath proposals — Gil Kalai @ 2:07 pm
Tags: ,

Tim Gowers Proposed and launched a new polymath proposal aimed at a certain approach he has for proving that $NP \ne P$.

## September 20, 2013

### Polymath8 – A Success !

Filed under: news — Gil Kalai @ 5:58 pm
Tags:

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 ${H}$, and then to lower that value of ${H}$ 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.)

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