Abstract:

In a series of 2 talks I will try to explain that in the function field case the unramified global class field theory has a simple geometric interpretation and a conceptual proof. We will only consider the unramified case (see, for example, https://arxiv.org/pdf/1507.00104.pdf or https://dspace.library.uu.nl/handle/1874/206061) Key words: Abel-Jacobi map, l-adic sheaves, sheaf-function correspondence. P.S. Michael will continue his series of lectures on May 6.

Geometric class field theory is an analog of the classical class field theory over function fields in which functions are replaced by sheaves. In the first part of my talk, I will formulate the result and explain its proof over C (the field of complex numbers).  

In the  second part of the talk, I will try to outline the proof in the case of finite fields and indicate how this result implies the classical unramified global class field theory over function fields. 

Most of the talk will be independent of the first one. 

The talk will be based on work done by Furstenberg, taken mainly from his paper "Randon Walks and Discrete Subgroups of Lie Groups". We will present the idea of a boundary attached to a random walk on a group, and explain intuitively how it can be applied to prove that SL2(R) and SLn(R) - for n greater than 2 - do not have isomorphic lattices. Then we focus on a key step in that proof: Constructing a random walk on a lattice in SLn(R) that has the same boundary as a "spherical" random walk on SLn(R) itself.

Old and new on the de Rham-Witt complex

Abstract: After reviewing the definition and the basic properties of the de Rham-Witt complex for smooth schemes over a perfect field, I will discuss the new approach to the subject developed by Bhatt, Lurie and Mathew.

I will explain the main results and sketch work in progress on the problems raised by this theory.

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.

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

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, solving Hilbert's problem 12 will be left to the audience as an exercise.

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.