## Sebastien Vasey: Chains of saturated models in Abstract Elementary Classes

Model Theory Seminar

Sebastien Vasey (joint work with Will Boney)

CMU

Title:   Chains of saturated models in Abstract Elementary Classes, Part 1

Abstract:   Fix a first-order theory $T$ and a cardinal $\lambda > |T|$. Is the union of a chain of $\lambda$-saturated models of $T$ $\lambda$-saturated? By a classical result of Saharon Shelah, the answer is positive when $T$ is superstable. When $T$ is only stable, this also holds provided that the length of the chain has cofinality at least $|T|^+$. In both cases, the proofs depend on the heavy machinery of forking and averages.

We prove versions of these two results in the general framework of tame abstract elementary classes. For a suitable definition of superstability, we manage to fully generalize Shelah’s result (for high-enough $\lambda$). We also have a theorem in stable AECs but use cardinal arithmetic assumptions on $\lambda$. Our main tool is a generalization of averages to abstract elementary classes. The starting point is Shelah’s work on averages in the framework of “stability theory inside a model”.
Date: Monday, February 2, 2015

Time: 5:00 – 6:30 PM
Location: Wean 8220

——————————————————————-
Model Theory Seminar

Sebastien Vasey (joint work with Will Boney)

CMU

Title:   Chains of saturated models in Abstract Elementary Classes, Part 2

Abstract:   Fix a first-order theory $T$ and a cardinal $\lambda > |T|$. Is the union of a chain of $\lambda$-saturated models of $T$ $\lambda$-saturated? By a classical result of Saharon Shelah, the answer is positive when $T$ is superstable. When $T$ is only stable, this also holds provided that the length of the chain has cofinality at least $|T|^+$. In both cases, the proofs depend on the heavy machinery of forking and averages.

We prove versions of these two results in the general framework of tame abstract elementary classes. For a suitable definition of superstability, we manage to fully generalize Shelah’s result (for high-enough $\lambda$). We also have a theorem in stable AECs but use cardinal arithmetic assumptions on $\lambda$. Our main tool is a generalization of averages to abstract elementary classes. The starting point is Shelah’s work on averages in the framework of “stability theory inside a model”.
Date: Monday, February 9, 2015

Time: 5:00 – 6:30 PM
Location: Wean 8220

## Nazanin Roshandel Tavana: Computable analysis and its applications

Invitation to the Logic Seminar at the National University of Singapore

Date: Wednesday, 28 January 2015, 17:00 hrs

Room: S17#05-11, Department of Mathematics, NUS

Speaker: Nazanin Roshandel Tavana
Amirkabir University of Technology and IPM

Title: Computable analysis and its applications

URL: http://www.comp.nus.edu.sg/~fstephan/logicseminar.html

Computable analysis is a branch of computability theory studying those
real functions and the related sets which can be computed by machines
such as digital computers. The increasing demand for reliable software
in scientific computation and engineering requires a sound foundation
not only of the analytic/numerical but also of the computational
aspects of real number computation. The central subject of this talk
is one of the approch of computable analysis called “Type Two Theory
of Effectivity (TTE)”. It is based on definitions of computable real
numbers and functions by A. Turing, A. Grzegorczyk and D. Lacombe.
First, computability on finite and infinite sequences of symbols is
introduced. Then, this kind of computability can be transfered to the
other sets by means of names or codes. After, the framework of
computability is settled down, we can talk about the effectiveness of
some other spaces as metric spaces. At the end, complexity of this
approach and its two applications are discussed. One application is
for effectiveness in metric model theory and the other one is in
measure theory.

## Andrés Caicedo: Topological partition properties of $\omega_1$, part II

Wednesday, January 28 from 3 to 4pm
Room: Math 124
Speaker: Andrés Caicedo (BSU)
Title: Some topological partition properties of $\omega_1$, part II

Abstract: We discuss some new results on the topological partition calculus of ordinals less than or equal to $\omega_1$. This is joint work with Jacob Hilton.

## 22nd Boise Extravaganza in Set Theory, June 15-17, 2015

The BEST conference (Boise Extravaganza in Set Theory) will be held this June 15–17 at San Francisco State University. It is part of the AAAS Pacific Division annual meeting. Further announcements will be made, but for now…

Save the date!

## Robin Tucker-Drob: Treeability and planarity in measured group theory

Mathematical logic seminar – January 27, 2015
Time:     12:30 – 13:30

Room:     Wean Hall 8201

Speaker:         Robin Tucker-Drob
Department of Mathematics
Rutgers University

Title:     Treeability and planarity in measured group theory

Abstract:

A probability measure preserving (p.m.p.) action of a group G is said to be treeable if the orbits of the action can be measurably structured by trees. A countable group G is called treeable if it has a freeÂ p.m.p. action which is treeable. The group G is called strongly treeable if all of its free p.m.p. actions are treeable. I will discuss recent joint work with C. Conley, D. Gaboriau, and A. Marks in which we show that finitely generated groups with planar Cayley graphs (e.g., surface groups) are strongly treeable. This provides the first examples of nonamenable strongly treeable groups with one end.

## Marcin Michalski: I-Luzin sets

Tuesday, January 27, 2015, 17:15
Wrocław University of Technology, 215 D-1

Speaker: Marcin Michalski (Wrocław University of Technology)

Title: I-Luzin sets

Abstract:

We will present some results obtained with Szymon Żeberski involving I-Luzin sets in Euclidean spaces. We shall construct a quite decent (non-trivial, possesing Borel base and translation invariant) sigma-ideal I of sets such that there exists an I-measurable I-Luzin set. We give also sufficient condition of I-nonmeasurability of I-Luzin sets involving Smital Property (precisely- it’s weaker version). We also discuss briefly Stenihaus and Smital Properties of Fubini product of sigma-ideals.

## Egbert Thümmel: Semiselective and distributive ideals on omega

Wednesday, January 28, 2015, 11:00
Prague – CTS

Speaker: Egbert Thümmel

Title: Semiselective and distributive ideals on omega

## David Fernandez: A model of ZFC with strongly summable ultrafilters, small covering of meagre and large dominating number

Place: Fields Institute, Room 210
Date and time: Friday 23 January 2015 (13:30-15:00)
Speaker: David Fernandez
Title: A model of ZFC with strongly summable ultrafilters, small covering of meagre and large dominating number.

Abstract: Strongly summable ultrafilters are a variety of ultrafilters that relate with Hindman’s finite sums theorem in a way that is somewhat analogous to that in which Ramsey ultrafilters relate to Ramsey’s theorem. It is known that the existence of these ultrafilters cannot be proved in ZFC, however such an existencial statement follows from having the covering of meagre to equal the continuum. Furthermore, using ultraLaver forcing in a short finite support iteration, it is possible to get models with strongly summable ultrafilters and a small covering of meagre, and these models will also have small dominating number. Using this ultraLaver forcing in a countable support iteration to get a model with small covering meagre and strongly summable ultrafilters is considerably harder, but it can be done and in this talk I will explain how (it involves a characterisation of a certain kind of strongly summable ultrafilter in terms of games). Interesingly, this way we also get the dominating number equal to the continuum, unlike the previously described model.

## Forcing and its applications: Retrospective Workshop, March 29 – April 2, 2015

The meeting will take place at the Field Institute, 222 College Street, Toronto.

In the years 1963-1964, Paul Cohen developed the method of forcing in order to settle Cantor’s Continuum Problem.  Since then, the method of forcing has been adapted into a broadly applicable technique for showing that a given statement is
consistent with the axioms of mathematics.

OVERVIEW

The fall 2012 thematic program at the Fields Institute on Forcing and its Applications witnessed breakthroughs in both the internal study of set theory as well as its applications to fields such as analysis and the general theory of topological groups. This workshop will bring together experts studying different aspects of set theory related to the program — large cardinals, singular cardinals, infinite combinatorics, and forcing axioms — as well as closely related fields of mathematics such as functional analysis and topolog

Participants List as of January 20:
*Pending Confirmation

 Full Name University/Affiliation Brech, Christina University of São Paulo Cox, Sean Virginia Commonwealth University Dow, Alan UNC Charlotte Hrusak, Michael UNAM Krueger, John University of North Texas Lopez-Abad, Jordi Instituto de Ciencias Matematicas (ICMAT) *Magidor, Menachem Hebrew University of Jerusalem Malliaris, Maryanthe University of Chicago Mildenberger, Heike Albert-Ludwigs-Universität Freiburg Mota, Miguel Ángel University of Toronto Neeman, Itay University of California, Los Angeles Peng, Yinhe Chinese Academy of Sciences Rinot, Assaf Bar-Ilan University Sabok, Marcin McGill University Sinapova, Dima University of Illinois at Chicago Viale, Matteo University of Torino Zapletal, Jindrich University of Florida

Organizing Committee

• Justin Moore (Cornell University)
• Stevo Todorcevic (University of Toronto)

## Assaf Rinot: Putting a diamond inside the square

Infinite Combinatorics Seminar (BIU)

25/January/2015, 10:15-12:00,
Room 201, Building 216, Bar-Ilan University

Speaker: Assaf Rinot

Title: Putting a diamond inside the square

Abstract: Gray’s combinatorial principle SD_k is a strong combination of Jensen’s $\square_\kappa$ and $\diamondsuit(\kappa^+)$ principles. This principle proved itself very useful in constructing uncountable graphs of counter-intuitive nature.

By a 35 year old theorem of Shelah, $\square_\kappa+\diamondsuit(\kappa^+)$ does not imply SD_k for regular uncountable cardinals $\kappa$. In this talk, I will prove that they are equivalent whenever $\kappa$ is singular.