Models and Sets Seminar

Location: MALL
Title: Regular solutions of systems of transexponential-polynomial equations

It is unknown whether there are o-minimal fields that are transexponential, i.e., that define functions which eventually grow faster than any tower of exponential functions. In past work, I constructed a Hardy field closed under a transexponential function $E$ which satisfies $E(x+1) = \exp E(x)$. Since the germs at infinity of unary functions definable in an o-minimal structure form a Hardy field, this can be seen as evidence that the real field expanded by $E$ could be o-minimal. To prove o-minimality, a better understanding of definable functions in several variable is likely needed. I will discuss one approach using a criterion for o-minimality due to Lion. This ongoing work is joint with Vincent Bagayoko and Elliot Kaplan.

Location: Roger Stevens LT 16
Title: Building generalized indiscernibles in nonelementary classes
NOTE location change: Roger Stevens LT 16

Generalized indiscernibles can be built in first-order theories by generalizing the combinatorial Ramsey’s Theorem to classes with more structure, which is an active area of study. Trying to do the same for infinitely theories (in the guise of Abstract Elementary Classes) requires generalizing the Erdos-Rado Theorem instead. We discuss various results about generalizations of the Erdos-Rado Theorem and techniques (including large cardinals and forcing) to build generalized indiscernibles.

Location: MALL
Title: Pseudo-finite semigroups and diameter

A semigroup $S$ is said to be (right) pseudo-finite if the universal right congruence $S \times S$ can be generated by a finite set $U$ of pairs of elements of $S$ and there is a bound on the length of derivations for an arbitrary pair as a consequence of those in $U$ . The diameter of a pseudo-finite semigroup is the smallest such bound taken over all finite generating sets.

The notion of being pseudo-finite was introduced by White in the language of ancestry, motivated by a conjecture of Dales and Zelazko for Banach algebras. The property also arises from several other sources.

Without assuming any prior knowledge, this talk investigates the somewhat unpredictable notion of pseudo-finiteness. Some well-known uncountable semigroups have diameter $1$; on the other hand, a pseudo-finite group is forced to be finite. Actions, presentations, Rees matrix constructions and some good old-fashioned semigroup tools all play a part.

This research sits in the wider framework of a study of finitary conditions for semigroups.

Location: MALL
Title: Combining Manin-Mumford and weak Zilber-Pink

I will introduce some classical notions and problems in Diophantine geometry, including the Manin-Mumford and Zilber-Pink conjectures, and explain how model-theoretic tools are used to approach them. I will then talk about one of my recent theorems establishing a new partial result towards Zilber-Pink by combining Manin-Mumford and a weak version of Zilber-Pink (both are theorems). I am going to start with very basic things, give quite a few examples and define/explain all concepts that I am going to use, so I hope that most of the talk will be accessible to a wide range of people including those who have not heard about Diophantine geometry before.

Location: MALL
Title: Indestructibility and $C^{(n)}$-supercompact cardinals
In the 70's Laver showed that a supercompact cardinal $\kappa$ may be made indestructible by a suitable class of forcings—namely, after a preparatory forcing, the supercompactness of $\kappa$ will not be destroyed by any further $<\kappa$-directed closed forcing. Many indestructibility results have since been written, as well as those demonstrating the impossibility of indestructibility (or even preservation) of many large cardinals. In this talk we will consider the case of $C^{(n)}$-supercompact cardinals—a stronger and more slippery variant of supercompact cardinals—and how they can be made indestructible for $n\leq 2$.

Location: MALL 1
Title: Purity in chains of modules

The model theory of modules extends naturally to certain functor categories. One such category is that of representations of the biinfinite quiver $A_{\infty}^{\infty}$, where each object can be thought of as a biinfinite chain of $R$-modules. This raises the question of how the objects and morphisms of (model theoretic) interest for this category relate to those of $\operatorname{Mod}\ R$. In the simplest case, we take $R$ to be von Neumann regular.

Location: MALL
Title: How far is almost strong compactness from strong compactness.
Almost strong compactness of $\kappa$ can be characterized as follows: for every $\delta < \kappa < \lambda$, there is an elementary embedding $j_{\delta,\lambda}: V \rightarrow M$ with critical point $\geq \delta$, so that $j_{\delta,\lambda}" \lambda \subseteq D \in M$ and $M \vDash |D|< j_{\delta,\lambda}(\kappa)$. Boney and Brooke-Taylor were wondering whether almost strong compactness is essentially the same as strong compactness. Recently, Goldberg showed that if $\kappa$ is of uncountable cofinality and SCH holds from below then these two closely related concepts are the same. In this joint work with Zhixing You, we show that these two can be different in general cases.

Location: MALL 1
Title: Differentially Large Fields and Taylor Morphisms

Differential largeness is a generalisation of the notion of largeness for pure fields, introduced by Leon-Sanchez and Tressl. This class of differential fields contains many of the model-theoretically tame classes, such as differentially closed fields, closed ordered differential fields, etc. One of the tools that have been developed to study such fields is known as the `twisted Taylor morphism’, which essentially transforms ring homomorphisms into differential ring homomorphisms into the ring of power series in a uniform way. We generalise this notion, and show that differential largeness can also be characterised in terms of generalised Taylor morphisms. If time allows, we will talk about the structure of these generalised Taylor morphisms.

Location: MALL
Title: Absolute Model Companionship, the AMC-spectrum of set theory, and the continuum problem
We introduce a classification tool for mathematical theories based on Robinson's notion of model companionship; roughly the idea is to attach to a mathematical theory $T$ those signatures $L$ such that $T$ as axiomatized in $L$ admits a model companion. We also introduce a slight strengthening of model companionship (absolute model companionship - AMC) which characterize those model companionable $L$-theories $T$ whose model companion is axiomatized by the $\Pi_2$-sentences for $L$ which are consistent with the universal and existential theory of any $L$-model of $T$. We use the above to analyze set theory, and we show that the above classification tools can be used to extract (surprising?) information on the continuum problem. Slides

Location: MALL 1
Title: Foundationless geology and a Foundation conservativity result
It is well-understood that the Axiom of Foundation has no "mathematical consequences" over ZFC - Foundation, since every mathematical structure is isomorphic to one whose universe is an ordinal by the well-ordering theorem. Over ZF - Foundation, there are mathematical consequences to adding Foundation, e.g. the sentence "if all orderable sets are well-orderable, then every set is well-orderable." In joint work with Asaf Karagila, we identify a precise sense in which there is no simpler consequence of adding Foundation. In particular, for any $\varphi$ a sentence in second-order logic, adding Foundation does not refute the existence of a set model of $\varphi.$ This talk will focus on applying techniques of set-theoretic geology in a context without Choice or Foundation, which is a key ingredient in the proof of this theorem. Slides