Let X be a stationary Z^d-process. We say that X is a factor of an i.i.d. process if there is a (deterministic and translation-invariant) way to construct a realization of X from i.i.d. variables associated to the sites of Z^d. That is, if there is an i.i.d. process Y and a measurable map F from the underlying space of Y to that of X, which commutes with translations of Z^d and satisfies that F(Y)=X in distribution. Such a factor is called finitary if, in order to determine the value of X at a given site, one only needs to look at a finite (but random) region of Y.
Abstract: In this talk, we will discuss the notion of small extensions in its various incarnations, from torsors under abelian groups to square-zero extensions of algebras. We will then focus on the somewhat less familiar case of small extensions of ∞-categories. Our main goal is to make this abstract concept concrete and intuitive through a variety of examples. In particular, we will advocate the point of view that small extensions of ∞-categories offer a unifying perspective in understanding many constructions appearing in obstruction, classification, and deformation theoretic problems
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.
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.
Dependent theories have now a very solid and well-established collection of results and applications. Beyond first order, the development of "dependency" has been rather scarce so far. In addition to the results due to Kaplan, Lavi and Shelah (dependent diagrams and the generic pair conjecture), I will speak on a few lines of current research around the extraction of indiscernibles for dependent diagrams and on various forms on dependence for abstract elementary classes. This is joint work with Saharon Shelah.
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).
The family of high rank arithmetic groups is a class of groups playing an important role in various areas of mathematics. It includes SL(n,Z), for n>2 , SL(n, Z[1/p] ) for n>1, their finite index subgroups and many more.
A number of remarkable results about them have been proven including; Mostow rigidity, Margulis Super rigidity and the Quasi-isometric rigidity.
Zilber's trichotomy conjecture, in modern formulation, distinguishes three flavours of geometries of strongly minimal sets --- disintegrated/trivial, modular, and the geometry of an ACF. Each of these three flavours has a classic ``template'' --- a set with no structure, a projective space over a prime field, and an algebraically closed field, respectively. The class of ab initio constructions with which Hrushovski refuted the conjecture features a new flavour of geometries --- non-modular, yet prohibiting any algebraic structure.
Weak Prediction Principles
Speaker: Yair Hayut
Abstract: Jensen's diamond is a well studied prediction principle. It holds in L (and other core models), and in many cases it follows from local instances of GCH.
In the talk I will address a weakening of diamond (due to Shaleh and Abraham) and present Abraham's theorem about the equivalence between weak diamond and a weak consequence of GCH. Abraham's argument works for successor cardinals. I will discuss what is known and what is open for inaccessible cardinals.
This is a joint work with Shimon Garti and Omer Ben-Neria.