Short Model Theory Huddle 2

16th - 17th - 18th of June 2021

All times are in CEST. See our page on ResearchSeminars.org for a schedule updated to your preferred time zone.

18/06/2021 (14:00-21:00 CEST)

• Kim-independence in positive logic - 14:00 - 14:45 [Slides]
  Mark Kamsma (University of East Anglia)
  This is joint work with Jan Dobrowolski. Positive logic is a proper generalisation of first-order logic where negation is not built in, but can be added as desired. In this talk I will give a brief introduction to positive logic. We will have a look at the challenges that positive logic presents us and how they can be solved. I will also explain why positive logic is useful and how we can use it to study structures that do not fit the usual framework of first-order logic. An important dividing line in the class of unstable theories is being NSOP1, which is more general than being simple. There has been a lot of recent work on the class of NSOP1 theories for first-order logic. The natural independence relation in this class is given by Kim-independence, generalising forking independence from stable and simple theories. We have generalised work on Kim-independence to the setting of positive logic. I will assume no prior knowledge of this topic and will introduce the definitions of NSOP1 and Kim-independence, and how we can make sense of this in positive logic. Our results can be summarised as a Kim-Pillay style theorem: a thick positive theory is NSOP1 if and only if there is a nice enough independence relation, and in this case the independence relation is given by Kim-independence.

• How many quantifiers are needed to existentially define a given subset of a field? - 15:00 - 15:45 [Slides]
  Nicolas Daans (University of Antwerp)
  When a subset of a field is existentially definable, one can ask what the smallest number of quantifiers is needed for an existential formula defining this set. This question turns out to be quite hard in general, since answering it requires a thorough understanding of the arithmetic of the field in question. On the other hand, if one asks how many quantifiers are needed to existentially define a definable class of subsets (e.g. the set of sums of 4 squares) uniformly across fields, then the problem can be tackled via a model-theoretic approach, which I sketch in this talk.
  Based on joint work with Arno Fehm and Philip Dittmann.

• A nondefinability result for Weierstrass $\wp$ functions - 16:00 - 16:45 [Slides]
  Raymond McCulloch (University of Manchester)
  The model theory of the structure $\mathbb{R}_{\exp}=(\bar{\mathbb{R}},+,\times,0,1,<,\exp)$ is well known. A result of Bianconi in this area states that no restriction of the sine function is definable in $\mathbb{R}_{\exp}$. This can be restated to give that no restriction of the complex exponential function is definable in $\mathbb{R}_{\exp}$. The Weierstrass $\wp$ function shares many of the properties of $\exp$ and this motivates the study of the model theory of $\wp$. In 2005 Macintyre observed that for certain special lattices, the Weierstrass $\wp$ function restricted to any disc in the complex plane is definable in the structure given by expanding the ordered real field by the $\wp$ function restricted to a real interval. These special lattices are real and have complex multiplication. In this talk I shall state the converse to this result, which is in fact a version of Bianconi’s theorem for the $\wp$ function and describe the proof. I will conclude by stating further results on the nondefinability of $\wp$ and the modular $j$ function.

• Forking in valued fields - 18:00 - 18:45 [Slides]
  Akash Hossain (Université Paris-Saclay)
  I will start this talk by discussing forking, its importance in model theory, and the informal ideas behind abstract independence relations. Then, I will give an introduction to the model theory of valued fields, in particular the least complicated ones : algebraically closed valued fields of residue characteristic zero (ACVF0). The subject of my thesis is the study of forking in valued fields, so I will eventually give the main results I obtained (still in ACVF0 for the sake of simplicity). I may also talk about the questions I am currently working on, related to forking in valued fields.

• Finding Dividing Lines in Graph Theory - 19:00 - 19:45 [Slides]
  Aris Papadopoulos (University of Leeds)
  An important theme in modern graph theory is the study of the density/sparsity dichotomy. Intuitively, a graph is dense if it has a lot of edges and sparse if it does not. Even though there is no precise line separating the two classes, there are properties that make us call a class of graphs sparse or dense. An example of such a property comes from the notion of "nowhere-density" which has been rediscovered by various authors (Podewski and Ziegler in 1978, Nešetřil and Ossona de Mendez in 2010) and is, for hereditary classes of graphs, equivalent to stability, in the model-theoretic sense. Even more surprisingly, in the context of monotone classes of binary structures, it turns out that this notion is equivalent to NIP (Adler and Adler, 2014). Recent work (Nešetřil and Ossona de Mendez 2020, Simon and Toruńczyk 2021, and others) has uncovered more connections between graph classes with "nice structural properties" (e.g. classes of bounded rankwidth/twinwidth) and dividing lines à la Shelah. The aim of this expository talk is to give a brief introduction/review of some relevant terminology and some useful tools, such as first-order transductions, and explain some of the more interesting results in this intersection of model theory and graph theory.

• Keynote lecture: Diophantine subsets and subfields of large fields - 20:00 - 20:45 [Slides]
  Sylvy Anscombe (Université de Paris, IMJ-PRG)
  The class of large fields includes plenty of familiar examples: algebraically closed, separably closed, real closed, p-adically closed, pseudo algebraically closed, pseudo real closed, etc. Many of these fields are well-understood model theoretically -- for example, by a quantifier-elimination. In his 2010 paper, Arno Fehm showed any infinite diophantine subset of a perfect large field is not contained in any proper subfield.
  We will give a proof of the analogous result for the imperfect case, and explore the link with definability in henselian valued fields.

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