Abstracts
Talks by Invited Speakers
Shaun Allison
Title: A rank determining when a Polish automorphism group has less-than-maximal classification strength
Abstract:
The classification strength of a Polish group roughly measures how much it can "emulate" other orbit equivalence relations in the context of Borel reductions. We would say that a Polish group $G$ emulates an equivalence relation $E$ living on a Polish space if there is a Polish space $Y$ and a continuous action of $G$ on $Y$ such that $E$ Borel- reduces to the resulting orbit equivalence relation. In this talk, we focus on the class of Polish automorphism (non-Archimedean) Polish groups. At the top of the classification strength hierarchy for these groups is the class of "maximal strength" groups which can emulate any orbit equivalence relation induced by a Polish automorphism group.
In analogy with our work with Aristotelis Panagiotopoulos, where we developed a rank function inspired by work of Deissler and Malicki to detect when a given Polish group is cli (has a complete left-invariant metric), we develop here a rank function which fully captures the class of Polish automorphism groups with less-than-maximal classification strength, a class which strictly contains the cli Polish automorphism groups. This leads to a natural but very novel independence notion which weakens the notion of disjoint amalgamation in Fraïssé theory. We will start with an analysis of a structure due to Julia Knight whose automorphism group has low classification strength yet fails to be cli. Then we will show how this suggests a rank function, and we will show how it has the desired properties. Time permitting, we will gather many different equivalent conditions for a Polish group to have less-than-maximal classification strength, showing that this class is quite robust and interesting.
Omer Ben-Neria
Title: Advances in Iterated Forcing Methods
Abstract:
We shall describe advances in the theory of iterated forcing, including certain technical insights and applications.
Michal Doucha
Title: Global properties of transitive and minimal actions of groups on
the Cantor space
Abstract:
I will present joint work in progress with Julien Melleray and
Todor Tsankov on descriptive set theoretic properties of actions of
countable groups on the Cantor space. This includes eg the existence
of comeager and dense conjugacy classes in the space of transitive and
minimal actions for various countable groups, genericity of minimality
resp. transitivity in the space of all actions, and the description of
the closure of the set of transitive actions in the space of all
actions.
Eyal Kaplan
Title: Failure of GCH on a measurable with the Ultrapower Axiom
Abstract:
The Ultrapower Axiom (UA) roughly states that any pair of ultrapowers can be compared by internal ultrapowers. The Axiom was extensively studied by Gabriel Goldberg, leading to a series of striking results.
Goldberg asked whether UA is consistent with a measurable cardinal that violates GCH. The main challenge is that UA is not easily preserved under forcing constructions, especially ones that achieve violation of GCH on a measurable from large cardinal assumptions. For example, such forcings might create normal measures which are incomparable in the Mitchell order – a property that negates UA.
In this talk, we show that the failure of GCH on the least measurable cardinal can indeed be forced while preserving UA, starting from the minimal canonical inner model carrying a $(\kappa,\kappa^{++})$-extender. We will present the forcing construction and provide a sketch of the main proof ideas. This is a joint work with Omer Ben-Neria.
Heike Mildenberger
Title: Namba Forcing and Singular Cardinals
Abstract:
Namba forcing was discovered independently by Bukovský and Namba to address
questions about singularizing $\aleph_2$ and the minimality of such extensions.
We present new results on singular Namba forcing. Our work pertains to minimality,
in some cases seemingly requiring a large cardinal assumption, and also to variations
of the Hamkins approximation property. Notably we deal with the challenges presented by the Laver versions of these forcings. The emphasis on this case is motivated in part by questions regarding the compactness of Jensen's square principle in the presence of good scales at $\aleph_\omega$. This is a joint work with Maxwell Levine.
Aristotelis Panagiotopoulos (University of Vienna)
Title: The class and dynamics of α-balanced Polish groups
Abstract:
A Polish group is TSI if it admits a two-side invariant metric. It is CLI if it admits complete and left-invariant metric. The class of CLI groups contains every TSI group but there are many CLI groups that fail to be TSI. In this talk we will introduce the class of α-balanced Polish groups where α ranges over all countable ordinals. We will show that these classes completely stratify the space between TSI and CLI. We will also introduce "generic $\alpha$-unbalancedness": a turbulence-like obstruction to classification by actions of $\alpha$-balanced Polish groups. Finally, for each $alpha$ we will provide an action of an $alpha$-balanced Polish group whose orbit equivalence relation is not classifiable by actions of any $\beta$-balanced Polish group with $\beta<\alpha$. This is a joint work with Shaun Allison.
Alejandro Poveda
Title: Axiom $\mathcal{A}$ and supercompactness
Abstract:
In this presentation I would like to report on some recent results on the large cardinal hierarchy between the first supercompact cardinal and Vopenka's Principle. Specifically, we will explore the possible configurations among the classes of supercompact, $C^{(n)}$-supercompact and $C^{(n)}$-extendible cardinals. We will present several consistency results as well as a conjecture about how the large-cardinal hierarchy of $\mathrm{Ultimate-}L$ at these latitudes looks like. One of our main theorems is that (consistently) every supercompact cardinal is $C^{(1)}$-supercompact (ie, supercompact with strong limit targets). This configuration is consistent with (virtually) all large cardinals, yet at odds with the predictions made by the $\mathrm{Ultimate-}L$ program for the class of $C^{(1)}$-supercompacts. This new configuration is a consequence of a new axiom we introduce (named $\mathcal{A}$) which regards the mutual relationship between superstrong and tall cardinals with strong limit targets. Time permitting we shall also propose open problems and discuss possible strengthenings of axiom $\mathcal{A}$.
Grigor Sargsyan
Title: Recent advances in descriptive inner model theory
Abstract:
In recent years, there have been major advances in descriptive
inner model theory. The younger generation has advanced the subject both inward and
outward, proving some striking results of theoretical interest as well as building
bridges with other topics such as forcing axioms and ideal hypotheses.
In some cases, they have found ways of expressing older results without using the
technical language of inner model theory (eg Siskind's characterization of the Core
Model or Kasum's characterization of the minimal model of AD$_{\mathbb{R}}$ + $ \Theta$ is regular).
Such reformulations allow a wider use of these rudimentary descriptive inner model
theoretical structures. In this talk we will outline the contributions made to descriptive
inner model theory by Benjamin Siskind (Characterization of the core model, full
normalization), Takehiko Gappo (focusing over models of determinacy, building higher
models of determinacy), William Chan (AD combinatorics) , Douglas Blue (forcing over
models of determinacy), Obrad Kasum (combinatorial characterization of models of AD),
Derek Levinson (unreachability within the projective hierarchy), Lukas Koschat
(models of the Largest Suslin Axiom via purely large cardinal techniques) and
Jan Kruschewski (an answer to a question of Steel) in collaboration with Jackson,
Larson, Muller, Neeman, the speaker, Schlutzenberg, Steel and Trang.
John Steel
Title: The Comparison Lemma
Abstract:
What makes a canonical inner model $M$ canonical is the existence of
an iteration strategy for $M$. This implies that $M$
can be compared with other models of its kind, a key fact is known as The Comparison Lemma
for the models in question.
It turns out that there is a comparison lemma for iteration strategies too. As the models become
more complicated, so do their iteration strategies. In both cases, comparison establishes
a fine, well-ordered complexity hierarchy. We shall outline
what is known about it at present.
Sarka Stejskalova
Title: Kurepa trees
Abstract:
In the talk we will survey recent results related to Kurepa trees, non-saturated trees and almost Kurepa Suslin trees at $\omega_1$. In the first part, we will discuss the solution of the question of Jin and Shelah who asked whether one could add a Kurepa tree with a small (size $\omega_1$) ccc forcing over models with no Kurepa trees. In the second part of the talk we will focus on some natural generalizations of Jin and Shelah's question; for instance, we will consider models where the property that there are no Kurepa trees is a consequence of the Guessing model principle (GMP).
Asger Tornquist
TBA
Anush Tserunyan
Title: Quasi-treeable equivalence relations are treeable
Abstract:
A result from geometric group theory says that if a Cayley graph of a
finitely generated group is tree-like (more precisely, is quasi-isometric to a tree)
then the group is virtually free. We prove the analogue of this in the context of
countable Borel equivalence relations, thus answering a question of R. Tucker-Drob
from 2015. Our method exploits the Stone duality between certain families of cuts
in a graph and median graphs ($1$-skeleta of CAT(0) cube complexes) via cloning
ultrafilters on cuts. This is a joint work with R. Chen, A. Poulin, and R. Tao.
Ur Ya'ar
Title: Iterating the construction of inner models from extended logics
Abstract:
One way to generalize Gödel's constructible universe $L$ is to replace the notion of definability used at successor stage, by taking all subsets which are definable using a logic $\mathcal{L}^*$ extending first order logic. This will result in a model of ZF, denoted $C(\mathcal{L}^*)$. In many cases it will also be a model of AC.
Specific cases of this kind of construction were studied by Chang and by Myhill and Scott in the past, but recently Juliette Kennedy, Menachem Magidor and Jouko Väänänen have initiated a systematic study of these models, leading to many new results.
The topic of this talk will be iterating this construction. As in the case of $L$, we can formulate the axiom ``$V=C(\mathcal{L}^*)$'', but unlike $L$, $C(\mathcal{L^*}) $ might not satisfy it -- when we construct $C(\mathcal{L}^*)$ inside $C(\mathcal{L}^*)$ we might get a smaller model.
When this happens we can iterate the construction, and if the iteration does not stabilize, we continue transfinitely by taking intersections at limit stages.
We will present some results concerning these sequences of iterated constructible models, focusing mainly on the case of the logic obtained by adding the cofinality-$\omega$ quantifier, and the case of Stationary logic. We will show, as time permits, that in the first case the possibilities can be rather limited, while in the second case we can get almost everything we want.
Contributed Talks
The deadline for applying to give a contributed talk is July 15.
