# Joel David Hamkins’ 1000th MO Answer is Coming

Update (May 2014): The second MO contributor to answer 1000 questions is another distinguished mathematician (and a firend) Igor Rivin.

Joel David Hamkins’ profile over MathOverflow reads: “My main research interest lies in mathematical logic, particularly set theory, focusing on the mathematics and philosophy of the infinite. A principal concern has been the interaction of forcing and large cardinals, two central concepts in set theory. I have worked in group theory and its interaction with set theory in the automorphism tower problem, and in computability theory, particularly the infinitary theory of infinite time Turing machines. Recently, I am preoccupied with the set-theoretic multiverse, engaging with the emerging field known as the philosophy of set theory.”

Joel is a wonderful MO contributor, one of those distinguished mathematicians whose arrays of MO answers in their areas of interest draw coherent deep pictures for these areas that you probably cannot find anywhere else. And Joel is also a very highly decorated and prolific MO contributor, whose 999th answer appeared today!!

Here is a very short selection of Joel’s answers. To (MO founder) Anton Geraschenko’s question What are some reasonable-sounding statements that are independent of ZFC? Joel answered; “If a set X is smaller in cardinality than another set Y, then X has fewer subsets than Y.” Joel gave a very thorough answer to  my question on Solutions to the Continuum Hypothesis; His 999th answer is on the question Can an ultraproduct be infinite countable? (the answer is yes! but this is a large cardinal assumption.) Update: Joel’s 1000th answer on a question about logic in mathematics and philosophy was just posted.

Joel also wrote a short assay, the use and value of MathOverflow over his blog. Here it is:

The principal draw of mathoverflow for me is the unending supply of extremely interesting mathematics, an eternal fountain of fascinating questions and answers. The mathematics here is simply compelling.

Mathoverflow has also taught me a lot about good mathematical exposition, both by the example of other’s high quality writing and by the immediate feedback we all get on our posts. This feedback reveals what kind of mathematical explanation is valued by the general mathematical community, in a direct way that one does not usually get so well when writing a paper or giving a conference talk. This kind of knowledge has helped me to improve my mathematical writing in general.

So, thanks very much mathoverflow! I am grateful.

# Polymath 8 – a Success!

### Yitang Zhang

Update (July 22, ’14). The polymath8b paper “Variants of the Selberg sieve, and bounded intervals containing many primes“, is now on the arXiv. See also this post on Terry Tao’s blog. Since the last update, we also had here at HUJI a beautiful learning seminar on small gaps between primes. James Maynard gave a series of three lectures and additional lectures were given by Zeev Rudnick and Tamar Ziegler.

Update (Jan 9, ’14, corrected Jan 10):  Polymath8b have just led to an impressive progress: Goldston, Pintz, and Yıldırım showed that conditioned on the  Elliott-Halberstam conjecture (EHC) there are infinitely many primes of bounded gap below 16. Maynard improved it to 12. Polymath8b have just improved it based on a generalized form of the EHC (proposed in 1986 by Bombieri, Friedlander, and  Iwaniec) further to 8.  [Further update:  6 and there are reasons so suspect that further improvement requires major breakthrough – namely getting over the “parity problem”.] The unconditional bound for gaps stands now on 270.

Update: A paper by James Maynard entitled “Small gaps between primes” proved that for  every k there are infinitely many intervals of length f(k) each containing at least k primes. He also reduced the gap between infinitely many pairs of primes to 600. The method is also (said to be) much simpler. Amazing! Similar results were obtained independently by Terry Tao.

Terry Tao launched a followup polymath8b to  improve the bounds for gaps between primes based on Maynard’s results.

### Zhang’s breakthrough and Polymath8

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.

The polymath project gave opportunity to a number of researchers to understand Zhang’s proof and the earlier breakthrough by Daniel Goldston, János Pintz, and Cem Yıldırım. It also gave an opportunity to a larger number of mathematicians to get some feeling about the involved mathematics.

### The story

Twin primes are two primes p and p+2. The ancient twin prime conjecture asserts that there are infinitely many twin primes. The prime number theorem asserts that there are (asymptotically)  n/log n primes whose value is smaller than a positive integer n, and this implies that we can find arbitrary large pairs of consecutive primes  p and q such that q-p is at most (log p). Until a few years ago nothing asymptotically better was known. Goldston, Pintz, and Yıldırım (GPY), showed in 2005 that there infinitely many pairs of primes p and q such that q-p is $O(\sqrt {\log n})$. A crucial idea was to derive information on gaps of primes from the distribution of primes in arithmetic progressions. GPY showed that conditioned on the  Elliott-Halberstam conjecture (EHC) there are infinitely many primes of bounded gaps (going all the way to 16, depending on a certain parameter in the conjecture, but not to 2). Yitang Zhang did not prove the EHC but based on further understanding of the situation found a way to shortcut the conjecture and to prove that there are infinitely many primes of with bounded gaps unconditionally!

Here is a very nice 2007 survey article by Kannan Soundararajan on this  general area of research and the GPY breakthrough. (One thing I recently learned is that  Soundararajan is called by friends and colleagues “Sound”. ) This article starts with a very thoughtful and endearing answer to the quastion: “Why do we care at all? After all primes were meant to be multiplied, not subtracted (or added).”

Here is a short list of thoughts (things I learned, things I wish to understand better…) from following (from distance) Polymath8 and related Internet activity.

1) How information on primes in arithmetic progressions leads to information on gaps between primes?

I do not really understand why the information on primes in arithmetic progressions e.g. the Elliott-Halberstam conjecture lead to the conclusion regarding primes with bounded gaps. I would be very happy to get a feeling for it.

2) The three-primes barrier.

Already GPY  tried to extend their methods to show the existence of three primes in a bounded interval of integers. So far, it is not known how to show that intervals of the form [n,n+o(log n)] contain triples of primes infinitely often. Perhaps, to actually solve the twin prime conjecture we will need to get a breakthrough for triples of primes, but maybe not. See also this MO question asked by Noam Elkies.

Update: Here is another interesting MO question Quantitative lower bounds related to Zhang’s theorem on bounded gaps, asked by Eric Naslund. Eric asks: what can be say based on Zhang’s work about the smallest value  of a pair of primes of distance k apart?

3) Cauchy-Schwarz everywhere;

This may sound silly but the way Cauchy-Schwarz (C-S) inequality is used again and again make you wonder again why C-S is it so useful, and why it is mainly C-S which is so useful.

4) Can detailed statistical understanding of primes in sets other than AP  be useful?

In recent years there was much activity (and I also was interested) in Mobius randomness and analogs of the prime number theorem for various more complicated subsets of integers. (E.g., subsets defined by various properties of the digital expansion.) Can understanding of this kind  also be used for the prime-gaps questions?

5) Usefulness of Deligne’s work on Riemann’s hypothesis for functions fields for questions in analytic number theory.

I knew, of course that Deligne famously proved analogs of the Riemann hypothesis for function fields in great generality but I was not aware that these results have applications to “ordinary” analytic number theory. Again, this is something I would be happy to know a little more about. There is a nice recent post on the Riemann hypothesis in various settings on “What’s new”.

6) Parity problem.  (Added Nov 27) There is a difficult “parity problem” which seems to be a difficult obstacle for getting the gap to two. (And for various related goals). Terry Tao wrote about it in 2007 in this post. In polymath8b VII an attempt to cross the “parity barrier” was  made but (as people expected) it turned out that the parity barrier indeed shows up causes this attempt to fail. (Update July 14:) This is further explained in this new post over Tao’s blog.

7) (Added Nov 27) One thing I am curious about is the following. Consider a random subset of primes (taking every prime with probability p, independently, and say p=1/2). Now consider only integers involving these primes. I think that it is known that this system of “integers” satisfies (almost surely) PNT but not at all RH. We can consider the properties BV (Bombieri Vinogradov), or more generally EH(θ) and the quantities $H_m$. For such systems does BV typically hold? or it is rare like RH. Is Meynard’s implication applies in this generality? Nicely here we can hope even for infinite consecutive primes. Update: after thinking about it further and a little discussion over polymath8b it looks that current sieve methods, and some of the involved statements, rely very strongly on both the multiplicative and additive structure of the integers and do not allow extensions to other systems of “integers.”

Update (August 23): Before moving to small gaps, Sound’s 2007 survey briefly describes the situation for large gaps. The Cramer probabilistic heuristic suggests that there are consecutive primes in [1,n] which are $c(\log n)^2$ apart, but not $C (\log n)^2$ apart where c and C are some small and large positive constants.  It follows from the prime number theorem that there is a gap of at least $\log n$. And there were a few improvements in the 30s ending with a remarkable result by Rankin who showed that there is a gap as large as $\log n$ times $\log \log n \log \log \log \log n (log log log n)^{-2}$. Last week Kevin Ford, Ben Green, Sergei Konyagin, and Terry Tao and independently James Maynard were able to  improve Rankin’s estimate by a function that goes to infinity with n.  See this post on “What’s new.”

# Open Collaborative Mathematics over the Internet – Three Examples

After much hesitation, I decided to share with you the videos of my lecture: Open collaborative mathematics over the internet – three examples, that I gave last January in Doron Zeilberger’s seminar at Rutgers on experimental mathematics. Parts of the 47-minutes talk is  mathematical, while other parts are about mathematics on the Internet, blogs,  the polymath projects, MathOverflow, etc.

Here is the video of part I (30 minutes) and here is the video of  part II (17 minutes).

I tried to give some homage to Doron’s own lecture style, but when I saw the video, I could not ignore some aspects of my own style – complete indifference between plus and a minus, between $x^2$ and $\sqrt x$, between multiplying by something and dividing by the same thing, between subscripts and superscripts, between what are “rows: and what are “columns”, etc., and, in addition, randomly ending English words with an ‘s’ regardless of what English grammar dictates. Apologies. This was a rare occasion of giving a talk about “meta” matters of doing mathematics.

My first (and main) example was Erdős discrepancy’s problem, and I mentioned some experimental aspects, and heuristic arguments. The second example was Möbius randomness,  continued with some comments on MathOverflow, some  of the “goodies” one can earn participating in MathOverflow, and some comments on a debate regarding polymath projects organized by I.A.S a few years ago.  The third example was about mathematically oriented skepticism. This time not about my debate with Aram Harrow and quantum computing skepticism (that I briefly mentioned), but about my “Angry Birds” skepticism. The lecture was a mixture of a blackboard talk and presentation of various Internet site.

Here are the links to the Internet pages I presented with an outline of the lecture:

Video I (Erdős discrepancy problem)

00:00-00:43 Doron’s introduction; 00:43-4:00 On the screen: Erdős discrepancy problems: showing this post (EDP22) from Gowers’s blog. I talked  about the chaotic nature of mathematics on the Internet. Then I explained what are polymath projects, mentioned polymath1 and density Hales-Jewett, other polymath projects, and polymath5.

4:00-12:00 Polymath5, discrepancy of hypergraphs, and The Erdős Discrepancy problem. [6:06 “There is nothing more satisfying in a lecture than seeing attempts by the speaker to move an unmovable blackboard”]; Some basic observations, random signs, Mobius functions, the log n example.

12:00-16:00 Plan for the next fifteen minutes: a) Greedy methods, b)Heuristic approaches, c) Hereditary discrepancy

17:00 – 18:40 Alex Nokolov and Kinal Talwar’s work on hereditary discrepancy. On the screen: Talwar’s post discrepancy to privacy and back on Windows on Theory

18:40 -23:00   Greedy approaches. On the screen:  My question on MathOverflow – on a certain greedy algorithm for Erdős discrepancy problem; The MO answer by ‘rlo'; a follow up question

23:00 – 27:00 What is MathOverflow. On the screen: my old MO page. (Here is the new one.) What is  “MO- reputation,” which “badges” you can earn and for what; Earning “hats” in more advanced site: On the screen:  hat champions from TCS-Stackexchange, winter 2012. (This webpage is no longer available.)

27:00 – 30:00 My heuristic approach for EDP

Video II (Möbius randomness, and angry birds)

00:00-02:00 Diversion: The IAS debate on polymath projects between Gowers and Sarnak, and comments  by me and by Noga Alon.

02:00-05:40  Möbius randomness and computational complexity. What is Mõbius randomness (MR);  Peter Sarnak’s Jerusalem talk on MR, his belief regarding the hardness of factoring, and my opinion; The $AC^0$ prime number conjectures and Walsh functions. My MO question; Ben Green’s MO-answer to one problem and Bourgain’s result answering a second question.

05:40-7:50 A remaining question on MR of Rudin-Shapiro sequences. On the screen – my MO question Mobius randomness of the Rudin-Shapiro sequence; What is a bounty in MathOverflow; Diversion: The power that comes with high reputation on MO. Who won my bounty.

7:50-09:30  My debate with Aram Harrow on my Quantum computers skepticism. On the screen the debates first post and then the post “can you hear the shape of a quantum computer?

09:30-14:30 Example 3: Another mathematically related skepticism – Angry Birds. On the screen this page from Arqade : Most voted questions (a few are very amusing); Introduction to the game “Angry Birds”; My skeptical theory: “Angry Birds” is becoming easier with new versions, and the statistical argument for it. I mentioned the Arqade’s question “Is angry Birds deterministic” that got (then) 143 upvotes, and was a role-model for me.

14:30-17:00 My Arqade question, my hopes, and how it was accepted by the computer-games community;  On the screen: my Arqade question:

# Polymath8: Bounded Gaps Between Primes

Yitang Zhang’s very recent shocking paper demonstrated that bounded gaps between primes occur infinitely often, with the explicit upper bound of 70,000,000 given for this gap. Polymath8 was launched for the dual purpose of learning Zhang’s proof and improving the upper bound for the gaps. Here are links for three posts (I, II, III) on Terry Tao’s blog, a post on the secret blogging seminar,  and for a post on the polymath blog. And here is the table for the world records so far.

Updates: Record for June 16 – 60,744

# Looking Again at Erdős’ Discrepancy Problem

Over Gowers’s blog Tim and I will make an attempt to revisit polymath5. Last Autumn I prepared three posts on the problems and we decided to launch them now. The first post is here. Here is a related MathOverflow question. Discrepancy theory is a wonderful theory and while I am not an expert we had several posts about it here. (This post on Beck-Fiala and related matters; and this news item on Beck’s 3-permutation conjecture.) I am aware of at least one important recent development in the theory that I did not report.  My posts go around the problem and do not attack it directly but I hope people will have a chance to think about the problem again and perhaps also about polymathing again. Meanwhile, polymath7 (over the polymath blog) is in a short recess but I hope a new research thread will start soon.

# The Internet, Journals and all that.

Tim Gowers wrote an interesting post where he proposed in surprising many details an Internet mechanism (mixing ingredients from the arXive, blogs, MathOverflow and polymath projects) to replace Journals. Noam Nisan (who advocated similar changes over the years) wrote an interesting related post entitled the problems with Journals

A subsequent post by Gowers proposed a much less radical proposal. And Noam also wrote an interesting subsequent post entitled the good things about Journal.

My favorite post on this issue from Izabella Laba’s blog The accidental mathematician is entitled Random thoughts on publishing and the internet .