Mathematics in Science & Society
Spring 2000

Archive

Tuesday, February 22, 245 Altgeld Hall, 4:00 p.m.

Speaker: Steve Zelditch, Johns Hopkins

Title: Quantum Chaos, Random Polynomials, and Complex Geometry

Abstract: Quantum chaos is about the eigenvalues and eigenfunctions of quantum Hamiltonians or maps when the classical limit system is chaotic (i.e. mixing, ergodic, positive entropy...) I will discuss the setting in which the classical phase space is a complex manifold and the wave functions are holomorphic. Chaotic eigenfunctions can then be modelled by random holomorphic polynomials (or more generally, holomorphic sections). I will discuss the configurations of zeros of typical holomorphic wave functions and the probabilities that sections do various things in the semiclassical limit as the degree (or inverse Planck constant) tends to infinity. The main result is that behaviour on length scales of order \sqrt{N} becomes universal in the semiclassical limit. Connections to symplectic geometry will also be mentioned.

Tuesday, March 28, 245 Altgeld Hall, 4:00 p.m.

Speaker: David Heath, Carnegie Mellon University

Title: The Quantification of Financial Risk

Abstract: We consider the problem of how to measure and limit risk in financial institutions. Because of several spectacular losses, regulators (the SEC, the Fed, the Bank of International Settlements, and others) are now beginning to require financial institutions to carefully measure and control the financial risks they take. The measures which they have proposed are regarded by many as crude and ineffective.

We present (and defend) a set of properties which are desirable for a risk measure, and obtain a representation for all risk measures having these properties. We compare our results with some of the methods in current use, and propose a modification of the popular "VaR" risk measure.

Last modified March 13, 2000