the spectrum of the existence of a universal model
תמצית/abstract: קיוּם מוֹדל כולל של תורה בעצמה נתוּנה זו שאלה טבעית בתוֹרת המוֹדלים ובתוֹרת הקבוּצוֹת. נטפל בתנאים מספיקים לאי קיוּם, אין צוֹרך בידיעוֹת מוּקדמוֹת.
The existence of a universal model (of a theory T in a cardinal lambda) is a natural question in model theory and set theory. We shall deal with new sufficient conditions for non-existence. No need of previous knowledge
Generic derivations on o-minimal structures Antongiulio Fornasiero
A derivation on a field K is a map d from K to K such that d(x + y) = d(x) + d(y) and d(x y) = x d(y) + d(x) y.
Given an o-minimal structure M in a language L, we introduce the notion L-derivation, i.e derivation compatible with L. For example, if M is the field of reals with exponentiation, then we further require that the derivation d satisfies d(exp x) = exp(x) d(x).
The purpose of this talk is to survey several results from Hjorth's theory of turbulent polish group actions.
We will start by discussing certain classification problems associated with Borel equivalence relations, and present the notions of Borel reductions and smooth relations, and the E_0 dichotomy theorem of Harrington-Kechris-Louveau.
For $\kappa < \lambda$ infinite cardinals let us consider the following generalization of the Lowenheim-Skolem theorem: "For every algebra with countably many operations over $\lambda^+$ there is a sub-algebra with order type exactly $\kappa^+$".
We will discuss the consistency and inconsistency of some global versions of this statement and present some open questions.
Model theory and geometry of fields with automorphism
I will review some of the model-theoretic geometry of difference varieties, and some open problems. A difference variety is defined by polynomial equations with an additional operator $\si$ interpreted as a field automorphism.
Randomisations, coheir sequences and NSOP1
[Joint with A Chernikov and N Ramsey]
Recall that if T is a theory, then its Keisler randomisation, T^R, is the theory of spaces of random variables which take values in a model of T .
It was show some time ago that if T has IP (e.g., simple unstable), then T^R has TP2, and in particular not simple.
In Eilat I announced the following result [with Chernikov and Ramsey] :
A. If T is NSOP1, then its randomisation T^R is NSOP1
Abstract: The stable fields conjecture asserts that every infinite stable field is separably closed. We will talk a bit about the history of this conjecture, its connection to an analogous conjecture on dependent fields and some of their consequences. Finally, we will end by proving the conjecture for fields of finite dp-rank.
Abstract: A meet-tree is a partial order such that the set of vertices below any vertex is linearly ordered, and for every pair of vertices there is a greatest element smaller than or equal to each of them. I'll talk on a work in progress with Itay Kaplan and Tomasz Rzepecki mainly showing that the universal homogeneous countable meet-tree admits generic automorphisms.
We develop a correspondence between the study of Borel equivalence relations induced by closed subgroups of $S_\infty$, and the study of symmetric models of set theory without choice, and apply it to prove a conjecture of Hjorth-Kechris-Louveau (1998).
Abstract: Let V be an irreducible algebraic subvariety of C^n X C^n of
dimension n.
If Schanuel Conjecture holds, under some natural conditions on V, we
show that, if V is defined over the rationals, there exists a in C^n
such that (a, exp(a)) is a generic point of V.
