Modern financial portfolio construction uses mean-variance optimisation that requiers the knowledge of a very large covariance matrix. Replacing the unknown covariance matrix by the sample covariance matrix (SCM) leads to disastrous out-of-sample results that can be explained by properties of large SCM understood since Marcenko and Pastur.  A better estimate of the true covariance can be built by studying the eigenvectors of SCM via the average matrix resolvent. This object can be computed using a matrix generalisation of Voiculescu’s addition and multiplication of free matrices.  The original result of Ledoit and Peche on SCM can be generalise to estimate any rotationally invariant matrix corrupted by additive or multiplicative noise. Note that the level of rigor of the seminar will be that of statistical physics.

This is a joint applied math/probability seminar.

I will first overview the classical holomorphic isometry problem between complex manifolds, in particular between bounded symmetric domains. When the source is the unit ball, in general the characterization of holomorphic isometries to bounded symmetric domains is not quite clear. With Shan Tai Chan, we recently characterized the holomorphic isometries from the Poincare disc to the product of the unit disc with the unit ball and it  provided new examples of holomorphic isometries from the Poincare disc into irreducible bounded symmetric domains of rank at least 2.

This talk will illustrate some topological properties of the space F_k(M) of ordered k-tuples of distinct points in a manifold M. For a fixed manifold M, as k increases, we might expect the topology of the configuration spaces F_k(M) to become increasingly complicated. Church and others showed, however, that when M is connected and open, there is a representation-theoretic sense in which these configuration spaces stabilize. In this talk I will explain these stability patterns, and describe a higher-order “secondary representation stability” phenomenon among the unstable homology classes. These results may be viewed as a representation-theoretic analogue of current work of Galatius–Kupers–Randal-Williams. The project is joint with Jeremy Miller.