CUNY Graduate Center
Hybrid
Organized by Russell Miller
Fall 2022
December 9
Daniel Turetsky
Victoria University of Wellington
Wadge degrees, games, and the separation and reduction properties
Abstract
In this talk, I will give an overview of the picture of the Borel Wadge degrees. Our system of descriptions allows us to describe their Delta-classes, as well as specify which degrees have the separation or reduction properties. Part of our analysis is based on playing games along our descriptions, and so I will explain how these games are played and what they can tell us.
Video
December 2
Michał Tomasz Godziszewski
University of Vienna
Cardinal characteristics of the Calkin algebra and other interactions between logic and operator algebras
Abstract
In recent years we have been witnessing a dynamic and fertile connection between logic and operator algebras. Many methods from set theory and model theory have been successfully applied to the investigations of $C^\ast$-algebras and other topics in abstract functional analysis (with a brilliant textbook on the 'Combinatorial Set Theory of $C^\ast$-algebras' by I. Farah presenting the current state of the art in this developing field). The purpose of this talk is to provide an introduction to this fruitful interplay with a focus on a certain set-theoretic problem concerning cardinal characteristics of the Calkin algebra which is a structure that may be thought of as a quantum counterpart of the Boolean algebra of subsets of natural numbers modulo finite sets.
Namely, I will present a result concerning possible sizes of families of projections (on a certain Hilbert space) that are mutually pairwise almost orthogonal, which informally means that they are orthogonal modulo 'compact perturbation'. The aforementioned result is joint work with V. Fischer (Vienna).
Video
November 18
Dima Sinapova
Rutgers University
Prikry sequences and square properties at $\aleph_\omega$
Abstract
It is well known that if an inaccessible cardinal $\kappa$ is singularized to countable cofinality while preserving cardinals, then $\square_{\kappa_\omega}$ holds in the outer model. Moreover, this remains true even when relaxing the cardinal preservation assumption a bit. In this talk we focus on when Prikry forcing adds weaker forms of square in a more general setting. We prove abstract theorems about when Prikry forcing with interleaved collapses to bring down the singularized cardinal to $\aleph_\omega$ will add a weak square sequence. This can be viewed as a partial positive result to a question of Woodin about whether the failure of SCH at $\aleph_\omega$ implies weak square.
Video
November 11
No seminar
November 4
Dave Marker
University of Illinois at Chicago
Automorphisms of differentially closed fields
Abstract
Answering a question of Russell Miller, we show that there are differentially closed fields with no non-trivial automorphisms.
Video
October 28
Corey Switzer
University of Vienna
Ideal Independence, Filters and Maximal Sets of Reals
Abstract
A family $\mathcal I \subseteq [\omega]^\omega$ is called ideal independent if given any finite, distinct $A, B_0, ..., B_{n-1} \in \mathcal I$, the set $A \setminus \bigcup_{i \lt n} B_i$ is infinite. In other words, the ideal generated by $\mathcal I \setminus \{A\}$ does not contain $A$ for any $A \in \mathcal I$. The least size of a maximal (with respect to inclusion) ideal independent family is denoted $\mathfrak{s}_{mm}$ and has recently been tied to several interesting questions in cardinal characteristics and Boolean algebra theory. In this talk we will sketch our new proof that this number is ZFC-provably greater than or equal to the ultrafilter number – the least size of a base for a non-principal ultrafilter on $\omega$. The proof is entirely combinatorial and relies only on a knowledge of ultrafilters and their properties. Time permitting, we will also discuss some interesting new applications of ideal independent families to topology via a generalization of Mrowka spaces usually studied for almost disjoint families. This is joint work with Serhii Bardyla, Jonathan Cancino and Vera Fischer.
Video
October 21
Philipp Rothmaler
CUNY
Generalized Bass modules
Abstract
Over half a century ago Hyman Bass proved that all flat left modules are projective precisely when the underlying ring satisfies the descending chain condition on right principal ideals. He called such rings left perfect. Gena Puninski noticed that this can be given a model theoretic proof. Every infinite descending chain of principal right ideals gives rise to a descending chain of (pp) formulas which, in turn, gives rise to a direct limit of finitely generated projective modules that is not projective. Such a module is flat and not projective, and called a Bass module.
I demonstrate how this construction is elementary model theory and at the same time generalizes to other classes of (pp) formulas and modules, which, among other things, yields a new proof of the late Daniel Simson’s result that all left modules are Mittag-Leffler iff the ring is left pure-semisimple (which, to model theorists, means that all left modules are totally transcendental).
I will emphasize the model theoretic ideas and explain the connection with the algebraic concepts. This is part of ongoing work with Anand Pillay.
Video
October 14
Chris Conidis
CUNY
The computability of Artin-Rees and Krull Intersection
Abstract
We will explore the computational content of two related algebraic theorems, namely the Artin-Rees Lemma and Krull Intersection Theorem. In particular we will show that, while the strengths of these theorems coincide for individual rings, they become distinct in the uniform context.
Video
October 7
Krzysztof Krupiński
University of Wroclaw
Some Ramsey theory and topological dynamics for first order theories
Abstract
I will discuss a theory developed in my joint paper with Junguk Lee and Slavko Moconja. One can view it as a variant of Kechris, Pestov, and Todorčević theory in the context of (complete first order) theories. I will discuss several 'definable' Ramsey-theoretic properties of first order theories and their dynamical characterizations. The point is that all the Ramsey-theoretic properties that we introduce involve 'definable colorings' and the dynamical characterizations are 'dynamical properties of the theories', i.e. they are expressed in terms of the action of the group of automorphisms of a monster (i.e. sufficiently saturated and homogeneous) model of the theory in question on the appropriate space of types. One of the basic results says that a theory has the definable Ramsey property iff it is extremely amenable (as defined by Hrushovski, Pillay and myself). But there are various other results, some of which are essentially new and may be surprising in comparison with the Kechris, Pestov, Todorčević theory. One of the motivations to study those properties was to understand better the so-called Ellis group of a theory (which was used by Pillay, Rzepecki, and myself to explain the nature of the Lascar Galois groups of first order theories and spaces of strong types, and led E. Hrushovski to some original development with striking applications to approximate subgroups). Using our dynamical characterizations, we obtain several criteria for profiniteness and for triviality of this Ellis group, with many examples where they apply. I will try to discuss it during my talk. If time permits, I may very briefly mention an abstract generalization of the above considerations and results, which also applies both to the context of definable groups as well as to the classical context of Kechris Pestov, Todorčević theory, leading to some new notions, results, and questions.
Video
September 30
Due to travel disruptions from Hurricane Ian, the speaker must give this talk remotely. The audience plans to meet in GC 6417 as usual to watch the talk live via zoom.
Hans Schoutens
CUNY
The model-theory of categories
Abstract
One could make the claim that category theory is as foundational as set-theory or model-theory. So, we should be able to transfer from one perspective to the other. In this talk, I will consider one aspect of this meta-equivalence, by introducing a theory in a very simple, one-sorted(!) language, whose models are all categories admitting a terminal object (many categories do). Many categorical constructions then turn out to be first-order. But something even more strange happens: standard categories (like the category of Abelian groups) become actually universal models! I'll explain this apparent contradiction.
In the second part of the talk, I will concentrate on one particularly interesting category: that of compact Hausdorff spaces. I will show that we can recover the natural numbers $N$ and the reals $R$, or rather, (the isomorphism classes of) their compactifications $\bar N$ and $\bar R$, by parameter-free definitions, including their order relation, addition and multiplication. Moreover, in any category that is elementary equivalent to the category of compact Hausdorff spaces, the resulting objects are then a model of PA and a real closed field respectively. Full disclosure: while I have a complete proof for the first assertion, the second is still conjectural.
Apart from some basic model-theory, category theory and topology, everything else will be explained in the talk and so it should be accessible to many.
Video
September 23
Russell Miller
CUNY
Interpreting a field in its Heisenberg group
Abstract
The Heisenberg group G(F) of a field F is the group of upper triangular matrices in GL_3(F), with 1's along the diagonal and 0's below it. This group is obviously interpretable (indeed definable) in the field F. Mal'cev showed that one can recover F from G(F), and indeed that there is an interpretation of F in G(F) using two parameters. Any two noncommuting elements of G(F) can serve as the parameters, but Mal'cev was unable to produce an interpretation without parameters.
After introducing the notions of a computable functor and an effective interpretation, we will present joint work showing that there is an effective interpretation of each countable field in its Heisenberg group, without parameters, uniformly in F. (That is, the same formulas give the interpretation, no matter which field F we consider.) Moreover, from the effective interpretation we will then extract a traditional interpretation without parameters, in the usual model-theoretic sense. Finally we will note that, whereas Mal'cev's result actually gives a definition of F in G(F), there is no parameter-free definition of F there.
This work is joint with Rachael Alvir, Wesley Calvert, Grant Goodman, Valentina Harizanov, Julia Knight, Andrey Morozov, Alexandra Soskova, and Rose Weisshaar.
Video
September 16
Gunter Fuchs
CUNY
The blurry HOD hierarchy
Abstract
Classically, an object is ordinal definable if it is the unique one satisfying a formula with ordinal parameters. Generalizing this concept, given a cardinal $\kappa$, I call an object $\lt\kappa$-blurrily ordinal definable if it belongs to an ordinal definable set with fewer than $\kappa$ elements (called a $\lt\kappa$-blurry definition). By considering the hereditary versions of this notion, one arrives at a hierarchy of inner models, indexed by cardinals $\kappa$: the collection of all hereditarily $\lt\kappa$-blurrily ordinal definable sets, which I call $\lt\kappa$-HOD. In a ZFC-model, this hierarchy spans the entire spectrum from HOD to V.
The special cases $\kappa=\omega$ and $\kappa=\omega_1$ have been previously considered, but no systematic study of the general setting has been carried out, it seems. One main aspect of the analysis is the notion of a leap, that is, a cardinal at which a new object becomes hereditarily blurrily definable.
In this talk, I will focus on the ZFC-provable structural properties of the blurry HOD hierarchy, which turn out to be surprisingly plentiful. So for the most part, the talk will be forcing-free. Time permitting, I may hint at the result of the equiconsistency between the least leap being the successor of a singular strong limit cardinal and the existence of a measurable cardinal, for which, admittedly, forcing is used in one direction.
Video