Logic, Set Theory and Topology Ical Atom

Varieties of Topological Groups were introduced by the speaker almost 50 years ago and Pro-Lie Groups were introduced by Karl Heinrich Hofmann and the speaker early this century. This seminar will assume minimal background knowledge of topological groups. The thrust will be to describe key results in both topics and how our knowledge is less about the disconnected case. This talk was first given last year at the conference “Winter of Disconnectedness”.

For quite some time, partition relations were considered assuming the generalized continuum hypothesis. After forcing entered the stage, independence results were pursued as well. Even later, connections between them, cardinal characteristics and other combinatorial principles where started to be considered by Joji Takahashi, Stevo Todorcevic and Jean Larson. I will give an account of the history of these endeavours and recent advances made in collaboration with William Chen and Chris Lambie-Hanson. Towards the end I am going to give an outlook towards possibilities of future research.

Since the 1950s, many versions of the partition calculus and arrow notation, introduced by Erdős and Rado, were studied. One such variant, introduced by Baumgartner and recently studied by Caicedo and Hilton, is the closed ordinal Ramsey number. For this variant, we require our homogeneous subset to be both order-isomorphic and homeomorphic to a given ordinal.

In the talk we present an approach with which to tackle this flavour of partition calculus, and if time permits prove some results. The talk is elementary and self-contained.

The class of NSOP_1 theories was isolated by Džamonja and Shelah in the mid-90s and later investigated by Shelah and Usvyatsov, but the theorems about this class were mainly restricted to its syntactic properties and the model-theoretic general consensus was that the property SOP_1 was more of an unimportant curiosity than a meaningful dividing line. I’ll describe recent work with Itay Kaplan which upends this view, characterizing NSOP_1 theories in terms of an independence relation called Kim-independence, which generalizes non-forking independence in simple theories. I’ll describe the basic theory and describe several examples of non-simple NSOP_1 theories, such as Frobenius fields and vector spaces with a generic bilinear form.

I plan to discuss the construction, examples and some applications the Galois-type correspondence between subsemigroups of the endomorphism semigroup End(A) of an algebra A and sets of formulas. Such Galois-type correspondence forms a natural frame for studying algebras by means of actions of different subsemigroups of End(A) on definable sets over A. Between possible applications of this Galois correspondence is a uniform approach to geometries defined by various fragments of the initial language.

The next prospective application deals with effective recognition of sets and effective computations with properties that can be defined by formulas from a fragment of the original language. In this way one can get an effective syntactical expression by semantic tools.

Yet another advantage is a common approach to generalizations of the main model theoretic concepts to the sublanguages of the first order language. It also reveals new connections between well-known concepts. One more application concerns the generalization of the unification theory or more generally Term Rewriting Theory to the logic unification theory.

The strong non-independence property was introduced by Shelah in order to capture, within the class of theories without the independence property (aka dependent theories), an analogue of the class of super-stable theories. Shelah conjectured (roughly) that any infinite field with the strong non-independence property (aka strongly dependent) is either real closed, algebraically closed or supports a definable (henselian) valuation. The conjecture was solved (Johnson) in the very special case of dp-minimal fields, and otherwise remains wide open. In fact, most experts believe the conjecture (replacing “algebraically closed” with “separably closed”) to be true of all fields without the independence property, and the algebraic division line between the two classes of fields remains unclear.

In the talk we will show that strongly dependent ordered abelian groups do have a simple algebraic characterisation, and suggest the interpretability of ordered abelian groups which are not strongly dependent as a new (not yet fully satisfactory) conjectural division line.

If time allows we will draw from the classification of strongly dependent ordered abelian groups some conclusions concerning strongly dependent henselian fields (e.g., if K is strongly dependent then any henselian valuation v – not necessarily definable – on K has strongly dependent residue field and value group).

The talk will aim to be, more or less, self-contained and little use (if any) will be made of technical model theoretic terms.

Based (mostly) on joint work with Yatir Halevi.

More than 40 years ago, Jensen constructed a free Souslin tree of height $\omega_1$. We show how to construct a free $\kappa$-Souslin tree, where $\kappa$ is an arbitrary regular uncountable cardinal. This is joint work with Assaf Rinot.

The strong non-independence property was introduced by Shelah in order to capture, within the class of theories without the independence property (aka dependent theories), an analogue of the class of super-stable theories. Shelah conjectured (roughly) that any infinite field with the strong non-independence property (aka strongly dependent) is either real closed, algebraically closed or supports a definable (henselian) valuation. The conjecture was solved (Johnson) in the very special case of dp-minimal fields, and otherwise remains wide open. In fact, most experts believe the conjecture (replacing “algebraically closed” with “separably closed”) to be true of all fields without the independence property, and the algebraic division line between the two classes of fields remains unclear.

In the talk we will show that strongly dependent ordered abelian groups do have a simple algebraic characterisation, and suggest the interpretability of ordered abelian groups which are not strongly dependent as a new (not yet fully satisfactory) conjectural division line.

If time allows we will draw from the classification of strongly dependent ordered abelian groups some conclusions concerning strongly dependent henselian fields (e.g., if K is strongly dependent then any henselian valuation v – not necessarily definable – on K has strongly dependent residue field and value group).

The talk will aim to be, more or less, self-contained and little use (if any) will be made of technical model theoretic terms.

Based (mostly) on joint work with Yatir Halevi.

The stick principle asserts that there is a family of infinite subsets of $\omega_1$ of size $\aleph_1$ so that any uncountable subset of $\omega_1$ has some member of the family as a subset. We will give a forcing construction to separate versions of the stick principle which put a bound on the order-type of the subsets in the family. Time permitting, we will say a little about the relation of the stick principle with the existence of Suslin trees.