Models and Sets Seminar

Title: Spectra and definability
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In this talk, we will consider two aspects in the study of extremal sets of reals, sets like maximal families of eventually different functions, maximal cofinitary groups, or maximal independent families. On one side, we will discuss their spectrum, defined as the set of cardinalities of such families and on the other, the existence of witnesses of optimal projective complexity. We will emphasize recent developments in the area and indicate interesting remaining open questions.

Title: Measures on perfect PAC fields
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This is work in progress, joint with Nick Ramsey (UCLA).

A conjecture, now disproved by Chernikov, Hrushovski, Kruckman, Krupinski, Pillay and Ramsey, asked whether any group with a simple theory is definably amenable.

It is well known that the counting measure on finite fields gives rise to a non-standard counting measure on pseudo-finite fields (the infinite models of the theory of finite fields). It was unknown whether other PAC fields possessed a reasonable measure, and in this talk, we will show that some of them do, although the measure we define does not have all the nice properties of a counting measure when the field is not pseudo-finite. This result can be used to show that if $G$ is a group definable in an $e$-free perfect PAC field, then $G$ is definably amenable. It extends to groups definable in $\omega$-free PAC fields. I will also discuss possible extensions to wider classes of perfect PAC fields.

Title: Products of CW complexes
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It's been a couple of years since I've spoken about CW complexes in a Leeds seminar, so for the benefit of newcomers and with apologies to the old hands who've seen it all before, this talk will be about my result on products of CW complexes. CW complexes are "nice" spaces that are often seen as good spaces to focus on for algebraic topology, avoiding many point-set-theoretic "pathologies". However, the product (as topological spaces) of two CW complexes need not be a CW complex. After giving all the necessary definitions I will go through my characterisation of exactly when the product is a CW complex; this characterisation involves the uncountable cardinal $\mathfrak{b}$.

Title: Pillay’s Conjecture for groups definable in weakly o-minimal non-valuational structures
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Let $M$ be a weakly o-minimal non-valuational structure, and $N$ its canonical o-minimal extension (by Wencel). We prove that every group $G$ definable in $M$ is a dense subgroup of a group $K$ definable in $N$. As an application, we obtain that $G^{00}= G\cap K^{00}$, and establish Pillay's Conjecture in this setting: $G/G^{00}$, equipped with the logic topology, is a compact Lie group, and if G has finitely satisfiable generics, then $\dim(G/G^{00})= \dim(G)$.

Title: Around Generalised Indiscernibles and Higher-arity Independence Properties
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The machinery of generalised indiscernibles has played a key role in recent developments of stability theory. One of the most important applications of this machinery is characterising dividing lines by collapsing indiscernibles, a programme essentially tracing back to the early work of Shelah in the 1980s which has seen a resurgence lately, starting with the work of Scow.

In my talk, I will survey the main definitions and some important notions concerning these generalised indiscernibles and give some examples of characterising dividing lines by collapsing indiscernibles. Finally, if time permits, I will discuss an application of generalised indiscernibles to higher-arity independence properties, showing that IPkcan be witnessed by formulas in singleton variables if one allows parameters (from some model).

Title: Backward ergodic theorem along trees and its consequences
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In the classical pointwise ergodic theorem for a probability measure preserving (pmp) transformation $T$, one takes averages of a given integrable function over the intervals $\{x, T(x), T^2(x), ..., T^n(x)\}$ in the "future" of a point $x$. In joint work with Jenna Zomback, we prove a backward ergodic theorem for a countable-to-one pmp $T$, where the averages are taken over arbitrary trees of possible "pasts" of $x$. Somewhat unexpectedly, this theorem yields ergodic theorems for actions of free groups, where the averages are taken along arbitrary subtrees of the standard Cayley graph rooted at the identity. This strengthens results of Grigorchuk (1987), Nevo (1994), and Bufetov (2000).

Title: Twenty Years of Tameness
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In the 1970s Saharon Shelah initiated a program to develop classification theory for non-elementary classes, and eventually settled on the setting of abstract elementary classes. For over three decades, limited progress was made, most of which required additional set theoretic axioms. In 2001, Rami Grossberg and I introduced the model theoretic concept of tameness which opened the door for stability results in abstract elementary classes in ZFC. During the following 20 years, tameness along with limit models have been used by several mathematicians to prove categoricity theorems and to develop non-first order analogs to forking calculus and stability theory, solving a very large number of problems posed by Shelah in ZFC. Recently, Marcus Mazari-Armida found applications to Abelian group theory and ring theory. In this presentation I will highlight some of the more surprising results involving tameness and limit models.

Title: Set theory without powerset
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Many natural set-theoretic structures satisfy the basic axioms of set theory, but not the powerset axiom. These include the collections $H_{\kappa^+}$ of sets whose transitive closure has size at most $\kappa$, forcing extensions of models of ${\rm ZFC}$ by pretame (but not tame) forcing, and first-order models that are morally equivalent to models of the second-order Kelley-Morse set theory (with class choice). It turns out that a reasonable set theory in the absence of the powerset axiom is not simply ${\rm ZFC}$ with the powerset axiom removed. Without the powerset axiom, the Replacement scheme is not equivalent to the Collection scheme, and the various forms of the Axiom of Choice are not equivalent. In this talk, I will give an overview of the properties of a robust set theory without powerset, ${\rm ZFC}^-$, whose axioms are ${\rm ZFC}$ without the powerset axiom, with the Collection scheme instead of the Replacement scheme and the Well-Ordering Principle instead of the Axiom of Choice. While a great deal of standard set theory can be carried out in ${\rm ZFC}^-$, for instance, forcing works mostly as it does in ${\rm ZFC}$, there are several important properties that are known to fail and some which we still don't know whether they hold. For example, the Intermediate Model Theorem fails for ${\rm ZFC}^-$, and so does ground model definability, and it is not known whether ${\rm HOD}$ is definable. I will also discuss a strengthening of ${\rm ZFC}^-$ obtained by adding the Dependent Choice Scheme, and some rather strange ${\rm ZFC}^-$-models.

Title: Simple automorphism groups
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The automorphism groups of many homogeneous structures (Riemannian symmetric spaces, projective spaces, trees, algebraically closed fields, Urysohn space etc) are abstractly simple groups - or at least are simple after taking an obvious quotient.

We present criteria to prove simplicity for a broad range of structures based on the notion of stationary independence.

Title: Galvin's problem in higher dimensions
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This talk will discuss recent work on Galvin's conjecture in Ramsey theory. I will review the background and discuss previous work on the two dimensional case before focusing on the recent work on dimensions greater than 2. This is joint work with Stevo Todorčević.