I will describe joint work with François Loeser which was 

motivated by recent results with Yuri Tschinkel concerning the Manin conjecture about asymptotics of the number of points of bounded height in varieties over number fields.
Here, we investigate a geometric analogue. Namely,

we consider a projective variety $X$ over a projective curve $C$,

a normal-crossing divisor $D$ in $X$ such that $-(K_X+D)$ is big over the generic fiber. Given

a finite subset $S$ of $C$, we can then consider 

the moduli space $M_n$ of ``integral sections''  of $X\setminus D$ of degree $n$ with respect

to the log-anticanonical divisor $-(K_X+D)$ and form a generating series 

with coefficients in the Grothendieck group of varieties, the \emph{motivic height zeta function}.

Making use of Hrushovski-Kazhdan's motivic Poisson formula, we prove in some instances

($X\setminus D$ is a vector group, of which $X$ is an equivariant compactification)

of this setup that this series is rational, understand its denominator and its ``largest pole''.

As a consequence, we derive estimates for the dimension of $M_n$ and its number of irreducible components of maximal dimension.

I will report on recent developments in understanding to what extent a T-convex expansion of a real closed valued field is determined by its value group and residue field. This is joint work with C. Ealy and J. Marikova.

Please note that due to techincal difficulties the video was compromised for the first 11 minutes.

Let k be a field, t an indeterminate, and K=k(t). Assume given a variety V defined over K, together with a birational map f:V \to V, and fix an embedding of V into projective space. There is a notion of Weil height on points of V(K): if P is represented by (x_0:...:x_n), with the x_i in k[T] without common divisor, then h(P)=max deg(x_i).

Assume that for some large N, for all m, the set of points P of V(K) such that P, f(P),..., f^m(P) all have height < N, is Zariski dense in V.

Then we would like to conclude something about (V,f). In general, it is not true that (V,f) will "descend to k", but what we can prove, is that for some r>0, (V,f^r) has a (positive dimensional) quotient defined over k. (Here, f^r denotes the r-th iterate of f).

The hypotheses imply that for some r, (V,f^r) is dominated by an algebraic dynamics (W,g) defined over k. We translate the problem in terms of difference fields, and types (in some existentially closed difference field).

The problem boils down to the study of types over the fixed field which are internal to the fixed field. Use of definable Galois theory comes into play, as well as certain rigidity results. After giving all definitions, I will give some details on the structure of the proof.

The classical Ax-Schanuel theorem states that, in a differential field, any algebraic relations involving the exponential function must arise in a 'trivial'

manner. It turns out that one can formulate natural analogues of this theorem in the context of uniformization maps arising from Shimura varieties, the simplest case of which is the j-function. Besides their inherent appeal, such analogues have applications to the Zilber-Pink conjecture in number theory; a far reaching generalization of Andre-Oort.

We will explain these analogues and sketch a proof in the case of the j-function. This is joint work with J.Pila.

I discuss the problem of determining those (unary) functions definable in R_{an, exp} that take integer values for positive integer arguments, improving upon a result of Jones and Thomas. The key new technique involves extending such functions to sectors in the complex plane without increasing their rate of growth by too much.

Recent work of Malliaris and Shelah has given a characterization of simple theories in terms of saturation of ultrapowers. The proof involves developing a framework for classifying the complexity of amalgamation patterns in simple theories. The talk will be about the model-theoretic picture of complexity emerging from this analysis.

I will talk about definably amenable groups in NIP theories, that is NIP groups admitting a translation-invariant measure on the class of definable sets. Examples include stable groups, compact Lie groups and solvable NIP groups. In a joint work with Artem Chernikov, we characterize "generic types" in this context, describe the space of invariant measures and prove "generic compact domination". I will present this work and mention some more recent developments.

Let $K$ be the function field of a smooth and proper curve $S$ over an algebraically closed field $k$ of characteristic $p>0$. Let 

$A$ be an ordinary abelian variety over $K$. Suppose that the N\'eron model $\CA$ of $A$ over $S$ has some closed fibre $\CA_s$, which is 

an abelian variety of $p$-rank $0$. We show that in this situation the group $A(K^\perf)$ is finitely generated (thus generalizing a special case of the Lang-N\'eron theorem). Here $K^\perf=K^{p^{-\infty}}$ is the maximal purely inseparable extension of $K$. This result  implies in particular that the "full" Mordell-Lang conjecture is verified in the situation described above. The proof relies on the theory of semistability (of vector bundles) in positive characteristic and on the existence of the compactification of the universal abelian scheme constructed by Faltings-Chai. 

When $A$ is an elliptic curve, this result was proven by D. Ghioca using a different method