Happenings – 2011 Nov 12

This post is late going out– but at least it will go out Saturday night rather than Sunday. I had to do some homeowner things late this morning, and it took me all day to get back to this.

First off, the good news: I have a post for this coming Monday – how to eliminate multicollinearity from the Hald data. It awaits only final edits– barring an emergency, it should go out Monday evening. Finally, after 5 consecutive diary posts, I’m putting out a technical post again.

Let me add one more piece of mathematics which I think a math geek should know: there must be 12 pentagons. That is, we can build a surface from hexagons and pentagons subject to 2 constraints: there can be any number, including zero, of hexagons except just one; and there must be exactly 12 pentagons. We can make a soccer ball almost any way we like – but it must have exactly 12 pentagons and it cannot have only 1 hexagon.

Now, that’s not important mathematics – it’s just weird mathematics– one might say it’s more geeky than it is mathematical.

While I’m at it, let me add another topic for our well-rounded math geek: Fermat’s Last Theorem.

What is it? Although there are infinitely many sets of nonzero integers a, b, c such that

a^2 = b^2 + c^2\

(3,4,5; and 5, 12, 13 come to mind), there are no (nonzero) integers such that

a^n = b^n + c^n\

for n an integer greater than 2. In particular, there are no (nonzero) integer solutions to

a^3 = b^3 + c^3\

or

a^4 = b^4 + c^4\ .

Why is it called his last theorem? Because Fermat left several unproved, and this was the last one to be either proved or disproved – i.e. the last one to be settled.

What’s the big deal? Fermat wrote, in the margin of a book, that he had a proof, but the margin was too small to contain it. That was in the 1630s, more than 350 years ago.

Who proved it? Andrew John Wiles, published in 1995.

How did he prove it? He showed that every elliptic curve (with some restrictions) is modular.

What does that mean? Well, an elliptic curve is an equation saying that y^2 is a cubic polynomial in x: y^2 = a x^3 + b x^2 + c x + d\ .

That’s fine for part of the statement– as for “modular”, I’ve mentioned modular forms only once, I think, and I didn’t say much. And that’s all I’m ready to say. It’s too complicated.

(More interesting is to read the history of the Taniyama-Shimura conjecture – that every elliptic curve is modular – or to watch the PBS (Nova?) episode about the conjecture.

Do we really think Fermat had a proof? No – someone would have figured it out again over the years. (We do believe that Fermat had a proof for fourth powers, n=4.)

As for the rest of the week, I bought a bunch of books. It seemed like it had been too long.

My visitor and I hit a Barnes and Noble– I picked up 3 new fantasy and science fiction books by authors I like: Mercedes Lackey, Verner Vinge, and Laurel K. Hamilton. (Yes, I read Valdemar and Anita Blake stories!)

I also happened to check Amazon for volume 3 of Zeidler’s “quantum field theory” – and it existed– so I ordered it– and it came– and my kid has read the 70 page prologue– and now I want to do differential geometry again.

My friend and I also hit the Stanford University bookstore, for the Springer yellow sale. I grabbed 4 books– 3 of which look good. First was Perrin’s “Algebraic Geometry, An Introduction”; it reads well, and may be the introduction I’ve been looking for.

Second was Roman’s “Field Theory” (Field extensions and Galois Theory); just as I have too many books on quantum mechanics, I have too many books on Galois theory – but this one looks nice. Of course, most of the ones I have look nice – that’s why I bought them.

Third was “The Mathematical Coloring Book” by Soifer – coloring the plane, coloring graphs, coloring maps, and a whole lot more. It looks fascinating. (And it wasn’t on sale, but I got it for half price because I’m in the Stanford Book Club.)

I regret buying the 4th book, Serre’s “Algebraic Groups and Class Fields” – If I had read the opening paragraph of chapter 2… “For the definitions and elementary results related to algebraic varieties and sheaves, I refer to my memoir on coherent sheaves.”… I wouldn’t have bought it. Having looked on Abebooks, I doubt that the original French book on coherent sheaves is available, never mind an English translation.

So I have the second volume of a set. My bad.

Oh, and I bought and watched “Harry Potter and the Deadly Hallows, Part 2” last night, after watching Part 1 again. That was easy.

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