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.

On the cofinality of some classical cardinal characteristics.


We will try to prove two results about the possible cofinality of cardinal characteristics.
The first result is about the ultrafilter number, and this is a part of a joint work with Saharon Shelah.
The second is about Galvin's number, and this is a joint work with Yair Hayut, Haim Horowitz and Menachem Magidor.

Uniform definability of types over finite sets

Uniform definability of types over finite sets (UDTFS) is a property of formulas which implies NIP and characterizes NIP in the level of theories (by Chernikov and Simon).

In this talk we will prove that if T is any theory with definable Skolem functions, then every dependent formula phi has UDTFS. This result can be seen as a translation of a result of Shay Moran and Amir Yehudayof in machine learning theory to the logical framework.

Categoricity relative to order and order stability


In this talk we will show a generalization of the notion of stability and categoricity relative to the order. One of the natural questions is whether categoricity implies stability, just like in the regular case. We will show that this is not true generally, by using a result of Pabion on peano arithmetic. We are also going to see some specific cases where categoricity relative to the order implies stability.

Tight weakly o-minimal structures


We introduce a class of weakly o-minimal expansions of groups, called tight structures. We prove that the o-minimal completion of a tight structure is linearly bounded.