The exact solutions of the Korteweg-de Vries (KdV) equation obtained by travelling wave and similarity reductions may be expressed in terms of elliptic functions and Painleve transcendents respectively. Discrete versions of the KdV equation may be obtained from chains of commuting Backlund transformations of the KdV equation. These systems are considered integrable in their own right. This introductory talk will demonstrate how solutions obtained as reductions of the discrete KdV equation give us discrete analogues of elliptic equations and discrete Painleve equations, mimicking the case for the KdV equation.

We continue our discussion of perfect and scattered subsets in the generalized Cantor space. We we continue our study of the collection of topologies over 2^{\lambda} introduced last time. These topologies rely on the notion of a P_{\kappa}\lambda-forest, which is a natural generalization of a tree.

Credit ratings have been an important variable in the measurement and management of credit risk. In this talk I will present a Markovian model of credit risk that takes into account an individual's migration between different credit ratings. I will also discuss the portfolio case and introduce a model for the correlation that takes place in a portfolio. I will present a way of measuring the associated Value at Risk and using it to set interest rates. Finally, I will present some results using data.

In 1932 von Neumann proposed classifying the statistical behavior of physical systems. The idea was to take a diffeomorphism of a compact manifold and describe what one might observe as random (as in coin flipping) or predictable (as in a translation on a compact group), or even better have a dictionary  in which one could look up the precise behavior.

Remarkable progress was made on this problem; benchmarks include the Halmos-von Neumann theorem on discrete spectrum and the work of Kolmogorov on Entropy that culminated in the Ornstein classification of Bernoulli shifts. One genre of applications of this theory were the results of Furstenberg on Szemeredi’s theorem and eventually the work of  Green and Tao.

Still the problem resisted a complete solution. Strange examples of completely determined systems that showed completely random statistical behavior began to surface. Starting in the 1990’s anti-classification theorems began to appear. These results showed, in a rigorous way, that complete invariants for measure preserving systems cannot exist. Moreover the isomorphism relation itself is completely intractable. Very recently these results were extended to measure preserving diffeomorphisms of the 2-torus.