The Littlewood and the p-adic Littlewood conjectures are famous open problems on the border between number theory and dynamics. In a joint work with Faustin Adiceam and Fred Lunnon we show that the analogue of the p-adic Littlewood conjecture over \mathbb{F}_3((1/t)) is false. The counterexample is given by the Laurent series whose coefficients are the regular paper folding sequence, and the method of proof is by reduction to the non vanishing of certain Hankel determinants.

Title: p-adic equidistribution of CM points on modular curves
Abstract: Let X be a modular curve. It is a curve over the integers, whose complex points form a quotient of the upper half-plane by a subgroup of SL(2,Z). In X there is a natural supply of algebraic points called CM points. After an idea of Heegner, they can be used to construct rational points on elliptic curves.

I will discuss joint work with S. Starchenko, which combines dynamical systems in the nilmanifold setting with definable objects in o-minimal structures (e.g. semi-algebraic sets): Let G be a real algebraic unipotent group and let L be a lattice in G with p:G->G/L the quotient map. Given a subset X of G which is semi-algerbaic, or more generally definable in an o-minimal structure, we describe the closure of p(X) in terms of finitely many definable families of cosets of positive dimensional algebraic subgroups of G.

Type Definable Semigroups in Stable Structures


A semigroup is a set together with an associative binary operation. As opposed to stable groups, the model theory of stable semigroups is not so rich. One reason for that is their abundance.
We will review (and prove) some known results on type-definable semigroups in stable structures and offer some examples and counter-examples.

In a series of talks I will describe in the chronological order all cases where an explicit
construction of CFT is known:
0. The multiplicative group and Kronecker-Weber -- the case of Q.
1. Elliptic curves with complex multiplication and
Kronecker's Jugendraum -- the case of imaginary quadratic extensions.
2. Formal O-models of Lubin-Tate -- the local case.
3. Drinfeld's elliptic modules -- the function field case.
\infinity. Extending this to real quadratic fields and, more generally,

We'll talk about explicit class field theory of imaginary quadratic fields

Class field theory classifies abelian extensions of local and global fields
in terms of groups constructed from the base. We shall survey the main results of class
field theory for number fields and function fields alike. The goal of these introductory lectures
is to prepare the ground for the study of explicit class field theory in the function field case,
via Drinfeld modules.
I will talk for the first 2 or 3 times.

We will discuss growth functions of algebras and monoids. 

Singular vectors are the ones for which Dirichlet’s theorem can be infinitely improved. For example, any rational vector is singular. The sequence of approximations for any rational vector q is 'obvious'; the tail of this sequence contains only q. In dimension one, the rational numbers are the only singulars. However, in higher dimensions there are additional singular vectors. By Dani's correspondence, the singular vectors are related to divergent trajectories in Homogeneous dynamical systems. A corresponding 'obvious' divergent trajectories can also be defined.

The cost of a measure-preserving equivalence relation is a quantitative measure of its complexity. I will
explain what the cost is and then discuss a recent result of Tom Hutchcroft and Gabor Pete in which they construct,
for any group with property T, a free ergodic measure preserving action with cost 1.

Class field theory classifies abelian extensions of local and global fields
in terms of groups constructed from the base. We shall survey the main results of class
field theory for number fields and function fields alike. The goal of these introductory lectures
is to prepare the ground for the study of explicit class field theory in the function field case,
via Drinfeld modules.
I will talk for the first 2 or 3 times.

The discovery of the Jones polynomial in the early 80's was the beginning of ``quantum topology'': the introduction of various invariants which, in one sense or another, arise from quantum mechanics and quantum field theory. There are many mathematical constructions of these invariants, but they all share the defect of being first defined in terms of a knot diagram, and only subsequently shown by calculation to be independent of the presentation. As a consequence, the geometric meaning has been somewhat opaque.