Ars Mathematica

I’m intrigued by the beginning of a new series of posts at the Everything Seminar about harmonic analysis. This particular post talks about the relationship of singular integral operators and Carleson’s Theorem. Carleson’s Theorem (that Fourier series of functions in L^p for p > 1 converge pointwise almost everywhere) is a famously difficult result; the post gives some idea of where the difficulty lies.

For the ambitious, a complete proof is available in a preprint by Michael Lacey.

Calculus has been the subject of immense amounts of educational material, ranging from textbooks to blog posts. Unfortunately, that is now all obsolete. The definitive presentation of calculus is here: Calculus, the Musical.

Brad de Long, an economist, has a post up about the significance of how he dresses for specific audiences. In particular, the consequences of wearing ties:

With math-oriented students, however, a tie tells them that I spend too little time thinking about isomorphisms.

(This inspired n-category jokes in the comments.)

Charles at Rigorous Trivialities has written a post outlining the proof of two pretty theorems from the invariant theory of finite groups: Noether’s theorem that the ring of invariants is finitely generated, and Molien’s formula for the number of homogeneous invariants of a given degree.

Not only is John Armstrong a failed crackpot, he is wrong about statistics. Statistics is, from the mathematical point of view, a perfectly interesting subject; this fact is carefully concealed from us by statisticians. For example, most mathematicians know the central limit theorem, which says that the sum of large numbers of independent, identically distributed (iid) random variables tend to be normally distributed. This even has an elegant proof in terms of Fourier analysis, where addition of random variables because multiplication of Fourier transforms.

What mathematicians don’t know is that almost every other statistic ever defined also satisfies the central limit theorem. The median of a large number of iid random variables? Normally distributed. The mode of a large number of iid random variables (where the underlying distribution has a single mode)? Normally distributed. The cosine of the seventeenth percentile? Normally distributed. The simplest explanation for this cavalcade of normality involves the Gâteaux derivative in functional analysis.

Isabel at God Plays Dice finds the definitive review of Dummit and Foote, here at adequacy.org.

(If you have any questions as to the objectivity of adequacy.org, I suggest checking out their Wikipedia page.)

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.