Seminar on Free Probability
and Random Matrices

Winter 2018

Organizer: J. Mingo


Tuesday, January 23, 3:30 - 5:00, Jeff 319
Rob Martin (University of Cape Town)
Non-commutative Clark measures for Free and Abelian multi-variable Hardy space
In classical Hardy space theory, there is a natural bijection between the Schur class of contractive analytic functions in the complex unit disk and Aleksandrov- Clark measures on the unit circle. A canonical several-variable analogue of Hardy space is the Drury-Arveson space of analytic functions in the unit ball of d-dimensional complex space, and the canonical non-commuting or free multi- variable analogue of Hardy space is the full Fock space over d-dimensional complex space. Here, the full Fock space is naturally identified with a non- commutative reproducing kernel Hilbert space of free or non-commutative ana- lytic functions acting on a several-variable non-commutative open unit ball. We will extend the concept of Aleksandrov-Clark measure, the bijection between the Schur class and AC measures, Clark’s unitary perturbations of the shift, Lebesgue decomposition formulas and additional related results from one to several commuting and non-commuting variables.

Tuesday, January 16, 4:00 - 5:30, Jeff 319
Neha Phabu (Queen's University)
Semicircle distribution in number theory
In free probability theory, the role of the semicircle distribution is analogous to that of the normal distribution in classical probability theory. However, the semicircle distribution also shows up in number theory: it governs the distribution of eigenvalues of Hecke operators acting on spaces of modular cusp forms. In this talk, I will give a brief introduction to this theory of Hecke operators and sketch the proof of a result which is a central limit type theorem from classical probability theory, that involves the semicircle measure.

Previous Schedules

Fall 2010 Fall 2011 Fall 2012 Fall 2013 Fall 2014 Fall 2015 Fall 2016 Fall 2017
Winter 2011 Winter 2012 Winter 2013 Winter 2014 Winter 2015 Winter 2016 Winter 2017
Fall 2003 Fall 2004 Fall 2005 Fall 2006 Fall 2007 Fall 2008 Fall 2009
Winter 2004 Winter 2005 Winter 2006 Winter 2007 Winter 2008 Winter 2009 Winter 2010

Getting to Jeffery Hall from the Hotel Belvedere