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.
Abstract: The set theoretic generalizations of algebras have been
introduced in the 1960s to give a set theoretic interpretation of usual
algebraic structures. The shift in perspective from algebra to set
theory is that in set theory the focus is on the collection of possible
algebras and sub-algebras on specific cardinals rather than on
particular algebraic structures. The study of collections of algebras
and sub-algebras has generated many well-known problems in combinatorial
set theory (e.g., Chang’s conjecture and the existence of small singular
This talk will be largely based on a paper by Joseph Shipman with the same title. We will discuss some variations of Fubini type theorems. The focus will be on what is known as "strong Fubini type theorems". Apparently these versions were proved to be independent of ZFC,and our main aim will be to sketch a proof of this result. We will assume basic knowledge in measure theory. Aside from that, the material is rather self contained.
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.
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.
Abstract: my talk will be devoted to a basic theory of extensions of
complete real-valued fields L/K. Naturally, one says that L is
topologically-algebraically generated over K by a subset S if L lies
in the completion of the algebraic closure of K(S). One can then define
topological analogues of algebraic independence, transcendence degree, etc.
These notions behave much more wierd than their algebraic analogues. For example,
there exist non-invertible continuous K-endomorphisms of the completed
algebraic closure of K(x). In my talk, I will tell which part
Abstract: Inseparable morphisms proved to be
an important tool for the study of algebraic
varieties in characteristic p. In particular,
Rudakov-Shafarevitch, Miyaoka and Ekedahl
have constructed a dictionary between
"height 1" foliations in the tangent bundle
and "height 1" purely inseparable quotients
of a non-singular variety in characteristic p.
In a joint work with Eyal Goren we use this
dictionary to study the special fiber S of a
unitary Shimura variety of signature (n,m),
m < n, at a prime p which is inert in the
underlying imaginary quadratic field. We
I discuss some class of function of several elliptic variables,
this functions generalize multiple polylogarithms of D. Zagier.
I show some applications of developed formalism.
This is a joint work with F. Brown.
Abstract: I will start with a motivation of what algebraic (and model-theoretic) properties
an algebraically closed field of characteristic 1 is expected to have. Then I will explain
how these properties can be obtained by the well-known in model theory Hrushovski's
construction and then formulate very precise axioms that such a field must satisfy.
The axioms have a form of statements about existence of solutions to systems
of equations in terms of a 'multi-dimansional' valuation theory and the validity
of these statements is an open problem to be discussed.
Abstract: In the last five years Bridgeland stability has revolutionized
our understanding of the geometry of moduli spaces of sheaves on surfaces,
allowing us to compute ample and effective cones and describe different
birational models. In this talk, I will survey some of my joint work with
Daniele Arcara, Aaron Bertram, Jack Huizenga and Matthew Woolf on the
birational geometry of moduli spaces of sheaves on the plane. I will
describe the ample and effective cones of these moduli spaces,
concentrating on Hilbert schemes of points and concrete examples.