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Maggie McKee, Nature, First proof that infinitely many prime numbers come in pairs, here. Better than hitting the Powerball Lottery for Prof. Zhang if this holds. When is the Princeton University Public Lecture scheduled? I’m in for that.
The new result, from Yitang Zhang of the University of New Hampshire in Durham, finds that there are infinitely many pairs of primes that are less than 70 million units apart without relying on unproven conjectures. Although 70 million seems like a very large number, the existence of any finite bound, no matter how large, means that that the gaps between consecutive numbers don’t keep growing forever. The jump from 2 to 70 million is nothing compared with the jump from 70 million to infinity. “If this is right, I’m absolutely astounded,” says Goldfeld.
Zhang presented his research on 13 May to an audience of a few dozen at Harvard University in Cambridge, Massachusetts, and the fact that the work seems to use standard mathematical techniques led some to question whether Zhang could really have succeeded where others failed.
Erica Klarreich, Simons Foundation, Unheralded Mathematician Bridges the Prime Gap, here. UNH students, if Prof. Zhang has any class openings in the Fall you are advised to pack yourselves in those seats, Pronto. This is like the hottest Fantasy Basketball pick up in 50 years RUN, DO NOT WALK and pull this guy’s course off the enrollment wire. This holds then you got a MJ/LBJ level world-class stud teaching there now – this won’t last forever – carpe diem.
Zhang read the GPY paper, and in particular the sentence referring to the hair’s breadth between GPY and bounded prime gaps. “That sentence impressed me so much,” he said.
Without communicating with the field’s experts, Zhang started thinking about the problem. After three years, however, he had made no progress. “I was so tired,” he said.
To take a break, Zhang visited a friend in Colorado last summer. There, on July 3, during a half-hour lull in his friend’s backyard before leaving for a concert, the solution suddenly came to him. “I immediately realized that it would work,” he said.
Zhang’s idea was to use not the GPY sieve but a modified version of it, in which the sieve filters not by every number, but only by numbers that have no large prime factors.
“His sieve doesn’t do as good a job because you’re not using everything you can sieve with,” Goldston said. “But it turns out that while it’s a little less effective, it gives him the flexibility that allows the argument to work.”
While the new sieve allowed Zhang to prove that there are infinitely many prime pairs closer together than 70 million, it is unlikely that his methods can be pushed as far as the twin primes conjecture, Goldston said. Even with the strongest possible assumptions about the value of the level of distribution, he said, the best result likely to emerge from the GPY method would be that there are infinitely many prime pairs that differ by 16 or less.
But Granville said that mathematicians shouldn’t prematurely rule out the possibility of reaching the twin primes conjecture by these methods.
Emily Riehl, The n-Category Cafe, Bounded Gaps Between Primes, here. Notes from the Harvard talk.
Whether we grow up to become category theorists or applied mathematicians, one thing that I suspect unites us all is that we were once enchanted by prime numbers. It comes as no surprise then that a seminar given yesterday afternoon at Harvard by Yi Tang Zhang of the University of New Hampshire reporting on his new paper “Bounded gaps between primes” attracted a diverse audience. I don’t believe the paper is publicly available yet, but word on the street is that the referees at the Annals say it all checks out.
What follows is a summary of his presentation. Any errors should be ascribed to the ignorance of the transcriber (a category theorist, not an analytic number theorist) rather than to the author or his talk, which was lovely.
mathoverflow, Philosophy behind Yitang Zhang’s work on the Twin Primes Conjecture, here.
In the mid 2000′s Goldston, Pintz and Yildirim proved that if the Elliott–Halberstam conjecture holds for any level of distribution θ>1/2 then one has infinitely many bounded prime gaps (where the size of the gap is a function of θ, for θ>.971 they get a gap of size 16). Since Bombieri-Vinogradov tells us that EH(θ) holds for θ<1/2, in some sense the Goldston-Pintz-Yildirim arguments just barely miss giving bounded gaps.
On the other hand, in the 1980s Fouvry and Iwaniec were able to push the level of distribution in the Bombieri-Vinogradov theorem above 1/2 at the expense of (1) removing the absolute values, (2) removing the maximum over residue classes and, (3) weighting the summand by a `well-factorable’ function. This was subsequently improved in a series of papers by Bombieri, Friedlander and Iwaniec. For the Goldston-Pintz-Yildirim argument the first two restrictions didn’t pose a significant hurdle, however the inclusion of the well-factorable weight appeared to prevent one from using it with the Goldston-Pintz-Yildirim machinery.
This is where my knowledge becomes extremely spotty, but my understanding is that Zhang replaces the Goldston-Pintz-Yildirim sieve with a slightly less efficient one. While less efficient, this modification allows him some additional flexibility. He then is able to divide up the quantity that is estimated using (*) into several parts. Some of these are handled using well-established techniques, however in the most interesting range of parameters he is able to reduce the problem to something that involves a well-factorable function (or something similar) which he ultimately handles with arguments similar to those of Bombieri, Friedlander and Iwaniec.
Zhang’s argument reportedly gives a gap size that is around 70 million. My suspicion is that this gap will quickly decrease. In their theorem, Bombieri, Friedlander and Iwaniec give a level of distribution around 4/7, where (according to these notes) Zhang seems to be working with a level of distribution of the form1/2+δ for δ on the order of 1/1000, so there is likely room for optimization. As a reference point if one had the level of distribution 55/100 (<4/7) in the unmodified Elliot-Halberstram conjecture (without the well-factorable weight), the Goldston, Pintz and Yildirim gives infinity many gaps of size less than 2956.
The parity problem, however, is widely known to prevent approaches of this form (at least on their own) from getting a gap smaller than length 6. Of course, there are more technical obstructions as well and even on the full Elliott–Halberstam conjecture one can only get a gap of 16 at present.
evelynjlamb, AMS Blogs, This Week in Number Theory, here.
David Roberts and Terry Tao both wrote public Google Plus posts about the twin prime conjecture. The announcement also inspired Silveira Neto to create a twin prime version of the Ulam spiral, which you can see at the top of this post. Patrick Honner, a high school math teacher in Brooklyn, uses the result to look at how big the gaps between primes can get. “And while being 70 million away may not seem close as far as prime numbers go, consider the following amazing fact: given any number N, we can find a string of N consecutive numbers that contains no primes at all!”
Paul Krugman, Brookings, It’s Baaack: Japan’s Slump and the Return of the Liquidity Trap, here.
THE LIQUIDITY TRAP-that awkward condition in which monetary policy loses its grip because the nominal interest rate is essentially zero, in which the quantity of money becomes irrelevant because money and bonds are essentially perfect substitutes-played a central role in the early years of macroeconomics as a discipline. John Hicks, in intro- ducing both the IS-LM model and the liquidity trap, identified the assumption that monetary policy is ineffective, rather than the assumed downward inflexibility of prices, as the central difference between Mr. Keynes and the classics. ‘ It has often been pointed out that the Alice in Wonderland character of early Keynesianism-with its paradoxes of thrift, widows’ cruses, and so on-depended on the explicit or implicit assumption of an accommodative monetary policy; it has less often been pointed out that in the late 1930s and early 1940s it seemed quite natural to assume that money was irrelevant at the margin. After all, at the end of the 1930s interest rates were hard up against the zero con- straint; the average rate on U.S. Treasury bills during 1940 was 0.014 percent.
Peter Woit, Not Even Wrong, Number Theory News, here. Woit’s link to Caroline Chen’s The Paradox of the Proof (below) is pretty good for a Mochizuki/ABC conjecture summary and update.
A special seminar has been scheduled for tomorrow (Monday) at 3pm at Harvard, where Yitang Zhang will present new results on “Bounded gaps between primes”. Evidently he has a proof that there exist infinitely many different pairs of primes p,q with p-q less than
Whether this proof is valid should become clear soon, but there still seems to be nothing happening in terms of others understanding Mochizuki’s claimed proof of the abc conjecture. For an excellent article describing the situation, see here.
Ingrid Daubechies, What’s new, Planning for the World Digital Math Library, here.
This guest blog entry concerns the many roles a World Digital Mathematical Library (WDML) could play for the mathematical community worldwide. We seek input to help sketch how a WDML could be so much more than just a huge collection of digitally available mathematical documents. If this is of interest to you, please read on!
Michael Taylor, Bankers Anonymous, Part I – The Most Powerful Math in the Universe Goes Untaught, here. One vote for exp() and log() as being the most powerful math functions. Apparently the Goldman Sachs teaches this to new hires. Multipart series coming up.
And yet, nobody taught me compound interest or discounted cash flows in school. I’d be willing to bet that almost all of you reading this didn’t get taught these concepts in school. That knowledge had to wait until I started as a bond guy at Goldman. This, despite the fact that you only need junior high school level math – basically algebra and the concept of ‘X raised to the power of Y’ – to understand and use compound interest and discounted cash flows.
Jeremy Kun, Probability Theory – A Primer, here.
It is a wonder that we have yet to officially write about probability theory on this blog. Probability theory underlies a huge portion of artificial intelligence, machine learning, and statistics, and a number of our future posts will rely on the ideas and terminology we lay out in this post. Our first formal theory of machine learning will be deeply ingrained in probability theory, we will derive and analyze probabilistic learning algorithms, and our entire treatment of mathematical finance will be framed in terms of random variables.
And so it’s about time we got to the bottom of probability theory. In this post, we will begin with a naive version of probability theory. That is, everything will be finite and framed in terms of naive set theory without the aid of measure theory. This has the benefit of making the analysis and definitions simple. The downside is that we are restricted in what kinds of probability we are allowed to speak of. For instance, we aren’t allowed to work with probabilities defined on all real numbers. But for the majority of our purposes on this blog, this treatment will be enough. Indeed, most programming applications restrict infinite problems to finite subproblems or approximations (although in their analysis we often appeal to the infinite).
We should make a quick disclaimer before we get into the thick of things: this primer is not meant to connect probability theory to the real world. Indeed, to do so would be decidedly unmathematical. We are primarily concerned with the mathematical formalisms involved in the theory of probability, and we will leave the philosophical concerns and applications to future posts. The point of this primer is simply to lay down the terminology and basic results needed to discuss such topics to begin with.
So let us begin with probability spaces and random variables.
GLL, The Amazing Zeta Code, here. It’s like Zoolander, the analytic functions are in the Zeta function! Why am I learning about this now for the first time? Look at the link to the number theory and physics archive as well, here. Best thing you will browse for a week.
The Amazing Property
Let be any compact set of complex numbers whose real parts satisfy . Because is closed and bounded, there exists a fixed such that for all with real part . The part can be as close as desired to the critical line , but it must stay some fixed distance away.
We also need to have no “holes,” i.e., to be homeomorphic to a closed disk of radius . Often is simply taken to be the disk of radius centered on the -axis at , but we’ll also think of a square grid. Then Voronin proved:
Theorem 1Given any analytic function that is non-vanishing on , and any , we can find some real value such that for all ,
Gus Lubin and Joshua Berlinger, Business Insider, 13 Of The Greatest Idea Hunters Ever, here. Icky list includes both Erdos (ok and reasonable) and then inexplicably includes Jack Welch. Is this some irony play? Ewwwww Erdos and Jack Welch in the same book. Technical Foul on the bench at Business Insider. Or as Rasheed Wallace would say “Ball don’t Lie.”
That’s the message of The Idea Hunters, a new book by Andy Boynton of the Carroll School of Management at Boston College and Bill Fischer of IMD in Switzerland.
Recent advances in high-frequency financial trading have made light propagation delays between geographically
separated exchanges relevant. Here we show that there exist optimal locations from which to coordinate
the statistical arbitrage of pairs of spacelike separated securities, and calculate a representative map of such
locations on Earth. Furthermore, trading local securities along chains of such intermediate locations results in
a novel econophysical effect, in which the relativistic propagation of tradable information is effectively slowed
or stopped by arbitrage.
Solomon Feferman, Notices of the AMS, Turing’s Thesis, here.
Here, in brief, is the story of what led Turing to
Church, what was in his thesis, and what came
after, both for him and for the subject.
Kevin Hartnett, The Boston Globe, An ABC proof too tough even for mathematicians, here.
Mochizuki made a name for himself in his 20s based on a number of significant contributions to a relatively new, complex subfield of arithmetic geometry known as anabelian geometry. In 1998 he received one of the highest honors in math—an invitation to address the quadrennial International Congress of Mathematicians. He gave his talk in August 1998 in Berlin, then effectively went to ground, disappearing for 14 years to work on ABC.
“I guess he wanted to work on a problem worthy of putting his full powers on,” says Jeffrey Lagarias, a professor of math at the University of Michigan, “and the ABC conjecture fit beautifully.”
Before mathematicians can even start to read the proof, or understand his four papers, they need to wade through 750 pages of Mochizuki’s incredibly complicated foundational work in anabelian geometry. At the moment, there are only about 50 people in the world who know anabelian geometry well enough to understand this preliminary work. Then, the proof itself is written in an entirely different branch of mathematics called “inter-universal geometry” that Mochizuki—who refers to himself as an “inter-universal Geometer”—invented and of which, at least so far, he is the sole practitioner.
michaelnielsen.org, PolyMath, ABC Conjecture, here. Rolling log of blog posts and news. mathbabe explains the basics of the 1979 paper Social Processes and Proofs of Theorems and Programs.
Adam Fisher and Lindsey Drew, the guardian, Functional Programming Principles in Scala: First Impressions, here. Martin Odersky’s coursera class review.
This week we’ve set ourselves the challenge (along with most of the developers here at the Guardian) of undertaking Functional Programming Principles in Scala.Scala is the main programming language that we use at the Guardian, and through studying this course we hope to gain a deeper understanding of the language, learn new features and improve our programming style. The course is devised and delivered by Scala’s creator, Martin Odersky and we’re excited by the insights he’s sure to reveal.
Jeffrey Dow Jones, Cognitive Concord, Q-Infinity and Beyond! here.
The Federal Reserve has two mandates.
- Price stability
- Full employment.
Zerohedge, Will Nasdaq Be The Next Casualty Of The SEC’s Anti-Latency Arbitrage Push? here.
Back in 2009 Zero Hedge was first the only, and shortly thereafter, one of very few non-conformist voices objecting to pervasive high frequency trading and other type of quantitative market manipulation in the form of Flash Trading (which has recently reemerged in yet another form of frontrunning known as “Hide not Slide” practices) quote stuffing, and naturally latency arbitrage: one of the most subversive means to rob the less than sophisticated investor blind, due to an illegal coordination between market markers, exchanges and regulators, which effectively encouraged a two-tier market (one for the ultra fast frontrunning professionals, and one for everyone else). A week ago we were amused to see that the SEC charged the NYSE with a wristslap, one for $5 million dollars and where the NYSE naturally neither admitted nor denied guilt, accusing it of doing precisely what we said it, and all others, had been doing for years: namely getting paid by wealthy traders, those using the prop data feed OpenBook Ultra and other paid systems, to create and perpetuate a two-tiered market, all the while the regulator, i.e., the SEC was paid to look the other way.
Walter Hickey, Business Insider,Nate Silver Explains The Most Important Concept In Statistics, here.
Bayes theorem computes the posterior probability, or the probability that, given you found the underwear, your spouse is cheating.
The posterior probability is equal to:xy/[xy + z(1-x)]
In this case 29%, which is still fairly low.
Nate Silver — who founded the FiveThirtyEight blog, and has been analyzing polling data better than anyone else in his field for the past several years — has finally written a book, and it’s an achievement.
The Signal and the Noise is great for a number of different reasons. Silver aimed much higher than a Chicken Soup for the Statistician’s Soul.
Barry Cipra, Science, ABC Proof Could Be Mathematical Jackpot, here.
You can’t always solve a mathematical problem by reducing it to something you’ve already solved. Sometimes, you need to invent an entirely new field of mathematics. Last month, Shinichi Mochizuki of Kyoto University in Japan announced that a new field he’s been developing for several years—which he calls Inter-universal Teichmüller theory—has proved a famous conjecture in number theory known as the “abc conjecture.” But theabc conjecture is only the beginning: If Mochizuki’s theory proves correct, it will settle a raft of open problems in number theory and other branches of math.
Y combinator, Proof claimed for Deep Connection between Prime Numbers, here.
Not Even Wrong, Proof of the abc Conjecture, here.
Jordan Ellenberg at Quomodocumque reports here on a potential breakthrough in number theory, a claimed proof of the abc conjecture by Shin Mochizuki. More than five years ago I wrote a posting with the same title, reporting on a talk by Lucien Szpiro claiming a proof of this conjecture (the proof soon was found to have a flaw). One change over the last five years is that now there are excellent Wikipedia articles about mathematically important questions like this conjecture, so you should consult theWikipedia article for more details on the mathematics of the conjecture. To get some idea of the significance of this, that article quotes my colleague and next-door office neighbor Dorian Goldfeld describing the conjecture as “the most important unsolved problem in Diophantine analysis”, i.e. for a very significant part of number theory.
Quomodocumque, Mochizuki on ABC, here.
Shin Mochizuki has released his long-rumored proof of the ABC conjecture, in a paper called “Inter-universal Teichmuller theory IV: log-volume computations and set-theoretic foundations.”
I just saw this an hour ago and so I have very little to say, beyond what I wrote on Google+ when rumors of this started circulating earlier this summer:
I hope it’s true: my sense is that there’s a lot of very beautiful, very hard math going on in Shin’s work which almost no one in the community has really engaged with, and the resolution of a major conjecture would obviously create such engagement very quickly.
Well, now the time has come. I have not even begun to understand Shin’s approach to the conjecture. But it’s clear that it involves ideas which are completely outside the mainstream of the subject. Looking at it, you feel a bit like you might be reading a paper from the future, or from outer space.
mathoverflow, Philosophy behind Mochizuki’s work on the ABC conjecture, here. Gowers take:
I’ll take a stab at answering this controversial question in a way that might satisfy the OP and benefit the mathematical community. I also want to give some opinions that contrast with or at least complement grp. Like others, I must give the caveats: I do not understand Mochizuki’s claimed proof, his other work, and I make no claims about the veracity of his recent work.
First, some background which might satisfy the OP. For years, Mochizuki has been working on things related to Grothendieck’s anabelian program. Here is why one might hope this is useful in attacking problems like ABC:
Begin with the Neukirch-Uchida theorem. See “Über die absoluten Galoisgruppen algebraischer Zahlkörper,” by J. Neukirch, Journées Arithmétiques de Caen (Univ. Caen, Caen, 1976), pp. 67–79. Asterisque, No. 41-42, Soc. Math. France, Paris, 1977. Also “Isomorphisms of Galois groups,” by K. Uchida, J. Math. Soc. Japan 28 (1976), no. 4, 617–620.
Noam Elkies, The ABC’s of Number Theory, here.
The ABC conjecture is a central open problem in modern number theory, connecting results, techniques and questions ranging from elementary number theory and algebra to the arithmetic of elliptic curves to algebraic geometry and even to entire functions of a complex variable. The conjecture asserts that, in a precise sense that we specify later, if A, B, C are relatively prime integers such that A + B = C then A,B,C cannot all have many repeated prime factors. This expository article outlines some of the connections between this assertion and more familiar Diophantine questions, following (with the occasional scenic detour) the historical route from Pythagorean triples via Fermat’s Last Theorem to the formulation of the ABC conjecture by Masser and Oesterle ́. We then state the conjecture and give a sample of its many consequences and the few very partial results available. Next we recite Mason’s proof of an analogous assertion for polynomials A(t), B(t), C(t) that implies, among other things, that one cannot hope to disprove the ABC conjecture using a polynomial identity such as the one that solves the Diophantine equation x2 + y2 = z2. We conclude by solving a Putnam problem that predates Mason’s theorem but is solved using the same method, and outlining some further open questions and fragmentary results beyond the ABC conjecture.‡