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
Jonsson cardinals).
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.
Speaker: Shira Zerbib Gelaki (MSRI, University of Michigan)
Title: Colorful coverings of polytopes -- the hidden topological truth behind different colorful phenomena
Abstract:
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
game theory.
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: This talk will be about joint work with Eyal Goren about the
structure of Picard modular surfaces at a prime p which is inert in the
underlying quadratic imaginary field. The main tool for studying the bad
reduction of Shimura varieties is the theory of local models (due to de
Jong and Rapoport-Zink). Our results concern global geometric questions
which go beyond the theory of global models. For example, we are able to
count supersingular curves on the Picard surface. We also study certain
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
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),
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.
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: 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
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,
Many of the main conjectures in Iwasawa theory can be phrased as saying
that the first Chern class of an Iwasawa module is generated by a p-adic
L-series.
In this talk I will describe how higher Chern classes pertain to the higher
codimension behavior of Iwasawa modules. I'll then describe a template
for conjectures which would link such higher Chern classes to elements
in the K-theory of Iwasawa algebras which are constructed from tuples of
Katz p-adic L-series. I will finally describe an instance in which a result of
