Zilber introduced quasi-minimal classes to generalize the model theory of pseudo exponential
fields. They are equipped with a pregeometry operator and satisfy interesting properties such
as having only countable or co-countable definable sets. Differentially closed fields of
characteristic 0, rich examples of a \omega-stable structures, are good candidates to be
quasiminimal. The difficulty is that a differential equation may have uncountably many
solutions, and thus violate the countable closure requirement of quasiminimal structures.
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.
Abstract: Paul Larson proved that under Martin's axiom and large continuum there are no Laver ideals over aleph_1. He asked about weakly Laver ideals under some forcing axiom.
We shall address two issues:
1. Under Martin's axiom and the continuum is above aleph_2, there are no weakly Laver ideals over aleph_1..
2. Under Baumgartner's axiom, the parallel of Larson's theorem holds for ideals over aleph_2.
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.
I will review some recent results in the Borel reducibility on uncountable cardinals of the Helsinki logic group.
Borel reducibility on the generalised Baire space \kappa^\kappa for uncountable \kappa is defined analogously to that for \kappa=\omega. One of the corollaries of this work is that under some mild cardinality assumptions on kappa, if T1 is classifiable and T2 is unstable or superstable with OTOP, then the ISOM(T1) is continuously reducible ISOM(T2) and ISOM(T2) is not Borel reducible to ISOM(T1).
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.
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.
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.
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|.
We will present briefly the "multiverse view" of set theory, advocated by Hamkins, that there are a multitude of set-theoretic universes, and not one background universe, and his proposed "Multiverse Axioms". We will then move on to present the main result of Gitman and Hamkins in their paper "A natural model of the multiverse axioms" - that the countable computably saturated models of ZFC form a "toy model" of the multiverse axioms.
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.