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John Rickert

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Link via Ken Ono: Prime pair gaps down to 600 (and if the Elliott-Halberstam conjecture is assumed, 12), and prime k-tuples gap to about k^3*exp(4k). Achieved by generalizing the weights in the Goldston-Pintz-Yildirim method.

The last time the Pirates had a .500 season Bryce Harper and Jurickson Profar weren't born. #Pirates  

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Manjul Bhargava and Arul Shankar have shown that at least 12% of randomly selected elliptic curves have finitely many rational points.  and a preliminary paper
Bjorn Poonen's explanation of the work:
There's been a breakthrough in elliptic curves - a subject important in number theory and cryptography! An elliptic curve is defined by an equation like

y² = x³ + ax + b

for some choice of a and b.  If we graph the real numbers x and y that solve this equation, we get pictures like those here.  But the true beauty of an elliptic curve is revealed when we draw the complex solutions: we get a shape like the surface of a doughnut with one point removed, and people usually add in that extra point to make it nicer.

If a and b are rational numbers it's very interesting to look for solutions where both x and y are rational: these are called rational points of the elliptic curve.  They're hard to find, but if you find two there's a cool way to get more of them.  Sometimes there are infinitely many, sometimes just finitely many.

Now Manjul Bhargava (one of the youngest people to become a full professor at Princeton University) and Arul Shankar (his grad student) have shown that if you randomly choose rational numbers a and b, an elliptic curve has just finitely many rational points at least 12% of the time! 

This is a step toward proving the minimalist conjecture, which says - among other things - that in fact this happens exactly 50% of the time. 

I'm leaving out a lot of fun and interesting details, just trying to give you a precise statement of what was discovered, with a minimum of fancy jargon.   If you want more, this popularization is great:

Thanks to +David Roberts for pointing this out!

If you want even more, try this expository account:

• Bjorn Poonen, Average rank of elliptic curves,

#spnetwork #recommend #numberTheory #ellipticCurves #expository arXiv:1203.0809

If you want

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I should probably have my students read this this summer.
There are extremely few papers that I would tag as "must read" for all research mathematicians, but I would certainly include Thurston's classic article on what progress in mathematical research is, and how this differs from (but is certainly related to) the mere acquisition of proofs of theorems, among this very short list.  #spnetwork #mustread arXiv:math/9404236

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Information about the new selected papers network.
Here it is at last: the Selected Papers Network.   Given that social networks already exist, all we need for truly open scientific communication is a convention on a consistent set of tags and IDs for discussing papers.  Christopher Lee has developed software that makes this work.  Try it out!  

What's cool about this system is that it's federated.  Instead of “locking up” your comments within its own website—the “walled garden” strategy followed by many other services—it explicitly shares these data in a way that people not on the Selected Papers Network can easily see. Any other service can see and use them too!

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From 70,000,000 to less than 400,000 in less than a month. via +Qiaochu Yuan 
Good news, prime gap watchers! Due to successes on multiple fronts, we now have an upper bound on prime gaps of 388 188  387 982 (an improvement of a factor of ~180 on the original 70M). This is much better than I was hoping, and the progress mostly came from solving a variational calculus problem (at Terry Tao's blog) that replaced a naive monomial with one parameter l_0 by something involving a Bessel function , taking a factor of 10 off the final bound. Also, people have found better ways of constructing admissible sets using various sieving algorithms on asymmetric intervals around 0 (at Secret Blogging Seminar).

Now the next natural milestone to me feels like it should be 10 000. Will we get there? Could I even hope for lowering the bound below 1000?

Watch the progress here:

#primenumbers   #primegap  

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A big step toward proving the twin-prime conjecture.
No link to a paper available yet.
The conjecture on twin prime numbers -- which may date all the way back to Euclid -- is finally solved! Except for a factor 35 million, but that's just a minor detail...

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Harald Helfgott polishing off the proof that every odd number greater than five is the sum of three primes. More reading to look forward to after finals next week.
Busy day in analytic number theory; Harald Helfgott has complemented his previous paper (obtaining minor arc estimates for the odd Goldbach problem) with major arc estimates, thus finally obtaining an unconditional proof of the odd Goldbach conjecture that every odd number greater than five is the sum of three primes.  (This improves upon a result of mine from last year showing that such numbers are the sum of five or fewer primes, though at the cost of a significantly lengthier argument.) As with virtually all successful partial results on the Goldbach problem, the argument proceeds by the Hardy-Littlewood-Vinogradov circle method; the challenge is to make all the estimates completely effective and to optimise all parameters (which, among other things, requires a certain amount of computer-assisted computation).

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Not a big surprise that the mathematics dissertations are among the shortest.
The average economics dissertation is much shorter than a history dissertation.
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