Sean at Cosmic Variance has written a nice introduction to symmetry breaking, one of the oddest (and most successful) ideas in particle physics.
Circle Packing Contest
Al Zimmermann is running a programming competition to see who can:
Pack n non-overlapping circles with radii from 1 to n into as small a circle as possible.
You have until mid-January to come up with solutions for n between 5 and 50.
(via Lamba the Ultimate).
Famous Errors?
The discussion about Goedel’s theorem made me wonder about this question: when was the last time a widely quoted mathematical result turned out to be false? I’ve seen preprints that had proofs I didn’t believe, and I know that journals occasionally print results that are wrong, but does anyone know of a result that was once widely accepted, but then turned out to be wrong?
Goedel’s Theorem is True
I’m sorry if my last post misled anyone. Goedel’s theorem is true. Lots of people have checked the proof. I’ve checked the proof. There are multiple proofs (the usual proof is Rosser’s, not Goedel’s original proof), as well as vast generalizations.
Here’s a two line proof of the theorem (some details are left out):
- If Goedel’s theorem is false, then the Halting Problem for Turing machines is solvable.
- The Halting Problem is unsolvable, therefore Goedel’s theorem is true.
At the time, Goedel’s result was very surprising, but by modern standards it’s almost obvious. The set of provable theorems of arithmetic is recursively enumerable. It’s not that surprising that first-order arithmetic would be at least as expressive as Turing machines, which implies that the set of provable theorems is not recursive. Goedel’s theorem follows.
I intended the post to be tongue-in-cheek. My real point was that a certain number of papers on Arxiv are apparently written by cranks.
Goedel’s Theorem is Invalid
According to a new paper on Arxiv, Goedel’s theorem is false. There you have it.
Euler angles for SU(n)
This paper, On the Euler angles for SU(N), gives a definition/derivation of Euler angles for SU(n). The paper claims that the construction first appeared in 2002. Does anyone know if that’s true? I thought for sure that I’d seen Euler angles defined for SU(n) (and SO(n)) sometime in the 90s.
Baez Week 222
Week 222 of This Week’s Finds in Mathematical Physics is up.
The Hilbertization of Sports
I was websurfing a football site, Football Outsiders, when I came across a link to an essay called, of all things, Football’s Hilbert Problems. This is not an isolated phenomenon. It’s modeled after a essay from 2000, Baseball’s Hilbert Problems.
Nobel Prize in Economics
The (sort-of) Nobel Prize in Economics has been announced. The contributions of one of the two, Robert Aumann, are almost purely mathematical work in game theory.
His most interesting idea is that of correlated equilibria. The usual definition of equilibrium in a non-cooperative game, Nash equilibrium, rules out certain kinds of cooperation, even when that cooperation is in the self-interest of each player. Correlated equilibria allow randomized strategies which rely on a random event that is known to both players. Some details about correlated equilibria can be found
here.
(As an aside, the Wikipedia entry for Aumann is unusually bad, so it’s a good candidate for updating, if anyone’s interested. There’s also no entry for correlated equilibrium.)
November Notices of the AMS
The November issue of the Notices of the AMS is available.
Some highlights:
- What is the Role of Algebra in Applied Mathematics, by David A. Cox. Cos is the author of one of the best undergraduate introductions to algebraic geometry, Ideals, Varieties, and Algorithms. The book takes a computational approach based on the technique of Groebner bases.
- What are Groebner bases, you might ask? Bernd Sturmfels answers that question in the latest What is….
- The most intriguing article is Generalized Fourier Transforms, Their Nonlinearizations, and the Imaging of the Brain by Fokas and Sung
.