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.
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
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.
In this talk, I present an analogue of the Hardy-Littlewood conjecture on the asymptotic distribution of prime constellations in the setting of short intervals in function fields of smooth projective curves over finite fields.
I will discuss the definition of a "short interval" on a curve as an additive translation of the space of global sections of a sufficiently positive divisor E by a suitable rational function f, and show how this definition generalizes the definition of a short interval in the polynomial
In this talk we discuss some aspects concerning the arithmetic of
systems of quadratic forms. This includes a result on the frequency of
counterexamples to the Hasse principle for del Pezzo surfaces of degree
four (joint work with J. Jahnel), and a result on the representability of
integers by systems of three quadratic forms (joint work with L. B. Pierce
and M. M. Wood).
Motivated by understanding the action of Hecke operators on special sub-varieties of Shimura varieties, we consider the simplest possible case: the action of Hecke operators on the j-line, namely on the moduli space of elliptic curves, and in particular the action on singular moduli. Our interest is in this action considered in the p-adic topology. The emerging picture is surprisingly rich and the answers involve Serre-Tate coordinates, the Gross-Hopkins period map and finally involves random walks on GL_n.
This is joint work with Payman Kassaei (King's College).
Abstract: In 1981, Sansuc obtained a formula for Tamagawa numbers of reductive groups over number fields, modulo some then unknown results on the arithmetic of simply connected groups which have since been proven, particularly Weil's conjecture on Tamagawa numbers over number fields. One easily deduces that this same formula holds for all linear algebraic groups over number fields. Sansuc's method still works to treat reductive groups in the function field setting, thanks to the recent resolution of Weil's conjecture in the function field setting by Lurie and Gaitsgory.
Hebrew University, Givat Ram, Ross Building, room 63
A natural question is whether there exists a continuous p-adic analogue
for the classical local Langlands correspondence for GL_n(F) .
Namely, for a finite extension F of Q_p, we want to associate continuous p -adic representations of GL_n(F) to n-dimensional p-adic representations of the Weil group of F.
The particular case, where F=Q_p and n=2 , is now known. One of the main tools for establishing this correspondence was the existence of GL_2(Q_p)-invariant norms in certain representations of GL_2(Q_p).
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