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.

Abstract. This is a joint work with Linhui Shen. A decorated surface is an oriented surface with punctures and a finite collection of special points on the boundary, considered modulo isotopy. Let G be a split adjoint group. We introduce a moduli space Loc(G,S) of G-local systems on a decorated surface S, which reduces to the character variety when S has no boundary, and quantize it.

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.

There is a general slogan according to which the limit behaviour of a one-parameter family of complex algebraic varieties when the parameter t tends to zero should be (partially) encoded in the associated t-adic analytic space in the sense of Berkovich; this slogan has given rise to deep and fascinating conjecturs by Konsevich and Soibelman, as well as positive results by various authors (Berkovich, Nicaise, Boucksom, Jonsson...).

Abstract: We combine a technique of Steel with one due to Jensen and Steel to obtain a core model below singular cardinals kappa which are sufficiently closed under the beth function, assuming that there is no premouse of height kappa with unboundedly many Woodin cardinals. The motivation for isolating such core model is computing a lower bound for the strength of the theory: T = ''ZFC + there is a singular cardinal kappa such that the set of ordinals below kappa where GCH holds is stationary and co-stationary''.

Abstract: We combine a technique of Steel with one due to Jensen and Steel to obtain a core model below singular cardinals kappa which are sufficiently closed under the beth function, assuming that there is no premouse of height kappa with unboundedly many Woodin cardinals. The motivation for isolating such core model is computing a lower bound for the strength of the theory: T = ''ZFC + there is a singular cardinal kappa such that the set of ordinals below kappa where GCH holds is stationary and co-stationary''.