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.

In an attempt to classify the geometries arising in strongly minimal sets, Zil'ber conjectured them to split into three different types: Trivial geometries, vector space-like geometries and field-like geometries. Soon after, Hrushovski refuted this conjecture while introducing a new construction method, which has been modified and used a lot ever since.

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.

Reflections on the coloring and chromatic numbers Speaker: Chris Lambie-Hanson Abstract: Compactness phenomena play a central role in modern set theory, and the investigation of compactness and incompactness for the coloring and chromatic numbers of graphs has been a thriving area of research since the mid-20th century, when De Bruijn and Erdős published their compactness theorem for finite chromatic

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.

We shall prove that there is a sequence of Boolean algebras for which the ultraproduct of the lengths divided by an ultrafilter is strictly less than the length of the product algebra. This is a joint work with Saharon Shelah.

We will follow a short note by Artem Chernikov & Sergei Starchenko: "A note on the Erdos-Hajnal Conjecture." “In this short note we provide a relatively simple proof of the Erd ̋os–Hajnal conjecture for families of finite (hyper-)graphs without the m-order property. It was originally proved by M. Malliaris and S. Shelah”

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.

First speaker: Daniel kalmanovich, HU Title: On the face numbers of cubical polytopes Abstract: Understanding the possible face numbers of polytopes, and of subfamilies of interest, is a fundamental question. The celebrated g-theorem, conjectured by McMullen in 1971 and proved by Stanley (necessity) and by Billera and Lee (sufficiency) in 1980-81, characterizes the f-vectors of simplicial polytopes.

Expansivness is a fundamental property of dynamical systems. It is sometimes viewed as an indication to chaos. However, expansiveness also sets limitations on the complexity of a system. Ma\~{n}'{e} proved in the 1970’s that a compact metric space that admits an expansive homeomorphism is finite dimensional. In this talk we will discuss a recent extension of Ma\~{n}'{e}’s theorem for actions generated by multiple homeomorphisms, based on joint work with Masaki Tsukamoto. This extension relies on a notion called “topological mean dimension’’ , introduced by Gromov and