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.
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.
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.
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.
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.
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.
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.
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 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.
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.
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.
Speaker: Shira Zerbib Gelaki (MSRI, University of Michigan)
Title: Colorful coverings of polytopes -- the hidden topological truth behind different colorful phenomena
The topological KKMS Theorem is a powerful extension of the Brouwer's Fixed-Point Theorem, which was proved by Shapley in 1973 in the context of
We prove a colorful and polytopal generalization of the KKMS Theorem, and show that our theorem implies some seemingly unrelated results in
discrete geometry and combinatorics involving colorful settings.
Let L(E/Q, s) be the L-function of an elliptic curve E defined over the rational field Q. We examine the central value L(E, 1, χ) of twists of L(E/Q, s) by Dirichlet characters χ. We discuss the vanishing and non-vanishing frequencies of these values as χ ranges over characters of fixed order greater than 2. We also examine thee square-free part of the algebraic part of L(E/F, 1) for abelian fields F/Q when these values are non-zero.
Abstract: In modern algebraic geometry we encounter the notion of derived intersection of subschemes. This is a sophisticated way to encode what happens when two subschemes Y_1 and Y_2 of a given scheme X intersect non-transversely. The classical intersection multiplicity can be extracted from the derived intersection.
This talk is in natural in the context of the Zagier conjecture.
We express values of the Kronecker double series at CM points in
terms of values some version (Bloch-Wigner) of dilogarithm in algebraic
numbers. As zeta-function of the Hilbert class field of quadratic field can
be expressed as combination of the Kronecker double series at CM points
my result gives explicit form of the Zagier conjecture.
My technique is rather elementary and the proof is based on the introduction
some new function (elliptic (1,1)-logarithm) and comparisons with it.