Ars Mathematica

A new issue of the Bulletin of the AMS, Volume 32, Number 3, has hit the virtual newstands.

Charles Weibel, who wrote the best book on homological algebra is now working on a book on algebraic K-theory. He has posted drafts of the first four chapters at An introduction to algebraic K-theory.

Well, okay, not really lies, but I formed ideas in my abstract algebra class that I later had to unlearn:

• Most integral domains are unique factorization domains. In reality, integral domains are almost never UFDs. Beyond the examples usually taught, there is one additional large class of UFDs, regular local rings, and then just scattered examples. If you take a polynomial ring in two or more variables, and mod out by any random prime ideal, you will almost certainly not get a UFD. For example, in the ring k[x,y]/(x^2+y^2-1), x^2 also factors as (1-y)(1+y)
• Abelian groups are a direct sum of cyclic groups. While this is true for finitely-generated abelian groups, it is far from true for infinitely-generated abelian groups, even if you consider infinite direct sums. A typical infinitely-generated abelian group is Q, which cannot be written as a direct sum of any subgroups, but very much not cyclic.
• Finite-dimensional noncommutative division rings over the rationals are all subrings of the quaternions. The definition of the quaternions makes sense with coefficients in any subfield of the reals, and gives you a finite-dimensional division ring over that subfield. If the subfield is itself finite-dimensional over Q, this gives you a finite-dimensional division algebra over Q. I thought that this construction gave you all of the possibilities. This is far from the case. The quaternions are 4-dimensional over their center, but you can construct other division algebras of any dimension over their center, as long as that dimension is a perfect square.

Did this happen to anyone else?

Mathematical Sciences Research Institute (MSRI, usually pronouned “misery”) has put all of the books it has produced since 1995 online. They also have made available Thurston’s 1980 lecture notes on the Geometry and Topology of Three-Manifolds, which has long served as a major source for Thurston’s approach to 3-manifolds.

The American Mathematical Society has put many of its books online. Several of the books available are classics, such as Jacobson’s Structures and Representations of Jordan Algebras, Lefschetz’ Algebraic Topology and Birkhoff’s Dynamical Systems.

I found an elementary introduction to the Catalan numbers at Tom Davis’ web site. The Catalan numbers arise in several counting problems, such as counting the number of ways of parenthesizing an expression and the number of ways to cut up a polygon into triangles.

Zach Teitler has written some notes on how to explicitly compute Chern classes in algebraic geometry.

Hugh Woodin has two survey articles on recent work on the Continuum Hypothesis: I and II. Most mathematicians consider the continuum hypothesis as a settled question: since it is independent of ZFC, its truth is unknowable.

Set theorists, on the other hand, sometimes hold out the hope that new, intuitive axioms will be found that will provide a definite answer. Woodin thinks that we are close to finding such an axiom, and it seems to indicate that the cardinality of the reals is aleph two. (The continuum hypothesis states that it’s aleph one.)

There is a simpler example of an intuitive result that implies that the continuum hypothesis is false. Details can be found here and here.

E. Lee Lady, a mathematician at the University of Hawaii, has a terrific collection of lecture notes in algebra. He also has posted a draft manuscript of a book on torsion-free modules over Dedekind rings, which years of graduate school brainwashing will convince you are the natural generalization of the ring of integers.

Leonhard Euler is one of the most prolific mathematicians in history. So prolific, in fact, that he has posted 14 articles to ArXiv, despite being dead for 222 years.

The Euler Archive has many more papers of Euler’s, both in the original (either Latin or French) and in translation.