Single recurrence: overview, open problems, and mysterious examples

The Poincaré Recurrence Theorem says that whenever $(X,\mu,T)$
is a probability measure preserving system and $A\subset X$ has positive
measure, there is an $n\in \mathbb N$ such that $\mu(A\cap T^{-n}A)>0$. 
H. Furstenberg and A. Sárközy independently proved an appealing
refinement: under the same hypotheses, one can conclude that $\mu(A\cap
T^{-n^2}A)>0$ for some $n\in \mathbb N$.

One may ask, in general, for which sets $S\subset \mathbb N$ can one
conclude that $\mu(A\cap T^{-n}A)>0$ for some $n\in S$? This is the
study of single recurrence in measurable dynamics. Many examples and
non-examples are known, but a satisfying description remains elusive.

This talk will provide a broad overview of the subject, touching on some
classical results obtained by functional analytic methods.  Explicit
sets whose recurrence properties are unknown will be considered and,
time permitting, we will advertise some open problems.