Fixing a complete first order theory T, countable for transparency, we had known quite well for which cardinals T has a saturated model. This depends on T of course - mainly of whether it is stable/super-stable. But the older, precursor notion of having a universal notion lead us to more complicated answer, quite partial so far, e.g the strict order property and even SOP_4 lead to having "few cardinals" (a case of GCH almost holds near the cardinal). Note that eg GCH gives a complete
Minimal and non-minimal automorphism groups of homogeneous structures
A Hausdorff topological group G is said minimal if G does not admit any strictly coarser Hausdorff group topology.
Examples include the isometry group of the Urysohn sphere, due to Uspenskij, and Aut(M) for M stable and w-categorical, a deep fact due to Ben Yacov and Tsankov.
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.
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.
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.
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).
