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 (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.
Lachlan conjectured that any omega-categorical stable theory is even omega-stable. Later in 1980 it was shown that there is no omega-categorical omega-stable pseudo plane. In 1988, Hrushovski refuted Lachlan's conjecture by constructing an omega-categorical, strictly stable pseudo-plane.
We will give a quick overview of the construction and try to use this example to test if some properties of omega-categorical omega-stable theories lift to omega-categorical stable theories.
G-compactness, hereditary G-compactness and related phenomena
The notion of G-compactness, along with the Galois groups, was introduced by Lascar in order to find a sufficient condition under which a first order theory can be recovered from the category of its models. I will recall this notion. In order to do that, I will also recall various classical notions of strong types, and possibly the Galois group of the theory (and briefly discuss their importance).