Some specialization problems in Geometry and Number Theory

We shall survey over the general issue of `specializations which preserve a property', for a parametrized family of algebraic varieties. We shall limit ourselves to a few examples. We shall start by recalling typical contexts like Bertini and Hilbert Irreducibility theorems, illustrating some new result. Then we shall jump to much more recent instances, related to algebraic families of abelian varieties.

** Please note, this video was recorded using an older in room system and has substantially diminished video quality.**