## Nick Ramsey: Definability in the absolute Galois group of a PAC field

HUJI Logic Seminar

The next meeting of the Logic Seminar will be in 15/05, 14:00 – 16:00 in Sprinzak Building, Room 101.  Please note the change in time and place.
Definability in the absolute Galois group of a PAC field

Speaker: Nick Ramsey
Abstract:  A field K is called pseudo-algebraically closed (PAC), if every absolutely irreducible variety defined over K has a K-rational point.  This class of fields was introduced in 1968 by Ax on the way to his famous algebraic characterization of the pseudo-finite fields: a field is elementarily equivalent to an ultra-product of finite fields if and only if it is perfect, has free profinite absolute Galois group on a single generator, and is PAC.  In the decades that followed, the PAC fields were an intensive object of study within model-theoretic algebra.  Cherlin, van den Dries, and MacIntyre gave complete invariants for the first-order theory of a PAC field in terms of its characteristic, absolute numbers, and the ‘co-theory’ of its absolute Galois group, in a formalism they called ‘co-logic’.  This ‘co-logic’ approach was presented within first-order logic by Chatzidakis who, later on, showed how to connect model-theoretic properties of the theory of the absolute Galois group of a PAC field to the model-theoretic properties of the field itself.  We will survey this work in detail and give some recent applications to classification-theoretic questions concerning PAC fields.  Time-permitting, we will also talk about some work in progress and many open problems in this area.

## Grigory Mashevitzky: Action of endomorphism semigroups on definable sets

​​​​BGU seminar in Logic, Set Theory and Topology

Time: Tuesday, May 9th, 12:15-13:30.
Place: Seminar room -101, Math building 58.
Speaker: Grigory Mashevitzky (BGU)

Title: Action of endomorphism semigroups on definable sets

Abstract:
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.

## Asaf Karagila: The Bristol model: A few steps into a Cohen real

HUJI Logic Seminar

The next meeting will be at 08/05 (in one week), Ross 63, 12:00-14:00.

Title: The Bristol model: A few steps into a Cohen real
Speaker:
Asaf Karagila
Abstract: We will take a close look at the first few steps of the construction of the Bristol model, which is a model intermediate to L[c], for a Cohen real c, satisfying V\neq L(x) for all x.

## Ashutosh Kumar: On possible restrictions of null and meager ideal

BIU seminar in Set Theory

On 04/05/2017, 10-12, Building 604, Room 103

Speaker:  Ashutosh Kumar

Title: On possible restrictions of null and meager ideal

Abstract. Fremlin asked if the null ideal restricted to a non null set of reals could be isomorphic to the non stationary ideal on omega_1. Eskew asked if the null and the meager ideal could both be somewhere countably saturated. We’ll show that the answer to both questions is yes. Joint work with Shelah.

## Nicholas Ramsey: NSOP_1 Theories

BGU seminar in Logic, Set Theory and Topology.

Time: Tuesday, April 25th, 12:15-13:30.
Place: Seminar room -101, Math building 58.
Speaker: Nicholas Ramsey (UC Berkeley)

Title: NSOP_1 Theories

Abstract:
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.

## Thilo Weinert: Partitioning pairs of sigma-scattered linear orders

BIU seminar in Set Theory

On 20/04/2017, 10-12, Building 604, Room 103

Speaker:  Thilo Weinert

Title: Partitioning pairs of sigma-scattered linear orders

Abstract. We are going to continue the analysis of generalised scattered orders, proving the result described towards the end of Chris Lambie-Hanson’s talk. This states that consistently, for every sigma-scattered linear ordering there is a colouring of its pairs in black & white such that every triple contains a white pair and every copy of one of the following order-types contains a black pair:

• omega_1^omega
• (omega_1^omega)^*
• omega_1 * (omega * omega^*)^omega
• omega_1^* * (omega * omega^*)^omega
• (omega * omega^*)^omega * omega_1
• (omega * omega^*)^omega * omega_1^*

This generalises a 46-year-old Theorem of Erdős & Rado about ordinals. A sufficient hypothesis implying this theorem is the existence of a colouring of the pairs of omega_1 * omega in black & white such that every triple contains a black pair and every subset of full order-type contains a white one. Time permitting we may present a proof that stick = b = Aleph_1 implies the existence of such a colouring. Here b is the unbounding number and stick = Aleph_1 is a weakening of the club principle which was considered by Baumgartner 41 years ago, named by Broverman, Ginsburg, Kunen & Tall two years thereafter and twenty years ago reconsidered as a cardinal characteristic by Fuchino, Shelah & Soukup.

## Assaf Rinot: Distributive Aronszajn trees

Forcing Seminar (Tel-Aviv University)

Tuesday, 4/Apr/2017, 9-11.
Room 007, Schriber building, Tel-Aviv University.

Speaker: Assaf Rinot

Title: Distributive Aronszajn trees

Abstract: We address a conjecture asserting that, assuming GCH, for every singular cardinal $\lambda$, if there exists a $\lambda^+$-Aronszajn tree, then there exists one which is moreover $\lambda$-distributive.

## Omer Mermelstein: Closed ordinal Ramsey numbers below $\omega^\omega$

BGU seminar in Logic, Set Theory and Topology
Time: Tuesday, April 4th, 12:15-13:30.
Place: Seminar room -101, Math building 58.
Speaker: Omer Mermelstein (BGU)

Title: Closed ordinal Ramsey numbers below $\omega^\omega$

Abstract: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.

## Chris Lambie-Hanson: Partition relations and generalized scattered orders

BIU seminar in Set Theory

On 30/03/2017, 10-12, Building 604, Room 103

Speaker: Chris Lambie-Hanson

Title: Partition relations and generalized scattered orders
Abstract: The class of scattered linear orders, isolated by Hausdorff, plays a prominent role in the study of general linear orders. In 2006, Dzamonja and Thompson introduced classes of orders generalizing the class of scattered orders. For a regular cardinal kappa, they defined the classes of kappa-scattered and weakly kappa-scattered linear orders. For kappa = omega, these two classes coincide and are equal to the classical class of scattered orders. For larger values of kappa, though, the two classes are provably different. In this talk, we will investigate properties of these generalized scattered orders with respect to partition relations, in particular the extent to which the classes of kappa-scattered or weakly kappa-scattered linear orders of size kappa are closed under partition relations of the form tau -> (phi, n) for n < omega. We will show that, assuming kappa^{<kappa} = kappa, the class of weakly kappa-scattered orders is closed under all such partition relations while, for uncountable values of kappa, the class of kappa-scattered orders consistently fails to be closed. Along the way, we will prove a generalization of the Milner-Rado paradox and look at some results regarding ordinal partition relations. This is joint work with Thilo Weinert.

## Chris Lambie-Hanson: Trees with ascent paths

HUJI Logic Seminar

The next meeting of the Logic Seminar will be in Wednesday, 22/03/17, 16:00 – 18:00, Ross buiding.

Speaker: Chris Lambie-Hanson

Title: Reflections on the coloring and chromatic numbers

Abstract: The notion of an ascent path through a tree, isolated by Laver, is a generalization of the notion of a cofinal branch and, in many cases, the existence of an ascent path through a tree provides a concrete obstruction to the tree being special. We will discuss some recent results regarding ascent paths through $\kappa$-trees, where $\kappa > \omega_1$ is a regular cardinal. We will discuss the consistency of the existence or non-existence of a special $\mu^+$-tree with a $cf(\mu)$-ascent path, where $\mu$ is a singular cardinal. We will also discuss the consistency of the statement, “There are $\omega_2$-Aronszajn trees but every $\omega_2$-tree contains an $\omega$-ascent path.” We will connect these topics with various square principles and with results about the productivity of chain conditions.