L3

This talk will be about various ways in which a variety can be "connected by higher genus curves", mimicking the notion of rational connectedness. At least in characteristic zero, the existence of a curve with a large deformation space of morphisms to a variety implies that the variety is in fact rationally connected. Time permitting I will discuss attempts to show this result in positive characteristic.

We describe how one can recover the Mori--Mukai

classification of smooth 3-dimensional Fano manifolds using mirror

symmetry, and indicate how the same ideas might apply to the

classification of smooth 4-dimensional Fano manifolds. This is joint

work in progress with Corti, Galkin, Golyshev, and Kasprzyk.

A pseudofinite field is a perfect pseudo-algebraically closed (PAC) field which

has $\hat{\mathbb{Z}}$ as absolute Galois group. Pseudofinite fields exists and they can

be realised as ultraproducts of finite fields. A group $G$ is geometrically

represented in a theory $T$ if there are modles $M_0\prec M$ of $T$,

substructures $A,B$ of $M$, $B\subset acl(A)$, such that $M_0\le A\le B\le M$

and $Aut(B/A)$ is isomorphic to $G$. Let $T$ be a complete theory of

pseudofinite fields. We show that, geometric representation of a group whose order

is divisibly by $p$ in $T$ heavily depends on the presence of $p^n$'th roots of unity

in models of $T$. As a consequence of this, we show that, for almost all

completions of the theory of pseudofinite fields, over a substructure $A$, algebraic

closure agrees with definable closure, if $A$ contains the relative algebraic closure

of the prime field. This is joint work with Ehud Hrushovski.

I will explain how to define a notion of stable-independence in NIP

theories, which is an attempt to capture the "stable part" of types.