The Toronto Set Theory Seminar is normally held on Fridays, from 1:30 to 3pm, in room 210 of the Fields Institute. For listing of talks from years earlier than 2006, see this page.

## Zoltán Vidnyánszky: Anti-basis results for graphs of infinite Borel chromatic number

Place: Fields Institute (Room 210)

Date: March 24, 2017 (13:30-15:00)

Speaker: Zoltán Vidnyánszky, York University

Title: Anti-basis results for graphs of infinite Borel chromatic number

Abstract: One of the most interesting results of Borel graph combinatorics is the $G_0$ dichotomy, i. e., the fact that a Borel graph has uncountable Borel chromatic number if and only if it contains a Borel homomorphic image of a graph called $G_0$. It was conjectured that an analogous statement could be true for graphs with infinite Borel chromatic number. Using descriptive set theoretic methods we answer this question and a couple of similar questions negatively, showing that one cannot hope for the existence of a Borel graph whose embeddability would characterize Borel (or even closed) graphs with infinite Borel chromatic number.

## Marcin Sabok: Hyperfiniteness of boundary actions of cubulated hyperbolic groups

Place: Fields Institute (Room 210)

Date: March 3rd, 2017 (13:30-15:00)

Speaker: Marcin Sabok

Title: Hyperfiniteness of boundary actions of cubulated hyperbolic groups

Abstract:  An old result of Dougherty, Jackson and Kechris implies that the
boundary action of the free group F2 induces a hyperfinite equivalence
relation. During the talk, I will discuss generalizations of this theorem
to the class of hyperbolic groups. The examples discussed will include
groups acting properly and cocompactly on CAT(0) cube complexes. This is
joint work with Jingyin Huang and Forte Shinko.

## Sergio Garcia-Balan: On star selection principles

Place: Fields Institute (Room 210)

Date: February 17th, 2017 (13:30-15:00)

Speaker: Segio Garcia-Balan

Title: On star selection principles

Abstract: In the theory of selection principles, an important result (due to L. Aurichi), states that every Menger space is a D-space. Motivated by this result, we will discuss the star versions of the Menger property and some other selection principles in specific topological spaces. We will also talk about the game version of some of these principles. This is joint work with Javier Casas de la Rosa and Paul Szeptycki.

## Yinhe Peng: Product of countable Frechet spaces

Place: Fields Institute (Room 210)

Date: February 3rd, 2017 (13:30-15:00)

Speaker: Yinhe Peng, University of Toronto

Title: Product of countable Frechet spaces

Abstract: I will discuss the preservation of Frechet property in the
product, mainly in the class of countable spaces. Some result in the
higher powers will also be mentioned.

## Zoltán Vidnyánszky: Random elements of large groups

Place: Fields Institute (Room 210)

Date: January 20th, 2017 (13:30-15:00)

Speaker: Zoltán Vidnyánszky

Title: Random elements of large groups

Abstract: The automorphism groups of countable homogeneous structures are usually interesting objects from group theoretic and set theoretic perspective. The description of typical (with respect to category) elements of such groups is a flourishing topic with a wide range of applications. A natural question is whether there exist measure theoretic analogues of these results. An obvious obstacle in this direction is that such automorphism groups are often non-locally compact, hence there is no natural translation invariant measure on them. Christensen introduced the notion of Haar null sets in non-locally compact Polish groups which is a well-behaved generalisation of the null ideal to such groups. Using Christensen’s Haar null ideal it makes sense to consider the properties of a random element of the group. We investigate these properties, giving a full description of random elements in the case of the automorphism group of the random graph and the rational numbers (as an ordered set).

## Dilip Raghavan: A long chain of P-points

Place: Fields Institute (Room 210)

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

Speaker: Dilip Raghavan

Title: A long chain of P-points

Abstract: We show under MA that there is a chain of P-points of length ${\mathfrak{c}}^{+}$ under the Rudin-Keisler order. This answers an old question of Blass. The paper can be found at https://arxiv.org/abs/1607.07188. This is joint work with Borisa Kuzeljevic.

## Franklin Tall: On the sigma-compactness of definable Menger spaces

Place: Fields Institute (Room 210)

Date: December 2, 2016 (13:30-15:00)

Speaker: Franklin Tall, University of Toronto

Title: On the sigma-compactness of definable Menger spaces

Abstract:  In a previous seminar, we saw that an inaccessible sufficed for the consistency of “every projective Menger set of reals is sigma-compact”. The inaccessible is in fact necessary.

## Juris Steprans: Automorphisms of $P(\omega)/fin$

Place: Fields Institute (Room 210)

Date: November 25th, 2016 (13:30-15:00)

Speaker: Juris Steprans, York University

Title: Automorphisms of $P(\omega)/fin$

Abstract:

The question of whether all automorphisms of $P(\kappa)/fin$ are trivial has been settled in the following cases:

– For $\kappa =\omega$ by Rudin and Shelah
– For $\kappa$ between $\omega_2$ and the first inaccessible by work of Larson & McKenney and Shelah and me.

However the second result hinges on the question for automorphisms of $P(\omega_1)/fin$. I will discuss a resolution to this question stemming from recent work of Shelah and me.

## Haosui Duanmu: Nonstandard analysis and its application to statistical decision theory

Place: Fields Institute (Room 210)

Date: November 18 , 2016 (13:30-15:00)

Speaker: Haosui Duanmu

Title: Nonstandard analysis and its application to statistical decision theory

Abstract: Statistical decision theory has been serving as a rigorous foundation for statistics since its development in the mid 20th century. For statistical decision problem with finite parameter space, every admissible estimator is Bayes which is the well-known complete class theorem. However, such relation begins to break down for general parameter spaces. By using nonstandard analysis, we introduce the notion of hyperfinite statistical decision problem and develop the nonstandard complete class theorem. We show that if there exists a suitable hyperfinite representation of the original statistical decision problem then the nonstandard counterpart of every standard admissible estimator is nonstandard Bayes. We close with a standard complete class theorem for compact parameter spaces.

## Justin Moore: The subgroup structure of Thompson’s group F

Place: Bahen Centre (Room 2179)

Date: November 11, 2016 (15:30- 17:00)

Speaker: Justin Moore, Cornell University

Title: The subgroup structure of Thompson’s group F

Abstract: What is the structure of the finitely generated subgroups of Thompson’s group F, equipped with the embeddability relation? This is conjectured to be a well quasi-order. We prove that it contains a chain of length $\epsilon_0+1$. This is joint work with Matt Brin and Collin Bleak.