Participants can register here.

12:30-13:00 | Arrival |

13:00-14:00 | Mark Kamsma (Queen Mary University of London) Title: Dividing lines between positive theoriesAbstract: This is joint work with Francesco Gallinaro and Anna Dmitrieva. We give definitions of the properties OP, IP, k-TP, TP1, k-TP2, SOP1, SOP2 and SOP3 in positive logic, a proper generalisation of full first-order logic where negation is not built in, but can be added as desired. We prove various implications and equivalences between these properties, matching those known from full first-order logic. We will look at an example illustrating why these definitions have to be altered for positive logic. We will also give positive versions of the well-known results that a theory is stable iff it is NSOP and NIP, and that a theory is simple iff it is NTP1 and NTP2. In the positive versions of these theorems we replace NSOP and NTP1 respectively by NSOP1. This allows us to give completely new proofs based on Kim-independence. |

14:00-15:00 | Alberto Miguel Gómez (Imperial College, London) Title: A strictly \(NSOP_4\) structure without stationarity Abstract: 3-hypertournaments are combinatorial structures that generalise tournaments to the ternary relational case in a similar way that 3-hypergraphs generalise graphs. Recently, Cherlin, Hubička, Konečny, and Nešetřil have identified a countable homogeneous 3-hypertournament with some wild behaviour from the structural-Ramsey-theoretic point of view. In this talk, I will show that this behaviour has a model-theoretic counterpart: namely, its theory is strictly NSOP4. Furthermore, the usual criteria from the literature do not apply in this case, making this a novel example of an NSOP4 theory. In this talk, I will discuss the proof of this fact and how it relates to the other known examples of NSOP4 in light of recent developments in the area. |

15:00-15:30 | Coffee, Tea & Biscuits |

15:30-16:30 | Gabriel Ng (University of Manchester) Title: Differentially Large Fields and Taylor Morphisms Abstract: Differentially large fields are a class of differential fields introduced by León Sánchez and Tressl which generalise the notion of largeness to differential fields. In order to study these fields, León Sánchez and Tressl construct the Twisted Taylor Morphism, a functor which constructs differential ring homomorphisms into the ring of power series from ring homomorphisms into the base field. In this talk, we will begin by giving a brief overview of the theory of differentially large fields. We will then consider a generalised notion of a Taylor morphism, and give a complete characterisation of such objects in terms of a twisted Hurwitz construction. |

16:30-17:30 | Jonathan Kirby (University of East Anglia) Title: Independence Relations for Exponential Fields Abstract: In classical first-order logic, the presence of an independence relation on models of a complete theory T can be used to show that T is strongly minimal, stable, simple, or NSOP1. Something analogous works in various generalisations of first-order logic, including positive logic and omitting types classes. In this talk I will illustrate the general principle by constructing various independence relations on exponential fields, that is, fields equipped with a homomorphism from their additive group to their multiplicative group, like the usual real and complex exponential maps. These independence relations can be used to prove that various AECs of exponential fields are quasiminimal, stable, or NSOP1 . In some of the stable cases, there are open questions around extending from the countable models, which are well-understood, to the uncountable ones. |

17:30-18:30 | Pub |

18:30 | Dinner |

The 15th SEEMOD meeting was held at Keble College, Oxford on June 13th 2023.

12:00-13:00 | Lunch |

13:00-14:00 | Benedikt Stock (Oxford) Title: On Fields Elementarily Characterised by their Absolute Galois GroupsAbstract: Galois groups contain a lot of arithmetic information about their base fields. In this talk, I will report on fields that are elementarily completely characterised by their absolute Galois groups. We will also hint at some recent attempts and approaches to eliminate certain edge cases that appear in the classification of these fields. |

14:00-15:00 | Luka Ilic (Queen Mary University of London) Title: Internal Reasoning for Difference Tannakian DualityAbstract: The idea of this talk is to give a comprehensive introduction into internal logic for difference structures and illustrate its usefulness via the example of Tannakian duality. To achieve this, we will go over notions from categorical logic and explain what we mean by internal logic inside the topos of difference sets and what use it has for studying difference structures. We will talk about how to view difference fields, rings and modules and their properties in this way and finish by explaining a resulting version of Tannakian duality. |

15:00-15:30 | Coffee, Tea & Biscuits |

15:30-16:30 | Paolo Marimon (Imperial) Title: Measures in ternary homogeneous structuresAbstract: We study invariant Keisler measures and MS-measurability in the context of ternary homogeneous structures. We prove that the universal homogeneous two-graph has a unique invariant Keisler measure in spite of not having any invariant type. This is related to Jahel's result that this structure is uniquely ergodic. Moreover, we prove that the generic tetrahedron-free 3-hypergraph is not MS-measurable (i.e. it does not have a dimension and a set of measures satisfying various desirable properties, such as a version of Fubini's theorem). We wanted a better understanding of the interactions between measures and higher amalgamation properties for (simple) \(\omega\)-categorical structures. Our result on the universal homogeneous two-graph implies that the measures of higher amalgamations are less well-behaved than those of 3-amalgamations, which can be studied using results of Hrushovski. In spite of these negative results, we can still obtain equations which must be satisfied by the measures of higher amalgamations in an \(\omega\)-categorical MS-measurable context. With these, we show that the generic tetrahedron-free 3-hypergraph is not MS-measurable. This is the first known example of a supersimple \(\omega\)-categorical one-based structure which is not MS-measurable. |

16:30-17:30 | Sylvy Anscombe (Université Paris Cité) Title: Interpretations of fragments of theories of fieldsAbstract: In previous work with Fehm, and then Dittmann and Fehm, we found that the existential theory of an equicharacteristic henselian valued field is axiomatised using the existential theory of its residue field, conditionally, similar to an earlier theorem of Denef and Schoutens -- giving a transfer of decidability for existential theories. In this talk I’ll describe parts of ongoing work with Fehm (in the main different to those discussed recently at CIRM) in which we use an "abstract" framework for interpreting families of incomplete theories in others in order to find transfers of decidability in various settings. I will discuss consequences for theories of PAC fields and parts of the universal-existential theory of equicharacteristic henselian valued fields. |

18:00-20:00 | Dinner at the Giggling Squid |

12:00-13:00 | Lunch |

13:00-14:00 | Mark Kamsma (Imperial) Title: Categorical NeostabilityAbstract: Neostability focuses on developing the powerful tools and ideas for stable theories in broader, less well-behaved, classes of theories. This has been hugely successful in simple theories and more recently in NSOP1 theories. A central notion in this context is that of an independence relation, which in stable and simple theories comes from forking and in NSOP1 theories it comes from Kim-forking. There are many mathematically interesting structures that cannot be studied in the classical framework of first-order logic, but where ideas from neostability, such as independence, apply. For this reason, neostability has been developed in more general logical frameworks, such as continuous logic, positive logic and a very general category-theoretic approach that subsumes all these frameworks. In this talk I will focus on the categorical approach to independence relations, and how even in this general setting we get canonicity theorems that are surprisingly close to those we know from first-order logic. I will finish with recent examples of where the categorical framework can be applied. |

14:00-15:00 | Soinbhe Nic Dhonncha (Manchester) Title: Notions of flatnessAbstract: Flatness of an R-module is typically defined in relation to exactness of tensor products. There are many equivalent characterisations, however, whose generalisations may not always agree. One characterisation is in terms of pure epimorphisms and positive primitive formulae. Using these as our basis, we will explore flatness in categories whose objects can be seen as certain chains of R-modules. |

15:00-15:30 | Coffee Break |

15:30-16:30 | Chieu-Minh Tran (National University of Singapore) Title: Measure doubling of small sets in \(\mathrm{SO}(3,\mathbb{R})\) Abstract: Let \(\mathrm{SO}(3,\mathbb{R})\) be the 3D-rotation group equipped with the real-manifold topology and the normalized Haar measure \(\mu\). Confirming a conjecture by Emmanuel Breuillard and Ben Green, we show that if \(A \subseteq \mathrm{SO}(3,\mathbb{R})\) is open and has sufficiently small measure, then $$ \mu(A^2) > 3.99 \mu(A).$$ We also show a more general result for the product of two sets, which can be seen as a Brunn-Minkowski-type inequality for sets with small measure in \(\mathrm{SO}(3,\mathbb{R})\). (Joint with Yifan Jing and Ruixiang Zhang) |

16:00-17:00 | Charlotte Kestner (Imperial) Title: Generalised measurability and bilinear forms Abstract: In this talk I will briefly go over measurable and generalised measurable structures, giving examples and non-examples. I will then go on to consider the two sorted structure (V,F,β) where V is an infinite dimensional vector space over F an infinite field, and \beta a bilinear form on this vector space. In particular I will consider the interaction of different notions of independence when this structure is pseudo finite. I will finish with some questions around generalised measurable structures. |

18:00-20:00 | Pub and dinner |