Listen to pianists Uri Caine and Mario Laginha discuss music with Institute Artist-in-Residence Derek Bermel.
This talk will review some theorems and conjectures about phase transitions of interacting spin systems in statistical mechanics. A phase transition may be thought of as a change in a typical spin configuration from ordered state at low temperature to disordered state at high temperature. I will illustrate how the symmetry of a spin system plays a crucial role in its qualitative behavior. Of particular interest is the connection between supersymmetric statistical mechanics and the spectral theory of random band matrices.
The P vs. NP problem has sometimes been unofficially paraphrased as asking whether
it is possible to improve on exhaustive search for such problems as Satisfiability, Clique,
Graph Coloring, etc. However, known algorithms for each of these problems indeed are
substantially better than exhaustive search, if still exponential. Furthermore, although a
polynomial-time algorithm for any one of these problems implies one for all of them, these
improved exponential algorithms are highly specific, and it is unclear what the limit of
improvement should be.
On one side, we show that conjugacy of different domains can't be C^1 near the boundary. In particular, billiard maps of the circle and an ellipse are both analytically integrable, but not C^1 conjugate. On the other side, if conjugate near the boundary s smoother, then domains are the same up to isometry.
(This is joint work with A. Sorrentino.)