# Reforming the Calculus Class, Permanently

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.

# Dressing to Impress Mathematicians

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.)

# Noether and Molien’s Theorems

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 GÃ¢teaux derivative in functional analysis.

# Dummit and Foote panned

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.)

# Notre Dame Mathematical Lectures

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.

# Why the Riemann Hypothesis?

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.

# The Fourth Dimension

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.