Foreign Dispatches

John Tate Wins the 2010 Abel Prize

Seeing as I've been getting quite a lot of traffic of late from people looking for a "For Dummies"* version of Perelman's Poincare Conjecture proof, this seems an especially opportune time to mention that John Tate has just been awarded this year's Abel Prize.

If you've ever read a book on modern algebraic number theory or know what an elliptic curve is, you'll already know who I'm talking about; if not, let me point you in the direction of a very good book Tate co-authored with John Silverman - "Rational Points on Elliptic Curves". Yes, it's a textbook, and a math textbook at that, but as math textbooks go they don't get much more fun and readable than this one does, and unlike Terence Tao's Poincare conjecture outline, this is a book any reasonably intelligent person should be able to work through (I know as I did precisely that at the ripe old age of 17). There are good reasons why this book has been on my side panel for several years now, I assure you.

*By which I really meant " 'Dummies' smart enough to make it through mathematics graduate school" ... Sorry about that, folks.

Perelman's Poincaré Conjecture Proof for Dummies

Where by "dummies", I mean "readers with a background of at least 1 year of mathematics graduate school" (so I exaggerate slightly). Terence Tao has created an interesting, relatively brief document providing an outline of what Grigori Perelman's proof of Thurston's Geometrization Conjecture (which implies the Poincaré Conjecture) is all about, in terms that readers with a basic understanding of algebraic topology and differential geometry can understand (not that those outside this relatively small set of individuals are likely to care anyway).

Lectures on the Poincare Conjecture

And now for something completely different! It turns out that Terence Tao's recently begun teaching a class at UCLA on Grigori Perelman's proof of the Poincare Conjecture, and for the benefit of all those of us who can't be there to enjoy the benefit of his teaching, he's been putting up his lecture notes on his weblog. If you have the requisite background to appreciate what Tao's writing about, I think you'll find his notes wonderfully lucid. Highly recommended.

Proof of Russia's Rigged Elections

It's one thing for the likes of Gary Kasparov or observers from the European Union to complain about "irregularities" in the recent parliamentary elections held in Russia, and something else altogether when someone uses a little elementary statistics to confirm beyond all doubt that Putin and company engaged in fraud on a massive scale. The numbers are indisputable: the 2007 Russian parliamentary elections are as meaningless as any Soviet elections would have been in 1967.

[Via Slashdot.]

Absence of Evidence *is* Evidence of Absence

Or why Bayesian thinking strongly suggests that gods, space-faring aliens, yetis and race-linked "IQ" genes are mere figments of undisciplined imaginations.

Regexp Magic

Anyone care to guess what the following Perl code does?

perl -wle '$_ = 1; (1 x $_) !~ /^(11+)\1+$/ && print while $_++'

Impressive, isn't it?

A Really Simple Math Problem

It's been ages since I last posted any math problems on here, mostly because the response to previous attempts was almost always poor, but this time I've got a nice little challenge which is simple enough for almost anyone to get with just a little bit of thought. Here it goes:

When you divide 12 by 5, the remainder is 2; it's what's left over after you have removed all the 5s from the 12. When you raise 4 to the fifth power (that is, 4^5), you multiply four by itself five times: 4×4x4×4x4, which equals 1,024.

What is the remainder when you divide 100^100 by 11?

It's not a trick question, but the solution is as straightforward as it gets if you go about it correctly.

Now Everyone's a Blogger

Via Ars Mathematica I learn that Fields Medalists Alain Connes and Terence Tao are now venturing into blogging. It's nice to see people with things actually worth saying participating in this medium, but on the other hand I almost feel guilty for pointing out the existence of their blogs, as it's only a matter of time before they draw the usual allotment of axe-grinders, cranks and assorted nutjobs: there are better things for the finest minds of our day to be occupied with than troll management.

An Introduction to the Lambda Calculus

And now for something different. Are you a programmer who's made a run in or two with esoteric languages like Haskell, Erlang and OCaml, and do you find yourself confused by closures, maddened by monads and feeling downright cantankerous whenever you hear about currying? If you are, I have a suggestion for you: instead of wasting your time struggling through yet another functional programming tutorial whose writers implicitly assume that you already understand half of the new things they're supposedly trying to teach, you'd be better off giving a close reading to a paper by Henk Barendregt and Erik Barendsen called "Introduction to Lambda Calculus". This paper gives a ground-up, axiomatic treatment of the subject from first principles, and what you learn from it will have immediate application with any programming language with functional aspects, not just one particular dialect like Ruby or Lisp. You might also find this online tutorial useful.

On a tangential note, I definitely get the feeling these days that functional programming has finally come of age. First you have languages like Java and C# starting to tackle closures, then we see OCaml and (especially) Haskell showing up more and more on places like Google Blogsearch and programming.reddit.com. This is a Good Thing™ for software development as a whole (no side effects => many fewer obscure bugs), though I believe the additional intellectual sophistication required to truly understand what functional programming is all about will likely prove a barrier too high for many programmers to surmount, just as many a procedural coder never really figured out object-oriented programming. Not that "OOP only" types are going to be out of work any time soon - just look at how many PHP coders there are out there churning out write-only procedural spaghetti.

The Mathematics of Beer Goggles

Ever wanted to know why it is that the more you drink, the more attractive the people you see in a bar seem to be? Ever wanted to be able to precisely quantify the magnitude of this effect? Well, you need suffer from ignorance no longer: the site "Howstuffworks" provides precise answers to both of these vexing questions.

In fact, the answers provided are so precise that I'm calling "bullshit" on this one. Just look at the formula on that page and tell me that it looks even slightly plausible to you - who's ever known of a behavioral equation uniformly describing all humans without any variation whatsoever, and why does this one so serendipitously do so without a need for a fudge constant of some sort, something even theories as precise as general relativity and quantum mechanics can't do without? At best this is junk research, at worst a hoax perpetuated on gullible BBC reporters who seem to be the original sources for this nonsense; if the inventors of this prattle had really wanted to do a good job of fooling the less ignorant, they'd have used a linear regression of some sort with an error term, rather than an equation with no fudge constant, but quadratic terms in the numerator and the denominator (and the latter stuck next to a square root at that).

I predict that this ridiculous story will be propagated uncritically by many, many blogs over the next few days, just like the nonsense over "blondes dying out" from a while back. It will be interesting to see how my forecast pans out.