## Assaf Hasson: On weakly o-minimal structures and o-minimal traces.

Set Theory and Topology seminar (BGU)

Time: Tuesday, April 21st, 12:15-13:40.
Place: Seminar room -101, Math building 58.
Speaker: Assaf Hasson (BGU).
Title: On weakly o-minimal structures and o-minimal traces.

Abstract:
A linearly ordered structure (M,<,…) is weakly o-minimal if every definable subset of the line is a finite union of convex sets. It is o-minimal if every definable subset of the line is a finite union of intervals. A weakly o-minimal expansion of a group is non-valuational if every definable cut has 0 diameter (the canonical example of a valuational weakly o-minimal strcture being a real closed valued field, where the cut determined by the valuation ring has infinite diameter). By a theorem of Wencel, every weakly o-minimal non-valuational expansion of a group M embeds canonically as a dense sub-set of an o-minimal structutre M’, and the structure M’ induces on (the universe of) M is precisely M.

By a theorem of Poizat and Baizalov, if M is an o-minimal structure, N a proper elementary sub-structure then the structure induced on N by all M-definable sets (namely, all sets of the form S\cap N^n where S is an M^definable subset of M^n) is weakly o-minimal. In case M expands a group, and N is dense in M the resulting o-minimal structure is weakly o-minimal and non-valuational. We call a structure obtained in this way an o-minimal trace. The canonical example for such a structure is the field of real algebraic numbers expanded by a unary predicate for the cut at \pi.

In view of Wencel’s theorem it is natural to ask: are all weakly o-minimal non-valuational expansions of groups o-minimal traces. In the talk we will present examples showing that reducts of o-minimal traces need not be o-minimal reducts. We will also present examples of weakly o-minimal non-valuational structures that are not reducts of o-minimal traces.

This is part of a paper to appear jointly with P. Elftheriou and G. Keren.

## Robert Raphael: On the countable lifting property for C(X)

Place:  Fields Institute (Room 210)

Date: 17-April-2015 (13:30-15:00)

Speaker: Robert Raphael

Title: On the countable lifting property for C(X)
Abstract: Suppose that Y is a subspace of a Tychonoff space X so that the induced ring homomorphism $C(X) \rightarrow C(Y)$ is onto. We show that a countable set of pairwise orthogonal functions in C(Y) can be lifted to a pairwise orthogonal preimage in C(X). The question originally arose in vector lattices. Topping published the result for vector lattices using an erroneous induction, but two years later Conrad gave a counterexample. This is joint work with A.W.Hager.

## Jonathan Verner: P-points not containing towers

Wednesday, April 22, 2015, 11:00
Prague – IM AS CR, Zitna 25, seminar room, front building, third floor

Speaker: Jonathan Verner

Title: P-points not containing towers

## Mirna Dzamonja: WQOs, FACs and their width

Tuesday, April 21, 2015, 17:15
Wrocław University of Technology, 215 D-1

Speaker: Mirna Dzamonja (University of East Anglia)

Title: WQOs, FACs and their width

Abstract:

A quasi-order is WQO if it has no infinite antichains or infinite decreasing sequences. A partial order is FAC if it has no infinite antichains. These restrictions on the orders mean that there are several naturally defined ordinal valued ranks that can be used to study them, for example, the rank of the tree of antichains, called the width. These ranks have been studied from the point of view of order theory, Ramsey theory, and also the theory of algorithms, since it turns out that a large class of « well structured systems « of algorithms can be modeled using the wqo. We shall present certain structural results connecting FAC and WQO orders and then some calculations of the ranks. The new results presented in the talk come from a collaborative work with Schnoebelen and Schmitz.

## Thomas Johnstone: Strongly uplifting cardinals and the boldface resurrection axioms

All talks are in Hill

Logic Seminar at 5:00 Room 705

Monday 04/20 — Thomas Johnstone (CUNY)
Title: Strongly uplifting cardinals and the boldface resurrection axioms

## Julia Wódka: Some remarks on monotone spaces

Wednesday, April 15, 2015, 11:00
Prague – IM AS CR, Zitna 25, seminar room, front building, third floor

Speaker: Julia Wódka

Title: Some remarks on monotone spaces

## Krzysztof Krupiński: Topological dynamics and Borel cardinalities in model theory

Set Theory and Topology seminar (BGU)

Time: Tuesday, April 14, 12:15-13:50.
Place: Seminar room -101, Math building 58.
Speaker: Krzysztof Krupiński (UWr).
Title: Topological dynamics and Borel cardinalities in model theory.
Abstract: Newelski introduced methods and ideas from topological dynamics to the context of definable groups.

I will recall some fundamental issues concerning this approach, and I will present a few deeper results from my joined paper with Anand Pillay written last year, which relate the so called generalized Bohr compactification of the given definable group to its model-theoretic connected components. Then I will discuss more recent (analogous) results for the group of automorphisms of the monster model, relating notions from topological dynamics to various Galois groups of the theory in question. As an application, I will present a general theorem concerning Borel cardinalities of Borel, bounded equivalence relations, which gives answers to some questions of Kaplan and Miller and of Rzepecki and myself. This theorem was not accessible by the methods used so far in the study of Borel cardinalities of Borel, bounded equivalence relations (by Kaplan, Miller, Pillay, Simon, Solecki, Rzepecki and myself). The topological dynamics for the group of automorphisms of the monster model and its applications to Borel cardinalities are planned to be contained in my future joint paper with Anand Pillay and Tomasz Rzepecki.

## Nam Trang: Covering and more covering

Mathematical logic seminar – April 14, 2015
Time:     12:30 – 13:30

Room:     Wean Hall 8201

Speaker:         Nam Trang
Department of Mathematical Sciences
CMU

Title:     Covering and more covering

Abstract:

We present a couple of basic arguments of getting sharps for operators with nice properties from certain failures of covering. These arguments are featured in various constructions of canonical models of large cardinals from forcing axioms like PFA, the existence of strongly compact measures in ZF+DC etc.

## Katarzyna Chrząszcz: On some properties of microscopic sets

Tuesday, April 14, 2015, 18:45
Wrocław University of Technology, 215 D-1

Speaker: Katarzyna Chrząszcz (Technical University of Łódź)

Title: On some properties of microscopic sets

Abstract:

The notion of microscopic set appeared for the first time in paper ‘Insiemi ed operatori “piccoli” in analisi funzionale’ (APPELL, J., Rend. Istit. Mat. Univ. Trieste 33 (2001), 127–199).

Def. A subset A of reals is called microscopic if for every e>0 there exists a sequence of segments (I_n)_{nin N} such that A is covered by the union of I_n’s and the length of I_n is less or equal to e^n for every n.

We will give generalizations of given notion to the case of arbitrary metric space. We will analyze algebraic and set-theoretic properties of the family of microscopic sets.

## Marek Bienias: General methods in algebrability

Tuesday, April 14, 2015, 17:15
Wrocław University of Technology, 215 D-1

Speaker: Marek Bienias (Technical University of Łódź)

Title: General methods in algebrability

Abstract:

During last 15 years new idea of measuring sets appeared and become popular.

Def. Let k be a cardinal number and let L be a commutative algebra. Assume that A is a subset of L. We say that A is:
k-algegrable if Au{0} contains k-generated algebra B;
strongly k-algegrable if Au{0} contains k-generated free algebra B.

In many recent articles authors studied algebrability of sets naturally appering in mathematical analysis. It seems that required results are the general methods of algebrability which can cover known methods and give new constructions.

We will describe two methods: independent Bernstein sets and exponential like. They let us prove many results concerning algebrability and strong algebrability of subsets of algebras R^R, C^C, R^N, C[0,1], l_infty. Most of presented applications give the best possible result in terms of complication of built algebraic structure and cardinality of set of generators of this structure (in most cases continuum or 2^continuum).