Much of the early development of model theoretic stability theory was motivated by stable groups, which include algebraic groups as guiding examples. Later work of Hrushovski and Pillay showed that many tools from stable group theory can be adapted to the local setting, where one works around a single stable formula rather than a stable theory. More recently, groups definable in NIP theories have been intensively studied, bringing back the importance of measures in model theory. On the other hand, local NIP group theory is not as well understood.
Speaker: Elad Levi
Algebraic regularity lemma for hypergraphs
Abstract: Szemer´edi’s Regularity Lemma is a fundamental tool in graph theory. It states that for every large enough graph, the set of vertices has a partition A1,..,Ak, such that for almost every two subsets Ai,Aj the induced bipartite graph on (Ai,Aj) is regular, i.e. similar to a random bipartite graph up to a given error.
This talk will be largely based on a paper by Joseph Shipman with the same title. We will discuss some variations of Fubini type theorems. The focus will be on what is known as "strong Fubini type theorems". Apparently these versions were proved to be independent of ZFC,and our main aim will be to sketch a proof of this result. We will assume basic knowledge in measure theory. Aside from that, the material is rather self contained.
In my master thesis we (Prof' Kobi Peterzil and I) investigated a problem in combinatorial geometry using tools from model theory. Following the article of Chernikov and Starchenko, "Regularity lemma for distal structures", we consider the Strong Erdos-Hajnal property for the incidence relation of points and lines in R^2. In particular, we compute a constant d such that for every finite sets of points P and lines L, with |P|,|L| > 2, there are a subsets P' of P and L' of L such that no point in P' lies on a line from L', and such that
|P'|>d|P| , |L'|>d|L|.
Abstract: The set theoretic generalizations of algebras have been
introduced in the 1960s to give a set theoretic interpretation of usual
algebraic structures. The shift in perspective from algebra to set
theory is that in set theory the focus is on the collection of possible
algebras and sub-algebras on specific cardinals rather than on
particular algebraic structures. The study of collections of algebras
and sub-algebras has generated many well-known problems in combinatorial
Abstract: We will discuss the main steps in the proof of the theorem stating that if (G,+, ...) is a strongly minimal expansion of a group interpretable in an o-minimal expansion of a field, and \dim(G)=2 then G is a pure algebraic group.
Joint work with P. Eleftheriou and Y. Peterzil.
Abstract: We continue with the topic of the previous week. We will define the Radin forcing, discuss (without proof) and its basic properties. We will give Woodin's proof for the consistency of the existence of strong inaccessible without diamond and show how to strengthen it to the consistency of strong inaccessible without weak diamond.
This talk is about three published papers of mine that form my phd. In the first two chapters I focus in the model theory of real closed fields and in the third one I take one step back and investigate in greater genearility dependent theories.
The results are the following:
1. Boundedness criterion for rational functions over generalized semi-algebraic sets in real closed fields.
2. Positivity criterion for polynomials over generalized semi-algebraic sets in real closed valued fields.
Sela proved that the theory of free groups is stable. It is thus natural to wonder what the independence relation looks like. Together with Sklinos, we worked out a complete characterization of forking independence in the standard model (over any set of parameters) in terms of an algebraic-geometric object called the JSJ decomposition, which encodes all the splittings of the group as an amalgamated product or an HNN extension relative to the parameter set.
In the talk we will try to give an idea of the proof over some examples.
A special class among the countably infinite relational structures is the class of homogeneous structures. These are the structures where every finite partial isomorphism extends to a total automorphism. A countable set, the ordered rationals, and the random graph are all homogeneous.
We isolate the property of being a critical point, and prove some basic positive
properties of them. We will also prove a lifting property that allows lifting
elementary embedding to symmetric extensions, and outline a construction that
shows that it is consistent that a successor of a critical cardinal is singular.
This is a recent work with Yair Hayut.
Abstract: The notion of an ascent path through a tree, isolated by Laver, is a generalization of the notion of a cofinal branch and, in many cases, the existence of an ascent path through a tree provides a concrete obstruction to the tree being special. We will discuss some recent results regarding ascent paths through kappa-trees, where kappa > omega_1 is a regular cardinal. We will discuss the consistency of the existence or non-existence of a special mu^+-tree with a cf(mu)-ascent path, where mu is a singular cardinal.