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More narrow admissible sets June 5, 2013

Posted by Scott Morrison in polymath.
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It looks like it may be time to roll over the search for narrow admissible sets to a new blog post, as we’re approaching 100 comments on the original thread.

In the meantime, an official polymath8 project has started. The wiki page is a good place to get started. Work to understand and improve the bounds in Zhang’s result on prime gaps has split into three main areas.

1) A reading seminar on Zhang’s Theorem 2.
2) A discussion on sieve theory, bridging the gap begin Zhang’s Theorem 2 and the parameter k_0 (see also the follow-up post).
3) Efforts to find narrow admissible sets of a given cardinality k_0 — the width of the narrowest set we find gives the current best bound on prime gaps.

We started on 3) in the previous blog post, and now will continue here. I’ll try to summarize the situation.

Just recently there’s been a significant improvement in k_0, the desired cardinality of the admissible set, and we’re now looking at k_0 = 34,429. Hopefully there’s going to be a whole new round of techniques, made possible by the significantly smaller problem size.

As I write this, the narrowest admissible set of size 34,429 found so far, due to Andrew Sutherland, has width 388,118.

This was found using the “greedy-greedy” algorithm. This starts with some chosen interval of integers, in this case [-185662,202456], and then sieves as follows. First discard 1 mod 2, and then 0 mod p for p \leq b, for some parameter b. (I’m not actually sure of the value of this parameter in Andrew’s best set.) After that, for each prime we pick a minimally occupied residue class, and sieve that out. Assuming we picked a sufficiently wide interval to begin with, when we’re done the resulting admissible set with still have at least k_0 elements.

More generally, there are several directions worth pursuing

1. sharpening bounds on \rho^*(x), the maximal cardinality of an admissible set of width at most x,
2. finding new constructions of admissible sets of a given size (and also ‘almost-admissible’ sequences)
3. developing algorithms or search techniques to find narrow admissible sets, perhaps starting from a wider or smaller admissible set, or starting from an ‘almost-admissible’ set.

(If these questions carry us in different directions, there’s always more room on the internet!)

For sufficiently small sizes (at most 372), everything is completely understood due to exhaustive computer searches described at http://www.opertech.com/primes/k-tuples.html. At least for now, we need to look at much larger sizes, so obtaining bounds and finding probabilistic methods is probably the right approach.

I’m writing this on a bus, beginning 30 hours of travel. (To be followed by a short sleep then an intense 3 day conference!) So my apologies if I missed something important!

I just can’t resist: there are infinitely many pairs of primes at most 59470640 apart May 30, 2013

Posted by Scott Morrison in Number theory, polymath.
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Everyone by now has heard about Zhang’s landmark result showing that there are infinitely many pairs of primes at most 70000000 apart.

His core result is that if a set of 3.5 * 10^6 (corrected, thanks to comment #2) numbers H is admissible (see below), then there are infinitely many n so that $latex n+H$ contains at least two primes. He then easily constructs an admissible set wherein the largest difference is 7 * 10^7, obtaining the stated result.

A set $latex H$ is admissible if there is no prime $latex p$ so $latex H \pmod p$ occupies every residue class. For a given $latex H$ this is clearly a checkable condition; there’s no need to look at primes larger than $latex |H|$.

(While Zhang went for a nice round number, Mark Lewko found his argument in fact gives 63374611, if you’re being overly specific about these things, which we are right now. :-)

In a short note on the arXiv yesterday, Tim Trudgian (whose office is not far from mine) pointed out another way to build an admissible set, giving a smaller largest difference, obtaining the result that there are infinitely many pairs of primes at most 59874594 apart. He considers sets of the form $latex H_m = {p_{m+1}, \ldots, p_{m+k_0}}$ (where $latex k_0$ is Zhang’s constant 3.5 * 10^7). These aren’t necessarily admissible, but they are for some values of $latex m$, and both Zhang and Tim noticed certain values for which this is easy to prove. Zhang used $latex H_m$ with $latex m=k_0$, while Tim’s observation is that $latex m_0=250150=\pi(k_0)$ also works. (Comment #2 below points out this isn’t right, and Zhang also intended $m=\pi(k_0)$, and the slack in his estimate is coming from just looking at the largest element of H_m, rather than the largest difference.) Thus the bound in his result is $latex p_{m_0+k_0}-p_{m_0+1} = 59874594$.

It turns out that checking admissibility for a given $latex H_m$ isn’t that hard; it takes about an hour to check a single value for $latex m ~ m_0$ (but if you find a prime witnessing $latex H_m$ not being admissible, it very often gives you a fast proof that $latex H_{m+1}$ is not admissible either, so searching is much faster).

I haven’t looked exhaustively, but one can check that $latex m_1 = m_0 / 2 = 125075$ gives an admissible $latex H_m$, and hence there are infinitely many pairs of primes at most $latex p_{m_1 + k_0}-p_{m_1+1}=59470640$. (Sadly, it’s impossible to get below 59 million with this trick; no $latex m$ below 27000 works; all witnessed by $latex p=182887$ or $latex 378071$.)

I just couldn’t resist momentarily “claiming the crown” for the smallest upper bound on gap size. :-) Of course the actual progress, that’s surely coming soon from people who actually understand Zhang’s work, is going to be in reducing his 3.5 * 10^6. You can read more about prospects for that in the answers to this MathOverflow question.

University of Melbourne hiring May 26, 2013

Posted by Scott Morrison in jobs.
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Often Australian jobs don’t make it on to MathJobs.org, for various reasons, so I thought I’d help distribute this one — Melbourne is hiring, initially a full professor, and subsequently several tenure track assistant professor positions, all in pure mathematics.

Below the fold, Arun Ram’s message about this. Australia is a nice place to come and work!

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Conference videos April 30, 2013

Posted by Ben Webster in Uncategorized.
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Well, from my perspective at least, the conference was a success.  We all made it through in one piece, and no one got trapped on the subway. If any of you are looking for the videos of the talks, they can be downloaded from this page. That’s a only a temporary hosting solution, but at least they’re available for the moment.

More on shameless promotion April 23, 2013

Posted by Ben Webster in Uncategorized.
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As those of you who’ve scrolled down the page know, the conference I mentioned a few months ago (now sadly memorializing the life of Andrei Zelevinsky) is starting tomorrow. Of course, for those of you who don’t live in the Boston area, coming to conference isn’t an option unless you were already traveling today, but I do have a (somewhat belated) announcement. Assuming that the AV gods are kind and everything goes as planned, it should be possible to watch the talks live (of course, we’ll also make the videos available after the conference, in case you’re busy). The schedule is here; the talks start at 10am tomorrow.

New open access journal in algebraic geometry March 3, 2013

Posted by David Speyer in Uncategorized.
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I just received an e-mail announcing that Compositio has launched an Open Access journal entitled Algebraic Geometry. Their website is live and promises “Open access implies here that the electronic version of the journal is freely accessible and that there are no article processing charges for authors whatsoever. The printed version of the journal will be available at the end of the calendar year against printing costs.”

The editorial board looks great, including L. Caporaso, J. Ellenberg, D. Maulik and R. Pandharipande. They will definitely get my next algebraic geometry paper.

This is really good news. It’s seemed clear from the debates on journals of the last year that what is needed is for people and institutions of high reputation to commit to running open journals. Compositio, and the editors they have found, are top of the line. From a selfish perspective, what makes me really happy is that I didn’t wind up on the editorial board.

Good work, and good luck, to Algebraic Geometry.

UK Parliament seeking feedback on Open Access support January 15, 2013

Posted by David Speyer in Uncategorized.
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Last September, the UK parliament earmarked £10 Million pounds from the science budget to support open access scientific publishing. Earlier this week, the UK parliament announced that they are seeking feedback on this policy, including “how the Government should address the concerns raised by the scientific and publishing communities about the policy”. Details of how to submit comments are here. I don’t know much about UK politics, so I can’t give much advice about how to frame your response, but the deadline is this weekend (Jan 18) so it seems important to get the word out.

Note some bizarre statements in the comment submission guidelines: They want submissions in Word or, if Word is not acceptable, another editable electronic format; they specifically state not PDF. (<rant> Why, oh why has the world forgotten RTF? Or plain ASCII? Or HTML? I understand why most people don’t want to use LaTeX, but the way that the world acts as if Word is the most convenient format drives me nuts. </rant>) They also say “[s]ubmisions become the property of the Committee.” That’s a bizarrely vague statement from a committee discussion copyright policy. I assume they mean that you are implicitly granting them the right to publish it, but if I were phrasing that I would say “you retain copyright to your words but grant us permission to retain your file/manuscript and to publish…”.

I don’t think that the points in the preceding paragraph represent bad intent on the part of the committee, but I do think they show an ignorance of how things are done in the academic world. So let’s help them out!

Hat tips: I learned about this from this blogpost, which I learned about from this comment. I am not familiar with either Glyn Moody or Alex.

A calculus free proof of the spectral theorem December 3, 2012

Posted by David Speyer in Uncategorized.
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Let A be an n \times n real symmetric matrix. Let \lambda be an eigenvalue and v be a corresponding eigenvector. Then

\displaystyle{\bar{v}^T A v = \bar{v}^T \lambda v = \lambda \bar{v}^T v}.

But, also

\displaystyle{\bar{v}^T A v = (\bar{\lambda} \bar{v}^T) v = \bar{\lambda} \bar{v}^T v}.

We deduce that \lambda \bar{v}^T v  = \bar{\lambda} \bar{v}^T v. And \bar{v}^T v is clearly a positive real, so \lambda = \bar{\lambda}.

This immediately shows that the characteristic polynomial of A has only real roots.

The other parts of the spectral theorem aren’t much harder

The canonical model structure on Cat November 16, 2012

Posted by Chris Schommer-Pries in Algebraic Topology, Category Theory.
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In this post I want to describe the following result, which I think is pretty neat and should be more widely known:

Theorem: On the category of (small) categories there is a unique model structure in which the weak equivalences are the equivalences of categories.

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Shameless conference promotion November 14, 2012

Posted by Ben Webster in Uncategorized.
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So, obviously posting on this blog has ebbed a little (I keep hoping to reverse this trend, but I think we’ve all found that demands on our time that come before blog posting tend to ratchet upward, not downward).  I assume there are still a few people reading, though, and I wanted to do a little promotion.  

One of the thing that’s been demanding my time lately has been conference organization.  We’re planning a conference at Northeastern next spring in recognition of the 60th birthday of Andrei Zelevinsky, with the exteremely original name of “Algebra, Combinatorics and Representation Theory”. I think it’s going to be great, and I encourage any of you who are able and interested to attend; we have a really great line-up of speakers.  

For more information, see our website.  I want to particularly encourage young people to attend; we’re really hoping (cross your fingers) to have funding for grad students and postdocs.  (Furthermore, it will help us to obtain said funding if young people express an interest in coming.  So, if you would like to come, please register).

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