## Set Theory and its Applications in Topology, September 11-16, 2016

The meeting took place in Oaxaca, Mexico. The slides may be found below.

 08:45 – 09:00 Introduction and Welcome (Conference Room San Felipe) 09:00 – 10:00 Alan Dow: The even numbered problems (Conference Room San Felipe) 10:00 – 10:30 Rodrigo Jesus Hernandez Gutierrez: Spaces discretely generated at infinity (Conference Room San Felipe) 10:30 – 11:00 Coffee Break (Conference Room San Felipe) 11:00 – 11:30 Isván Juhász: Lindelöf spaces of countable pseudocharacter (Conference Room San Felipe) 11:30 – 12:00 Juris Steprans: PID and universal graphs (Conference Room San Felipe) 13:20 – 13:30 Group Photo (Hotel Hacienda Los Laureles) 13:30 – 15:00 Lunch (Restaurant Hotel Hacienda Los Laureles) 15:00 – 16:00 Itay Neeman: Forcing one instance of the Moore-Todorcevic principle (Conference Room San Felipe) 16:00 – 16:30 Coffee Break (Conference Room San Felipe) 16:30 – 17:00 James Cummings: Dowker and super-Dowker filters (Conference Room San Felipe) 17:00 – 17:30 Assaf Rinot: The $\omega_2$-Souslin problem (Conference Room San Felipe) 19:00 – 21:00 Dinner (Restaurant Hotel Hacienda Los Laureles)
Tuesday, September 13
07:30 – 09:00 Breakfast (Restaurant at your assigned hotel)
09:00 – 10:00 Christina Brech: Bases of Homogeneous families bellow the first Mahlo cardinal (Conference Room San Felipe)
10:30 – 11:00 Coffee Break (Conference Room San Felipe)
11:00 – 11:30 Piotr Koszmider: A non-commutative Mrówka’s $\Psi$-space (Conference Room San Felipe)
11:30 – 12:00 Asger Tornquist: Invariant descriptive set theory and almost disjointness modulo an ideal (Conference Room San Felipe)
13:30 – 15:00 Lunch (Restaurant Hotel Hacienda Los Laureles)
15:00 – 16:00 Alexander Shibakov: Sequential groups: large and small (Conference Room San Felipe)
16:00 – 16:30 Coffee Break (Conference Room San Felipe)
16:30 – 17:00 Jindrich Zapletal: Strong measure zero sets in Polish groups (Conference Room San Felipe)
17:00 – 17:30 Marcin Sabok: On hyperfiniteness of boundary actions of hyperbolic groups (Conference Room San Felipe)
19:00 – 21:00 Dinner (Restaurant Hotel Hacienda Los Laureles)
Wednesday, September 14
07:30 – 09:00 Breakfast (Restaurant at your assigned hotel)
09:00 – 09:30 Joerg Brendle: Q (Conference Room San Felipe)
09:30 – 10:00 Dilip Raghavan: More on the density zero ideal (Conference Room San Felipe)
10:00 – 10:30 Osvaldo Guzmán: Combinatorial properties of MAD families (Conference Room San Felipe)
10:30 – 11:00 Coffee Break (Conference Room San Felipe)
11:00 – 11:30 Victor Torres-Perez: Constructions with oppositions: Cardinal invariants and games (Conference Room San Felipe)
11:30 – 12:00 David Fernández Bretón: Strong failures of higher analogs of Hindman’s theorem (Conference Room San Felipe)
12:00 – 12:30 Natasha Dobrinen: Topological Ramsey spaces in some creature forcings (Conference Room San Felipe)
12:30 – 13:30 Lunch (Restaurant Hotel Hacienda Los Laureles)
13:30 – 17:30 Free Afternoon (Oaxaca)
19:00 – 21:00 Dinner (Restaurant Hotel Hacienda Los Laureles)
Thursday, September 15
07:30 – 09:00 Breakfast (Restaurant at your assigned hotel)
09:00 – 10:00 Slawomir Solecki: Monoid actions on left-topological compact semigroups (Conference Room San Felipe)
10:30 – 11:00 Coffee Break (Conference Room San Felipe)
11:00 – 11:30 Aleksandra Kwiatkowska: The Ramsey degree of the pre-pseudoarc (Conference Room San Felipe)
11:30 – 12:00 Dana Bartosova: Ultrafilter combinatorics in topological dynamics (Conference Room San Felipe)
13:30 – 15:00 Lunch (Restaurant Hotel Hacienda Los Laureles)
15:00 – 16:00 Jan van Mill: Erdős spaces (Conference Room San Felipe)
16:00 – 16:30 Coffee Break (Conference Room San Felipe)
16:30 – 17:00 Anush Tserunyan: Topological dimension and Baire category (Conference Room San Felipe)
17:00 – 17:30 Yinhe Peng: Weak network and the basis problem (Conference Room San Felipe)
19:00 – 21:00 Dinner (Restaurant Hotel Hacienda Los Laureles)
Friday, September 16
07:30 – 09:00 Breakfast (Restaurant at your assigned hotel)
09:00 – 09:30 Jeffrey Bergfalk: Walks… (Conference Room San Felipe)
09:30 – 10:00 Iian Smythe: A local Ramsey theory for block sequences (Conference Room San Felipe)
10:00 – 10:30 Noé de Rancourt: Ramsey theory with and without the pigeonhole principle (Conference Room San Felipe)
10:30 – 11:00 Coffee Break (Conference Room San Felipe)
11:00 – 11:30 Claribet Piña: Topological partition relations for $\omega^2$ (Conference Room San Felipe)
11:30 – 12:00 Carlos Uzcategui: Bases and selectors for cofinal families of countable sets (Conference Room San Felipe)
12:00 – 12:30 Carlos Di Prisco: Graphs on the Cantor set (Conference Room San Felipe)
12:30 – 14:30 Lunch (Restaurant Hotel Hacienda Los Laureles)

## Franklin Tall: Work in progress

Place: Fields Institute (Room 210)

Date: September 23, 2016 (13:30-15:00)

Speaker: Franklin Tall, University of Toronto

Title: Work in progress

Abstract: Depending on what I accomplish in the next week, I will either speak on the cardinality of regular Lindelof spaces with points $G_{\delta}$ or else on some connections among spaces satisfying the Baire Category Theorem, logics satisfying the Omitting Types Theorem, and non-meager P-filters.

## Joe Zielinski: Compact metrizable structures and classification problems

Mathematical logic seminar – September 20 2016
Time:     3:30pm – 4:30 pm

Room:     Wean Hall 8220

Speaker:         Joe Zielinski
Department of Mathematical Sciences
CMU

Title:     Compact metrizable structures and classification problems

Abstract:

We consider compact metrizable spaces with equipped with closed relations. Two such structures are considered equivalent when there is a homeomorphism between their domains that respects the relational structure. By representing other classes of objects as compact structures, we establish bounds for classification problems in Borel reducibility. Portions of this talk are based on joint work with C. Rosendal.

## Scott Schneider: Commuting endomorphisms and hypersmooth equivalence relations

Thursday, September 22, 4:00-5:30, East Hall 3096.

An equivalence relation E is hypersmooth (hyperfinite) if E is the union of an increasing sequence of smooth (finite) Borel equivalence relations.  In the mid 80s, Weiss proved that the equivalence relation generated by a finite family of commuting Borel automorphisms is hyperfinite, and in the mid 90s, Dougherty, Jackson, and Kechris proved that the equivalence relation generated by a single Borel endomorphism is hypersmooth.  We will generalize both results to show that the equivalence relation generated by a finite family of commuting Borel endomorphisms is hypersmooth.  As is typical in this area, the proof will involve the construction of a suitable family of Borel marker sets.  This talk will be part 1 of 2.

## Shashi Srivastava: Some Applications of Descriptive Set Theory to Transition Probabilities

Tuesday, September 20, 2016, 17:15
Wrocław University of Technology, 215 D-1

Speaker: Shashi Srivastava (Kalkuta)

Title: Some Applications of Descriptive Set Theory to Transition Probabilities

Abstract:

We use measurable selection theorems and prove several results on extensions and existence of transition probabilities with prescribed domain. This is part of joint work with E. E. Doberkat. The remaining part of the work will be presented at Mathematical Institute, University of Wroclaw on 21 September 2016.

## Samuel Coskey: Classifying automorphisms of countable trees

Tuesday, September 13 from 3 to 4pm
Room: MB 124
Speaker: Samuel Coskey (BSU)
Title: Classifying automorphisms of countable trees

Abstract: We summarize some of the results from Kyle Beserra’s master’s thesis. In Serre’s study of trees and their automorphisms, he observed that the automorphisms all lie in one of three classes: invert an edge, shift a bi-infinite path, or fix a subtree pointwise. But of course there are many types of automorphisms within each of these classes. So it is natural to ask just how complex is the classification of tree automorphisms? And what is the complexity of each of Serre’s three classes? We can make these questions formal using the language Borel complexity theory. In this talk we answer the question for regular trees.

## Ian Greig: Dense Subsets of $2^c$ and Independent Families

Place: Fields Institute (Room 210)

Date: September 9th, 2016 (13:30-15:00)

Speaker: Ian Greig

Title: Dense Subsets of $2^c$ and Independent Families

Abstract: We examine the topological space of all functions from the continuum into 2. Specifically, we show that there exists a countable dense subset of this space such that no point in the space is the limit of a sequence from our dense set. Additionally, under the assumption of Martin’s Axiom for Countable Partial Orders, we construct a countable dense subset D such that any discrete subset of D is closed.

## James Cummings: Dowker filters

Seminar will generally be meeting at 3:30pm in Wean Hall 8220 this semester.

Mathematical logic seminar – September 6 2016
Time:     3:30pm – 4:30 pm  (PLEASE NOTE CHANGED TIME FROM LAST YEAR)

Room:     Wean Hall 8220

Speaker:         James Cummings
Department of Mathematical Sciences
CMU

Title:     Dowker filters

Abstract:

Dowker filters are a class of filters introduced by Dowker in connection with a problem in topology. Their properties are quite mysterious. We will discuss some recent progress (in joint work with Charles Morgan) and mention a number of outstanding open questions.

## Ashutosh Kumar: Avoiding rational distances in plane

Place: Bahen Centre (Room BA 2135)

Date: August 26th, 2016 (13:30-15:00)

Speaker: Ashutosh Kumar

Title: Avoiding rational distances in plane

Abstract: We show that every graph of countable coloring number on a set of reals has an everywhere non meager independent set. In particular, every set of points in the plane has an everywhere non meager subset no two of whose points are at a rational distance.

## Secil Tokgoz: OCA and Menger’s Conjecture

Place: Bahen Centre (Room BA2135)

Date: August 19th , 2016 (13:30-15:00)

Speaker: Secil Tokgoz

Title: OCA and Menger’s Conjecture

Abstract: It was previously known that Projective Determinacy implies Menger projective sets of reals are sigma-compact. The hypothesis has considerable large cardinal strength; we are able to reduce the conclusion’s consistency strength to that of an inaccessible. In fact this is an equiconsistency result. We derive the conclusion from a perfect strengthening of OCA. Equiconsistency is proved by an argument involving the dominating number and the Covering Lemma. This is joint work by Secil Tokgoz, Frank Tall and S. Todorcevic.