On ArXiv there is a new survey paper on finite-dimensional algebras and quivers. The paper is rather dense, so it would be tough going for someone not already familiar with the vocabulary of quivers, but it covers some of the surprising connections with Kac-Moody Lie algebras.
Peter Woit’s weblog is an interesting source for information about the intersection of math and physics. His latest is a post on the early history of using group theory in quantum mechanics. While group-theoretic methods in physics (and chemistry) are uncontroversial these days, the original emergence of the subject was painful, with pro- and anti-group theory partisans. (Wolfgang Pauli termed group theory the “Gruppenpest”.)
About a month ago, Slashdot had linked to a completely opaque article about something called the Serre conjecture. The subsequent thread showed Slashdot at its worst, as people desperately typed “Serre conjecture” into Google to find links to stick in a comment to earn some undeserved karma. Unfortunately, Serre was an influential and productive mathematicians, so there are lots of Serre conjectures. One particularly unfortunate soul hit upon a link to the Quillen-Suslin theorem, which was conjectured by Serre in the 1950s but proven in 1976, and complained “do people not do research any more to see if their work has already been done?”
The Serre Conjecture in question is a completely separate conjecture in Galois theory. Many questions in number theory can be reduced to studying the Galois group of the algebraic closure of the rationals over the rationals. This group is gigantic and extremely complicated, so mathematicians try to understand it in pieces, but the pieces themselves are hard to come by. Serre conjectured one approach to understand certain small pieces. The paper in question, by Chandrashekhar Khare, proves a part of that conjecture. So Khare proved a part of a conjecture that only provides a small part of the solution of the general problem, but even that small part has remained unsolved for 18 years. So this is a hard problem we’re talking about. The truly macho can download the paper itself at ArXiv.
John Baez, of This Week’s Finds in Mathematical Physics fame, has a new article The Octonions.
The octonions are a mysterious example in mathematics that have been drawing attention in physics. Initially discovered by Graves in 1843, the octonions provided the first example of a number system with nonassociative multiplication: (ab)c is not equal to a(bc). Lots of examples of nonassociative multiplication are known, but the octonions remain the most interesting. For a more elementary introduction to the subject, there’s the article on Wikipedia.