MSI Weekly Bulletin - Week starting Monday 28 July, 2008

I will give an(other) introduction to Witt vectors and Lambda-rings. My main goal is to do this in the simplest possible way. The only prerequisites are the basic definitions of commutative algebra (rings, homomorphisms,...) and some categorical language, such as adjoint functors. In two weeks, I'll explain why all this is interesting.

By a "billiard" I mean a bounded plane domain D, with smooth (enough) boundary. Quantum billiards is the study of properties of eigenfunctions of the Laplacian on D, i.e. solutions of $\Delta u = Eu$, where $u$ is a function on D vanishing at the boundary and $E$ is a real number, in the limit as $E \to \infty$. This large-E limit is the "classical limit" in which eigenfunctions exhibit behaviour related to the classical billiard system (a billiard ball moving around inside D, bouncing off the boundary). I will talk about quantum Ergodicity, which is the property that "most of" the eigenfunctions become uniformly distributed in D, asymptotically as $E \to \infty$, i.e. they are the same size, on average, in all parts of the domain D; and the related property of Quantum Unique Ergodicity, which is the same property with the words "most of" deleted. There has been a conjecture open for the last 20 years or so, that certain domains called "stadium domains" are quantum ergodic but not quantum unique ergodic, which I solved very recently. I will motivate and discuss this conjecture and talk a little about the proof, which is surprisingly simple.