Location: MALL 1
Title: On the strength of some first-order problems corresponding to Ramseyan principles
Given a represented space $X$, we say that a partial multifunction $f$ with $\operatorname{dom}(f) ⊆ X$ is first-order if its codomain is the set of the natural numbers. In this talk, we will study, from the point of view of Weihrauch reducibility, some first-order problems corresponding to Ramseyan combinatorial principles. We will start by analyzing some problems that can be seen naturally as first-order: more specifically, we study some principles whose strengths, from a reverse mathematical perspective, lie around $IΣ_0^2$. We will then move to study the first-order part of problems which cannot be presented as first-order ones. The first-order part operator was introduced by Dzhafarov, Solomon and Yokoyama in unpublished work, and it has already proved to be a valuable tool to gauge the strengths of various problems according to Weihrauch reducibility. After giving some technical results on this operator, we will focus on the first-order part of Ramsey's theorem for pairs, presenting various results on the position of its degree in the Weihrauch lattice. The results presented are joint work with Arno Pauly, Pierre Pradic, and Manlio Valenti.
I will introduce the notion of "tame field" and sketch what is known and is not known about the model theory of tame fields. Further, I will introduce the notion of "defect" of finite extensions of valued fields and indicate why it is important for the model theory of valued fields with residue fields of positive characteristic. Tame fields are defectless fields, i.e., all of their finite extensions have only trivial defect.
Tame fields are the closest we have come to the open problem of the decidability of formal Laurent series fields over finite fields, such as $\mathbb{F}_p((t))$, where $\mathbb{F}_p$ is the field with $p$ elements; yet tame fields are perfect, while $\mathbb{F}_p((t))$ is not. We are still lacking a complete recursive axiomatization for $\mathbb{F}_p((t))$ with the $t$-adic valuation. It is known that a simple adaptation of the axiom system that works for $p$-adically closed fields is not complete.
In the year 2003 Yuri Ershov introduced the notion of "extremal valued field" and proved that $\mathbb{F}_p((t))$ is extremal, but this was wrong as the definition of the notion was incorrect. In joint work with Salih Azgin and Florian Pop, we corrected the definition, and thereby the proof, and partially characterized extremal valued fields. This work was then continued together with Sylvy Anscombe. It has remained an open problem whether adding the (very handy) axiom system expressing extremality to the aforementioned adapted system will render it complete.
Another important question is whether we can push things beyond the class of defectless fields. In joint work with Anna Rzepka I have studied the valuation theory of "deeply ramified fields", which admit only certain less harmful defects. Perfectoid fields belong to this (elementary) class of valued fields, as well as all perfect valued fields of positive characteristic, such as the perfect hull of $\mathbb{F}_p((t))$. Thus the theory of deeply ramified fields may offer some clues for the open problem of the decidability of the latter.
Location: MALL
Title: Cardinal characteristics modulo nice ideals on $\omega$
Many of the standard cardinal characteristics of the continuum are defined in terms of a relation holding almost everywhere, where "almost everywhere" means on all but a finite set. A very natural generalisation is to take "almost everywhere" to mean on all but a member of a given ideal. I will talk about what happens when we do this, with the density 0 ideal on $\omega$ as a focal example.
Location: MAGIC
Title: Ordering Ramsey Classes via the Modelling Property
NOTE room change: MAGIC
(Joint work with N. Meir) Generalised indiscernibles are one of the most useful tools in a model theorist's toolbox. Existence of generalised indiscernibles is closely connected with Ramsey's theorem, in a way made very explicit by Scow. In my talk will present and explain the basic background for generalised indiscernibles, Ramsey classes, and Scow's theorem. Then, I will discuss some techniques we recently introduced to generalise Scow's theorem, removing from it (almost) all technical assumptions. Finally, time permitting, I will mention how as an almost immediate corollary of our results, we obtain that every Ramsey class (not necessarily countable) admits a linear order which is the union of quantifier-free types.
Location: EC Stoner seminar room 8.90
Title: An unbounded version of Zarankiewicz's problem
NOTE room change: EC Stoner seminar room 8.90
Zarankiewicz's problem for hypergraphs asks for upper bounds on the number of edges of a hypergraph that has no complete sub-hypergraphs of a given size. Let $M$ be an o-minimal structure. Basit-Chernikov-Starchenko-Tao-Tran (2021) proved that the following are equivalent:
(1) "linear Zarankiewicz bounds" hold for hypergraphs whose edge relation is induced by a fixed relation definable in $M$
(2) $M$ does not define an infinite field.
We prove that the following are equivalent:
(1)' "linear Zarankiewicz bounds" hold for sufficiently "distant" hypergraphs whose edge relation is induced by a fixed relation definable in $M$.
(2)' $M$ does not define a full field (that is, one whose domain is the whole universe of $M$).
This is joint work (in progress) with Aris Papadopoulos.
Location: Roger Stevens LT 08 (9.08)
Title: Interaction of determinacy and forcing
NOTE room change: Roger Stevens LT 08 (9.08)
Determinacy principles provide a unified theory of definable sets of reals beyond Borel and analytic sets, while forcing is an important technique to study the independence of properties of sets of reals. This suggests studying the interaction of the two: how robust are determinacy principles under well behaved forcings? I will talk about the history of this problem as well as recent joint results with Jonathan Schilhan and Johannes Schürz on iterations of proper forcings. A sample application of our results is the following: starting from a model of analytic determinacy, one can construct a model of analytic determinacy and the Borel conjecture.
A structure expanding a dense linear order is weakly o-minimal if any definable set in one variable is a finite union of points and convex sets. This is a more general condition compared to regular o-minimality, where we require these sets to be a finite union of points and intervals.
As a result, weak o-minimality doesn't have all the useful properties that o-minimality has, for example it's not preserved under elementary equivalence, nor can we show that any definable function is piecewise monotone and continuous.
In this talk I'll highlight some of the nice properties o-minimal structures have, give examples of weakly o-minimal structures which break those properties, and then show how we can salvage weaker versions of the properties in a weakly o-minimal setting.
Most of this talk will be based on a paper by Dugald Macpherson, David Marker and Charles Steinhorn called "Weakly O-minimal structures and real closed fields".
