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.
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.
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.
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.
Chang's Conjecture is a strengthening of Lowenheim-Skolem-Tarski theorem. While Lowenheim-Skolem-Tarski theorem is provable in ZFC, any instance of Chang's Conjecture is independent with ZFC and has nontrivial consistency strength. Thus, the question of how many instances of Chang's Conjecture can consistently hold simultaneously is natural.
I will talk about some classical results on the impossibility of some instances of Chang's Conjecture and present some results from a joint work with Monroe Eskew.
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”
I'll show how the Vandermonde determinant identity allows us to
estimate the volume of certain spaces of polynomials in one variable
(or rather, of homogeneous polynomials in two variables), as the degree
goes to infinity.
I'll explain what this is good for in the context of globally valued
fields, and, given time constraints, may give some indications on the
approach for the "real inequality" in higher projective dimension.
Keisler measures were introduced in the late 80's by Keisler but they became central objects in model theory only recently with the development of NIP theories. This led naturally to the question of whether there might be a parallel theory of measures in other tame classes, especially in the simple theories where pseudofinite counting measures supply natural and interesting examples. We will describe some first steps toward establishing such a theory, based on Keisler randomizations and the theory of independence for NSOP1 theories in continuous logic.
We shall try to prove some surprising (and hopefully, correct) theorems about the relationship between the club principle (Hebrew: tiltan) and the splitting number, with respect to the classical s at omega and the generalized s at supercompact cardinals.
It is a familiar fact (sometimes attributed to Ahlbrandt-Ziegler, though it is possibly older) that two aleph0-categorical theories are bi-interpretable if and only if their countable models have isomorphic topological isomorphism groups. Conversely, groups arising in this manner can be given an abstract characterisation, and a countable model of the theory (up to bi-interpretation, of course) can be reconstructed.
Better lucky than smart: realizing a quasi-generic class of measure preserving transformations as diffeomorphisms.
Speaker: Matthew Foreman
Abstract: In 1932, von Neumann proposed classifying measure preserving diffeomorphisms up to measure isomorphism. Joint work with B. Weiss
shows this is impossible in the sense that the corresponding equivalence relation is not Borel; hence impossible to capture using countable methods.
In [Sh771] Shelah rediscovered an old result of Dudley on the non-admissibility of a Polish group topology on an uncountable free group. Crucial to his proof is a so-called Compactness Lemma for Polish groups, concerning satisfaction of algebraic equations for certain sequences of group elements converging to 0 (in distance).