I just ran across an interesting weblog. Henry Wilton taught a course in the spring on geometric group theory, which is the use of topological techniques to prove theorems in group theory. He had his students write up the lecture notes for each lecture as a separate post on the class weblog, 392C Geometric Group Theory. I’m sorry that I didn’t find this site when the course was on-going. It would have been interesting to follow along with the class in real time.

The tail spikes of stegosauruses are known in palaeontology as the thagomizer. Yes, after the Far Side.

The Polish Virtual Library of Science (which contains Banach’s book) has several other interesting mathematical works in English, French, German, and Polish. They have the archives for several journals, including Studia Mathematica. They also have several older but famous monographs, such as Zygmund’s Trigonometric Series.

Stefan Banach’s famous monograph, Théorie des opérations linéaires, which gave birth to the field of Banach spaces, is available online.

I’ve been doing some more online reading on the Positivstellensatz. I had blithely assumed that the polynomials that are non-negative for all real values were given by sums of squares of polynomials, but this is false. What is true is that a polynomial is non-negative if and only if it can be written as the sums of squares of rational functions, but this is a nontrivial result. In fact, showing this was Hilbert’s seventeenth problem. (You can derive it from the Positivstellensatz.)

If you were to try to axiomatize the idea of positivity inside a commutative ring in such a way that the same set of axioms cover both positive real numbers and non-negative polynomials, you would include axioms such as the sum and product of two positive elements are positive, and that squares are always positive. Hilbert’s seventeenth problem shows that you need an additional axiom: if a and ab are positive, then so is b.

In algebraic geometry, the Nullstellensatz gives an algebraic characterization of when a multivariate polynomial vanishes on a set of points defined by a system of (complex) polynomial equations.

I’ve just come across an analogue of this over the reals. In this case, it’s a purely algebraic characterization of when a multivariate polynomial is guaranteed to be positive over a set of points defined by a system of polynomial inequalities. In analogy with the complex case, the result is known as the Positivstellensatz.

Sorry about the long gap between posts. I’ve been unusually busy. I expect to be busy for the next month or so, and then I’ll try to resume a more-normal posting schedule.

Of course, by then, Ars Math’s entire readership will consist of people who forgot to delete us from their RSS readers…

Wow. The pharmaceutical company Merck put together a fake medical journal, the Australasian Journal of Bone and Joint Medicine. It was published by everybody’s favorite academic publisher, Elsevier.

Cognition and Culture, a weblog devoted to cognitive science and anthropology, has an interesting post about folk theories of physics. People naturally subscribe to a view they call “impetus physics”, where objects only move if they receive an impetus from an outside agent, which leads to a variety of false predictions. Physicists learn to give the right predictions, but a recent experiment showed that when asked to make intuitive predictions under some circumstances physicists will revert to the intuitive folk theory.

Did you ever wonder what Jim Stewart of Stewart’s Calculus did with his ill-gotten millions? Apparently, he built a really big house one with a concert hall in the middle. In addition to his textbook writing, Stewart is also a classical violinist, and he built the hall so that he and others could use it to perform.