Models and Sets Seminar

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

Location: MALL 1
Title: O-minimal tame set-theoretic topology
We give a positive answer in the o-minimal setting to a conjecture in set-theoretic topology and explore similar open problems in topology from the point of view of o-minimality. Slides

Location: MALL
Title: Wetzel's problem and the continuum

In the early 60's, John Wetzel came up with the following question in his PhD thesis on harmonic functions: If $\mathcal{F}$ is a family of entire functions (functions that are holomorphic on the complex plane) which at each point attains at most countably many values, is $\mathcal{F}$ itself necessarily countable? This question makes sense considering the quite restrictive nature of holomorphic functions. Not much thereafter, Erdős could show that a negative answer to Wetzel's Problem is in fact equivalent to the continuum hypothesis. His argument shows that any family of entire functions, that attains at each point less values than elements of that family, must have size continuum. Recently Kumar and Shelah have shown that consistently such a family exists while the continuum has size $\aleph_{\omega_1}$. We answer their main open problem by showing that continuum $\aleph_2$ is possible as well. This is joint work with Thilo Weinert.

Location: MALL
Title: Residue field domination in some theories of valued fields

A paraphrase of the Ax-Kochen-Ersov theorem for some theories of valued fields is that the elementary theory is determined by the theory of the value group and the residue field. At the level of types, the intuition is that a type should be controlled by its trace in each of the residue field and value group. In this talk, I will explore some ways in which this intuition can be made precise, and also some limitations to that preliminary intuition. I will try to give lots of examples to keep the discussion concrete.

Speaker's homepage

Location: MALL 1
Title: Two New Inequalities for Cardinal Characteristics of the Continuum

NOTE date and time change: Thu 1pm

Over the last decades the theory of cardinal characteristics of the continuum has emerged as one among several important subfields of set theory. Some of the classical results in it precede the invention of forcing and arguably the aforementioned emergence. Open problems in this field have inspired the invention over ever more versatile constructions of forcing notions and much of the progress has consisted of proving the values of cardinal characteristics not to be ZFC-provably related. A recent outlier has been the celebrated result by Malliaris and Shelah that $\mathfrak{p}$ is equal to $\mathfrak{t}$. I had guessed that there might be more ZFC-provable relations between the hitherto defined characteristics and I am going to talk about what I found up to now. This is to say that I am going to present some ZFC-provable inequalities. In particular I am going to show that the evasion number is at most the subseries number.

These cardinal characteristics have been introduced in work by Blass, Brendle, Brian, and Hamkins and originate from Algebra and Analysis, respectively. The proof interpolates via the pair-splitting number which is due to Minami.

Location: MALL 2
Title: Around definable types in valued fields

NOTE time change: 11am

Haskell, Hrushovski, and Macpherson showed that the theory ACVF of algebraically closed valued fields has elimination of imaginaries after adding the so-called "geometric sorts" to the language. The same result holds in $p$-adically closed fields ($p$CF) by work of Hrushovski, Martin, and Rideau. In the case of ACVF, one way to prove this is to encode imaginaries using definable types, and then encode definable types in the geometric sorts. While $p$CF does not have "enough" definable types to encode imaginaries, the encoding of definable types carries over. Surprisingly, the geometric sorts are unnecessary: any definable type in $p$CF has a code in the home sort (the field sort). This fact and its proof have some unexpected applications to definable groups and definable topological spaces in $p$CF. For example, certain quotient groups are definable rather than interpretable, and there is a unified notion of "definable compactness" for definable topological spaces. Parts of this talk are joint work with Pablo Andújar Guerrero.

Location: MALL
Title: Spotting rare Pokémon

Sometimes more abstract concepts in general topology are considered as having little relation with areas outside of topology. In this talk we will explore a beautiful construction that unexpectedly links the notion of n-Hausdorffness and a special topology in the dynamical systems setting.

Location: MALL 1
Title: Elimination of imaginaries in ordered abelian groups of bounded regular rank

NOTE time change: 4pm

In this talk I will present some results about elimination of imaginaries in pure ordered abelian groups. For the class of ordered abelian groups with bounded regular rank (equivalently with finite spines) we obtain weak elimination of imaginaries once we add sorts for the quotient groups $\Gamma/ \Delta$ for each definable convex subgroup $\Delta$, and sorts for the quotient groups $\Gamma/(\Delta+ \ell\Gamma)$ where $\Delta$ is a definable convex subgroup and $\ell \in \mathbb{N}_{\geq 2}$. We refer to these sorts as the quotient sorts. For the dp-minimal case we obtain a complete elimination of imaginaries if we also add constants to distinguish the cosets of $\Delta+\ell\Gamma$ in $\Gamma$, where $\Delta$ is a definable convex subgroup and $\ell \in \mathbb{N}_{\geq 2}$.


Speaker's homepage