Determinantal point processes: quasi-symmetries, completeness, interpolation

Take a unitary matrix and a collection of disjoint small intervals on the unit circle. The density of the probability that each of our intervals contains an eigenvalue of our matrix is then given by the determinant of the Dirichlet kernel.
Point processes whose correlation functions are given by determinants arise in the study of very different objects such as random matrices (Dyson), fermions (Macchi), random series (Peres-Virag, Krishnapur),
spanning trees (Burton-Pemantle, Benjamini-Lyons-Peres-Schramm), Young diagrams (Baik-Deift-Johansson, Borodin-Okounkov-Olshanski), representations of infinite-dimensional groups (Borodin-Olshanski).
At the same time, a rich theory has been built for general determinantal point processes: existence theorems (Macchi, Soshnikov, Shirai-Takahashi), description of Palm measures (Shirai-Takahashi, Lyons), the Central Limit Theorem (Soshnikov), rigidity (Ghosh-Peres).
The talk will start with an elementary introduction to determinantal point processes, proceeding to recent developments and open problems.