The Notre Dame Mathematical Lectures series is now freely available online. It was a fairly small series of lecture notes on various topics. The most famous was probably Emil Artin’s lectures on Galois Theory.

Thanks to Greg Muller, I’m looking at this paper by Dror Bar-Natan that reduces the Four Color Theorem to a plausible statement about Lie algebras. Now we just have to hope this new conjecture does not not require hundreds of pages of computer generated proof.

The 25th Carnival of Mathematics is up at Walking Randomly.

Way back when, I had a post about explaining the Riemann hypothesis in elementary terms. I thought I’d go into some more detail.

The Riemann hypothesis is regarded as one of the outstanding open problems in mathematics. Part of the reason is that it has a certain mystique, since Riemann conjectured it back in 1859, and it has withstood many attempts to prove it since then. A bigger reason is that it solution (either positive or negative) is the main obstacle to answering the question “How many primes are there?”

The fact that there are infinitely many primes goes back to Euclid. The next most logical question is to ask how many primes there are less than a given number. Thanks to the Prime Number Theorem, we know that there are approximately n / ln n primes less than a given number. But this is only an approximation. How good or bad of an approximation is it? We don’t know. That is the question the Riemann hypothesis is trying to answer.

On the off-chance anyone else comes along for the Carnival… Sometimes, when I’m asked what mathematicians do, I’ll start talking about something related to geometry in more than three dimensions. One question I occasionally get is “But what is the fourth dimension?” (The more physics literate will say something like “ I know that time is the fourth dimension, but what would be the fifth dimension?”) I usually try to explain that higher dimensions are abstract concepts, and that we understand them through analogies.

Along those lines, here are two facts about the fourth dimension that seemed inexplicable to me when I first heard them, but now seem obvious:

1. In three dimensions, if two planes intersect they must intersect in a line. In four dimensions, two planes can intersect at a single point.
2. In three dimensions, if you try to roll up a piece of paper into a torus, you have to crinkle the paper to close up the tube into a torus. In four dimensions, you could do it without crinkling the paper.

I thought, since for the next couple of days we’ll probably have a broader audience than we usually do, that I would do posts on some more elementary topics than usual. I’m going to try to explain an odd example from topology in elementary terms, without (hopefully) butchering the math too much.

Put a rubber band around a balloon, and then blow it up. By stretching the rubber band, you can take it off the balloon. Now imagine that you first tie the balloon in a complicated balloon animal shape. Can you still get the rubber band off? You can imagine that the balloon is tied so tightly that there isn’t enough room to squeeze the rubber band by, but this is a perfectly flat mathematical rubber band we’re talking about here; no spot is too tight to squeeze through. Given that, can you still get the rubber band off?

The mathematical answer is no. Alexander discovered a counterexample in 1924 now known as the Alexander horned sphere. You can twist the balloon into a strange fractal shape with infinitely many interlocking horns so that the rubber band cannot be pushed past all of the horns. (The practical answer is yes, since you can’t really twist a balloon into a fractal shape, and to interlock the horns I think you have to surreptitiously cut up the balloon and glue it back together when no one is looking.)

Welcome to the 24th Carnival of Mathematics!

24 dimensions is the home of the mysterious Leech lattice. The Leech lattice can be used to answer questions in 24 dimensions, such as the densest regular sphere packing or the kissing number. The answers to these questions are not known in any dimension larger than 8 other than 24. The Leech lattice can be used to construct other exotic objects in mathematics, such as sporadic simple groups.

Back here in three dimensions:

Mathew Needleman presents Kindergarten Math Skills Predict Future Success posted at Creating Lifelong Learners. Matthew comments on a study that shows that at the kindergarten level, success at mathematics best predicts future academic success.

Kevin OConnor presents Mental Maths Shortcut 5 squared Genius | MemoryMentor’s Blog posted at MemoryMentor’s Blog. Kevin shows how easy it is to square numbers that end in 5.

praveen presents How Long Are The Candles? posted at Math and Logic Play. This is a word problem first posed in Ask Marilyn.

Denise presents 2008 mathematics game posted at Let’s play math!. The game consists of asking how many numbers can you make out of the digits of 2008 using basic math operations such as division and factorial.

Sol Lederman presents Impressive Math magic with 16 index cards posted at Wild About Math!.

Maria H. Andersen presents Animated Demo of Domain and Range Projections posted at Teaching College Math Technology Blog. Maria shows you how to animate your Powerpoint slides to show students how to find the domain and range of a graph.

Mike surveys the mathematics software available on the Pocket PC, from calculators to spreadsheets to computer algebra systems, in Mathematics on the Pocket PC, posted at Walking Randomly.

Dave Marain presents M^2 – N^2 = 12…Prove there is only one solution in positive integers and much more and An Introduction to the Mathematics of Bingo – Part I: An Investigation for Grades 7-12 posted at MathNotations. The first post leads to a debate in comments as to whether the question is too hard for high school students. The second post illustrates the complexity of Bingo.

jonathan presents Gazinta – two remainder puzzles to kick things off posted at JD2718.

Brent Yorgey shows you an elegant enumeration of the rationals with Recounting the Rationals, part II (fractions grow on trees!) posted at The Math Less Traveled.

Blake Stacey is writing a series of posts Science after Sunclipse on supersymmetric quantum mechanics. The first post introduces the basics of superalgebra, while the second post uses this to solve some actual quantum mechanical systems.

Maurizio Monge points out that a new preprint has appeared at arXiv with the title Lindelof’s hypothesis is true and Riemann’s one is not. This paper has already been withdrawn by its author, so the Riemann hypothesis live to fight another day.

The next Carnival of Mathematics will be hosted at Walking Randomly.

Wikipedia has an interesting page on lacunary series, which are Taylor series or Fourier series with sparse coefficients. Lacunary series can be used to give examples of phenomena such as a complex analytic function which has essential singularities at every point of the unit circle, or a function whose Fourier series diverges almost everywhere.