J.H. Conway, Contemporary Mathematics, Universal Quadratic Forms and the Fifteen Theorem, here.

The representation theory of quadratic forms has a long history, start- ing in the seventeenth century with Fermat’s assertions of 1640 about the numbers represented by x2 + y2. In the next century, Euler gave proofs of these and some similar assertions about other simple binary quadratics, and although these proofs had some gaps, they contributed greatly to setting the theory on a firm foundation. Lagrange started the theory of universal quadratic forms in 1770 by proving his celebrated Four Squares Theorem, which in current language is expressed by saying that the form x2+y2+z2+t2 is universal. The eighteenth century was closed by a considerably deeper statement – Legendre’s Three Squares Theorem of 1798; this found exactly which numbers needed all four squares. In his Theorie des Nombres of 1830, Legendre also created a very general theory of binary quadratics.

Peter Woit, Not Even Wrong, More Quick Links, here. Tao on Navier Stokes story from Quanta and SCIgen papers in peer review journals.

First, a couple of examples of recent progress in mathematics

• Terry Tao has some new ideas about the Navier-Stokes equation. See his blog here, a paper here, and a story by Erica Klarreich at Quanta here.

Damiano Brigo, Imperial College, Interest Rate Models with Credit Risk, Collateral, Funding Liquidity Risk and Multiple Curves, here. Nice comprehensive slides.

Andy Kiersz, BI, A Kazakh Mathematician Claims To Have Solved An Enormously Important Equation, here.

A huge mathematical breakthrough might have just been made, but a language barrier is slowing things down.

New Scientist reports that Kazakh mathematician Mukhtarbay Otelbayev may have solved an extremely difficult and useful mathematics problem: the Navier-Stokes equations.

This is another example of why you simply go directly to Tao, Gowers, and Lipton to see what is actually going on. The drama here is off the charts huge, you have a couple weeks until Tao presents his take on the Navier Stokes approach. He is thinking counterexample based on the non geometric approach. From Tao’s comment log:

Villatoro’s blog has some detailed analysis, and has recently raised some serious issues with the paper as well, in the crucial Section 6. (It’s in Spanish, but this is easy to translate online.)

I can’t read Russian either, so I am happy to defer the detailed checking to others, but my feeling is that this sort of abstract approach to the regularity problem, using only the energy identity and harmonic analysis estimates on the nonlinearity rather than more precise geometric information specific to the Navier-Stokes equation (e.g. the vorticity equation) is necessarily doomed to failure. I think I can formalise a specific obstruction in this regard and hope to present it here in a couple weeks.

Seriously, Netflix guys, you need something to replace CSI: Miami. This is way better –  Clay Scene Investigation: Los Angeles, get Keanu to play Tao in the series. Get Goldblum to grow a beard to play Lipton. Jeremy Irons for Gowers? Get Shelby Lyman from Fischer Spassky 1 to be the announcer for the show.  I’ll even spot you the elevator pitch – Reality TV meets Police Investigation Drama. Just have your writers point their feedlys to these blogs and the screenplays just pop out. You got Perelman, Twin Primes,  the ABC  conjecture, that Deolalikar P!=NP proof what else do you need to get started? There’s a bunch of guys working right now deep inside Fine Hall to save the proof that Peano Arithmetic is inconsistent. If they get it, all the math proofs in all the journals and math books have to be rechecked 1 by 1. A generation of young mathematics PhDs will have to be consigned to the old proof checking mines, like in that Indiana Jones movie Temple of Doom. The proposed show goes  like that Vaya Con Dios scene in Point Break, with a little more math thrown in. I assume that the UCLA Math department is in Malibu, right? That is all.

Cory Doctorow, Boing Boing, Why the sum of all positive integers is -1/12, here.

Here’s a brain-meltingly cool proof of the bizarre mathematical truth that the sum of all positive integers (1 + 2 + 3 + 4 + 5….) is -1/12. This is not only provably true, it’s also foundational to certain testable elements of physics. In other words: not just a logical curiosity, but also the bedrock of real-world, useful stuff.

Michael Segal, Nautilus, The Twin Prime Hero, here. Total Stud

What would you do with the money?

Maybe the best way would be to give the money to my wife. Let her deal with this issue.

The paper is quite large now (164 pages!) but it is fortunately rather modular, and thus hopefully somewhat readable (particularly regarding the first half of the paper, which does not  need any of the advanced exponential sum estimates).  The size should not be a major issue for the journal, so I would not seek to artificially shorten the paper at the expense of readability or content.

The new issue of Nautilus has a wonderful story about Yitang Zhang, called The Twin Prime Hero, which includes a long interview with him. Zhang’s remarkable mathematical career includes several years working at a Subway in Kentucky. His sucessful work on the twin prime conjecture (see here) was done over four years, working seven days a week without almost any breaks, while teaching two classes at a time.

For the next two semesters, I organized a seminar with five graduate students (including Yitang) on Prof. Hironaka’s monumental papers on the theory of resolutions of singularities. I believed that we doubled the world population of those who had studied the papers after we finished two semesters. Prof. Grothendick once described those papers as among the most complicated thesises in the human history.