## High and low forcing

January 11 to January 15, 2016 at the
American Institute of Mathematics, San Jose, California

organized by

Itay Neeman and Dima Sinapova

This workshop, sponsored by AIM and the NSF, is devoted to new methods of forcing, in infinitary combinatorics, and in connection with axioms about the real line. There were exciting recent developments in each of the two areas, involving Prikry type forcing and principles related to the tree property for the former, and finite support iterations with side conditions for the latter. The goal of the workshop is to foster interaction between people working in these two areas, and initiate collaborations on the many problems that remain open.

The topics of the workshop are:

1. Infinitary combinatorics.
2. Forcing with side conditions in connections to axioms about the real line.

The workshop will differ from typical conferences in some regards. Participants will be invited to suggest open problems and questions before the workshop begins, and these will be posted on the workshop website. These include specific problems on which there is hope of making some progress during the workshop, as well as more ambitious problems which may influence the future activity of the field. Lectures at the workshop will be focused on familiarizing the participants with the background material leading up to specific problems, and the schedule will include discussion and parallel working sessions.

## Andy Zucker: Universal metrizable minimal flows

Mathematical logic seminar – November 24, 2015
Time:     12:30 – 13:30

Room:     Wean Hall 7201

Speaker:         Andy Zucker
Department of Mathematical Sciences
CMU

Title:     Universal metrizable minimal flows

Abstract:     Let G be a Polish group. We consider the following question: is it possible for the universal minimal flow M(G) to be non-metrizable, but for there to exist a metrizable minimal G-flow which maps onto every other metrizable minimal G-flow? We show that this cannot be the case. As a corollary, we also show that if G admits only countably many metrizable minimal flows, then M(G) is metrizable.

## Jordi Lopez-Abad: Approximate Ramsey property of matrices and f.d. normed spaces

Place: Fields Institute (Room 210)

Date: November 20th , 2015 (13:00-14:00)

Title: Approximate Ramsey property of matrices and f.d. normed spaces

Abstract: We present the approximate Ramsey property  of  the finite dimensional normed spaces. This is a particular case of a result concerning “metric Ramsey degrees” of matrices. This is a joint work with D. Bartosova and B. Mbombo (U. Sao Paulo)

## Julien Melleray: The simplex of invariant measures of a minimal homeomorphism

Place: Fields Institute (Room 210)

Date: November 20th, 2015 (14:00-15:00)

Speaker: Julien Melleray

Title: The simplex of invariant measures of a minimal homeomorphism

Abstract:  (joint work with Tomás Ibarlucia) We give a characterization
of all simplices of probability measures on a Cantor space X which may
be realized as the simplex of all invariant probability measures for
some minimal homeomorphism g of X. This extends a result of Akin for the
case when K is a singleton, and an unpublished result of Dahl when K is
finite-dimensional. All relevant notions of topological dynamics will be
recalled.

## Piotr Szewczak: Products of Menger spaces

Tuesday, November 24, 2015, 17:15
Wrocław University of Technology, 215 D-1

Speaker: Piotr Szewczak (Cardinal Stefan Wyszyński University in Warsaw)

Coauthor: Boaz Tsaban (Bar-Ilan University)

Title: Products of Menger spaces

Abstract:

A topological space $X$ is Menger if for every sequence of open covers $O_1, O_2,\ldots$ there are finite subfamilies $F_1$ of $O_1$, $F_2$ of $O_2$, . . . such that their union is a cover of $X$. The above property generalizes sigma-compactness.

One of the major open problems in the field of selection principles is whether there are, in ZFC, two Menger sets of real numbers whose product is not Menger. We provide examples under various set theoretic hypotheses, some being weak portions of the Continuum Hypothesis, and some violating it. The proof method is new.

## Marcin Michalski: A generalized version of the Rothberger theorem

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

Speaker: Marcin Michalski (Wrocław University of Technology)

Title: A generalized version of the Rothberger theorem

Abstract:

We call a set $X$ a generalized Luzin set if $|L\cap M|<|L|$ for every meager set $M$. Dually, if we replace meager set with a null set, we obtain a definition of a generalized Sierpiński set.

We will show that if $2^\omega$ is a regular cardinal then for every generalized Luzin set $L$ and every generalized Sierpiński set $S$ an algebraic sum $L+S$ belongs to the Marczewski ideal $s_0$ (i.e. for every perfect set $P$ there exists a perfect set $Q$ such that $Q\subseteq P$ and $Q\cap (L+S)=\emptyset$). To prove the theorem we shall prove and use a generalized version of the Rothberger theorem.

We will also formulate a series of results involving algebraic, topological and measure structure of the real line, that emerged during searching for a proof of the above theorem.

## David Fernández Bretón: An introduction to weak diamonds, II

Thursday, November 19, 2015; 16:00-17:30; East Hall 3096

After having introduced the basics of weak diamond principles, we will show their usage with some examples: construction of a Suslin tree, of strongly summable ulrafilters, and of gruff ultrafilters.

## Andres Villaveces: Model Theory in Abstract Elementary Classes

This month, Andres Villaveces will visit Iran and will give a series of lectures in Institute for Research in Fundamental Sciences (IPM) and Amirkabir University of Technology (AUT).
Here is the abstract of his talks according the last announcement:

Title: Model Theory in Abstract Elementary Classes

Abstract:

Stability theory for first order logic developed very quickly after 1970, prompted by the work of Morley, Lachlan, Baldwin, Shelah and Lascar, among others. Earlier on, model theory of infinitary logics, generalized quantifiers had been started (Keisler, Mostowski) but its stability theory was for a long time postponed. Around 25 years ago, a synthesis of many of these logics started, with semantic rather than syntactic emphasis, with focus on the classes of models rather than on the logics axiomatizing them. This synthesis, started by Shelah, is Abstract Elementary Classes. What followed was the development of a very rich and structural stability theory for these classes. Recently, the emphasis has gone back to the interplay with many logics, on the one hand, and with category theory, on the other hand.

There will be four sessions, organized as follows:

1- Sunday 22th Nov. 10-12 in IPM

The Basics: definitions of AECs, examples, the Presentation Theorem. Galois types, stability, independence notions (splitting), the categoricity conjecture and partial results.

2- Tuesday 24th Nov.13-15 in AUT

A proof of categoricity transfer: this part of the minicourse will have a sketch of a proof of categoricity transfer (Vaughtian pairs, rooted types).
3- Thursday 26th Nov. 14-16 in IPM.

Categoricity and Large Cardinals. More independence notions (connected to splitting, forking), tameness and large cardinals. The Categoricity Conjecture (consistency).

4- Sunday 29th Nov. 10-12 in IPM.

Category theoretic versions. This will be an exploration of recent category theoretic constructions connected with the model theory of AECs. This will also be a summary of the previous lectures, done through the adaptation to category theory.

## Chris Eagle: Definability in infinitary [0, 1]-valued logic

Place: Fields Institute (Room 210)

Date: November 13th, 2015 (13:30-15:00)

Speaker: Chris Eagle

Title: Definability in infinitary [0, 1]-valued logic

Abstract: In recent years there have been several proposals for the
“right” analogue of the infinitary logic $L_{\omega_1, \omega}$ for metric
structures.  I will present the three most recent candidates, and discuss
issues around definablity in each of those logics.  The main result is
that in the most expressive of these logics, a continuous [0, 1]-valued
function on a complete separable metric structure is definable if and only
if it is automorphism invariant.

## John Clemens: Relative primeness of equivalence relations

Wednesday, November 11 from 3 to 4pm
Room: MP 207
Speaker: John Clemens (BSU)

Title: Relative primeness of equivalence relations

Abstract: Let $E$ and $F$ be equivalence relations on the spaces $X$ and $Y$. We say that $E$ is prime to $F$ if: whenever $\varphi: X \rightarrow Y$ is a homomorphism from $E$ to $F$, there is a continuous embedding $\rho$ from $E$ to itself so that the range of $\varphi \circ \rho$ is contained in a single $F$ class. That is to say, $\varphi$ is constant (up to $F$-equivalence) on a set on which $E$ maintains its full complexity with respect to Borel reducibility. When $E$ is prime to $F$, $E$ fails to be Borel-reducible to $F$ in a very strong way. I will discuss this notion and show that many non-reducibility results in the theory of Borel equivalence relations can be strengthened to produce primeness results. I will also discuss the possibility of new types of dichotomies involving the notion of primeness.