The recent proofs of the Tate conjecture for K3 surfaces over finite fields start by lifting the surface to characteristic 0. Serre showed in the sixties that not every variety can be lifted, but the question whether every motive lifts to characteristic 0 is open. We give a negative answer to a geometric version of this question, by constructing a smooth projective variety that cannot be dominated by a smooth projective variety that lifts to characteristic 0.

Counting non-isomorphic finite nilpotent groups of order n is a very hard problem. One way to approach this problem is to count finite nilpotent groups of fixed nilpotency class c on d generators. The enumeration of such isomorphism classes of objects involves number theory and the theory of algebraic groups. However, very little is known about the explicit generating functions of these sequences of numbers when  c > 2 or d > 2. We use a direct enumeration of such groups that began in the works of M. Bacon, L. Kappe, et al, to provide a natural multivariable extension of the generating function counting such groups. Then we rederive the explicit formulas that are known so far.

A local-global principle is a result that allows us to deduce global information about an object from local information. A well-known example is the Hasse-Minkowski theorem, which asserts that a quadratic form represents a number if and only if it does so everywhere locally. In this talk, we'll discuss certain local-global principles in arithmetic geometry, highlighting two that are related to elliptic curves, one for torsion and one for isogenies. In contrast to the Hasse-Minkowski theorem, we'll see that these two results exhibit considerable rigidity in the sense that a failure of either of their corresponding everywhere local conditions must be rather significant.