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.

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