## Periodic Reporting for period 1 - MODFIN (Model theory of finite and pseudofinite structures)

Reporting period: 2016-06-01 to 2018-05-31

The Marie Skłowdowska-Curie Individual Fellowship MODFIN was held by the researcher Dr. Darío García in the School of Mathematics at the University of Leeds from 1 June 2016 to 31 May 2018. There was a 3-month secondment at the Institut Henri Poincaré in Paris from 8 January 2018 to 6 April 2018 to participate in the programme ‘Model theory, combinatorics and valued fields’.

The project MODFIN was in model theory. This is a branch of mathematical logic, which aims to study the structures of pure mathematics (graphs, groups, rings, fields, vector spaces, topological spaces, etc.) from the viewpoint of what can be said about them in a formal ‘first order’ logical language. Concepts such as ‘algebraic variety’ from algebraic geometry are generalised in model theory to the notion of ‘definable set’. The subject has a rich internal theory, applications throughout mathematics, and its fertility was testified by the success of the above programme at the Institut Henri Poincaré, attended by some 300 researchers from many parts of mathematics.

MODFIN was more particularly about the model theory of finite and ‘pseudofinite’ structures. It had close connections to, for example, combinatorics (via extremal graph theory) and theoretical computer science (via finite model theory and computational complexity). By a pseudofinite structure we mean a structure which is infinite but such that every statement in first order logic which is true of it also holds of some finite structure. These can be seen as logical smoothings or limits of finite structures, and give a bird’s-eye logical perspective on finite structures.

There were three main Workpackages, each with concrete objectives attached. The first concerned the `pure' model theory of pseudofinite structures, specifically on the geometry of definable sets (a definable set is a set of solutions of a logical formula) and on a notion of pseudofinite dimension. The second concerned pseudofinite groups, concerning a conjecture of Zilber on possible quotients of pseudofinite groups, and concerning connections to extremal combinatorics. The third concerned totally ordered pseudofinite structures, and possible connections to finite model theory.

The project MODFIN was in model theory. This is a branch of mathematical logic, which aims to study the structures of pure mathematics (graphs, groups, rings, fields, vector spaces, topological spaces, etc.) from the viewpoint of what can be said about them in a formal ‘first order’ logical language. Concepts such as ‘algebraic variety’ from algebraic geometry are generalised in model theory to the notion of ‘definable set’. The subject has a rich internal theory, applications throughout mathematics, and its fertility was testified by the success of the above programme at the Institut Henri Poincaré, attended by some 300 researchers from many parts of mathematics.

MODFIN was more particularly about the model theory of finite and ‘pseudofinite’ structures. It had close connections to, for example, combinatorics (via extremal graph theory) and theoretical computer science (via finite model theory and computational complexity). By a pseudofinite structure we mean a structure which is infinite but such that every statement in first order logic which is true of it also holds of some finite structure. These can be seen as logical smoothings or limits of finite structures, and give a bird’s-eye logical perspective on finite structures.

There were three main Workpackages, each with concrete objectives attached. The first concerned the `pure' model theory of pseudofinite structures, specifically on the geometry of definable sets (a definable set is a set of solutions of a logical formula) and on a notion of pseudofinite dimension. The second concerned pseudofinite groups, concerning a conjecture of Zilber on possible quotients of pseudofinite groups, and concerning connections to extremal combinatorics. The third concerned totally ordered pseudofinite structures, and possible connections to finite model theory.

García made significant progress on all the Workpackages, partly on his own, and partly in collaboration with Ghadernezhad (Imperial College London), Tingxiang Zou (Lyon 1), and his supervisor Dugald Macpherson in Leeds.

On Workpackage 1, García and Tingxiang Zou studied the abstract model theoretic meaning of pseudofinite dimension for ‘H-structures’, certain expansions of a structure by a predicate for a set H assumed to have ‘coarse’ pseudofinite dimension 0. They understood how this dimension behaves in enlightening key examples. García also formulated a fascinating conjecture that every stable omega-categorical pseudofinite structure is one-based. His close study with Ghadernezhad of an example stemming from graph-theoretic work of Shelah and Spencer provides strong evidence for this.

García worked with Macpherson on Workpackage 2. They noted that every pseudofinite omega-categorical group is nilpotent-by-finite – a straightforward consequence of known results, which invites closer study of omega-categorical nilpotent pseudofinite groups. They also picked up a recent concept of a ‘faux finite’ structure from a communication to them by A. Chernikov. This is essentially an infinite structure whose logic is determined by a single sentence σ (plus an axiom of infinite), where σ also holds of arbitrarily large finite structures. They mainly explored faux finite groups, classifying abelian pseudofinite groups of finite exponent, and obtaining partial results towards a general structure theory for faux finite groups.

Towards Workpackage 3, García has submitted for publication a paper ‘Ordered asymptotic classes of finite structures’, where he developed the model theory of finite totally ordered structures which satisfy a fascinating regularity property on the sizes of their definable sets. He had began this work in his PhD thesis, but extended it during the project, finding key examples and counter-examples which show the limits of the structure theory.

This paper has open access on https://arxiv.org/abs/1410.2615. We expect three further papers to be published containing the results described above (and extensions). All will be placed on arXiv and on other public repositories.

On Workpackage 1, García and Tingxiang Zou studied the abstract model theoretic meaning of pseudofinite dimension for ‘H-structures’, certain expansions of a structure by a predicate for a set H assumed to have ‘coarse’ pseudofinite dimension 0. They understood how this dimension behaves in enlightening key examples. García also formulated a fascinating conjecture that every stable omega-categorical pseudofinite structure is one-based. His close study with Ghadernezhad of an example stemming from graph-theoretic work of Shelah and Spencer provides strong evidence for this.

García worked with Macpherson on Workpackage 2. They noted that every pseudofinite omega-categorical group is nilpotent-by-finite – a straightforward consequence of known results, which invites closer study of omega-categorical nilpotent pseudofinite groups. They also picked up a recent concept of a ‘faux finite’ structure from a communication to them by A. Chernikov. This is essentially an infinite structure whose logic is determined by a single sentence σ (plus an axiom of infinite), where σ also holds of arbitrarily large finite structures. They mainly explored faux finite groups, classifying abelian pseudofinite groups of finite exponent, and obtaining partial results towards a general structure theory for faux finite groups.

Towards Workpackage 3, García has submitted for publication a paper ‘Ordered asymptotic classes of finite structures’, where he developed the model theory of finite totally ordered structures which satisfy a fascinating regularity property on the sizes of their definable sets. He had began this work in his PhD thesis, but extended it during the project, finding key examples and counter-examples which show the limits of the structure theory.

This paper has open access on https://arxiv.org/abs/1410.2615. We expect three further papers to be published containing the results described above (and extensions). All will be placed on arXiv and on other public repositories.

The results of the project, and the broader area of research, were disseminated widely. García gave lecture courses on pseudofinite structures at the IPM in Tehran and at the Institut Henri Poincaré in Paris, the latter course leading to a very active informal study group by researchers in the programme on pseudofinite groups. He gave many seminar talks on his research, in Leeds, Manchester, Preston, London, Oxford, Lyon, Paris, Freiburg, and Isfahan. He also, as planned in the proposal, organised a workshop ‘Model theory of finite and pseudofinite groups’ in Leeds April 9-11 2018. This had 28 participants including 15 speakers (5 female), and, in addition to funding from the project, received support from the University of Leeds and from the British Logic Colloquium, to ensure wider participation. In total, during the fellowship he participated in 14 conferences or workshops, giving talks in three.

Leeds was an ideal host for the project, since it has one of the leading logic groups in Europe, with breadth across logic (model theory set theory, proof theory and constructivism, computability theory, logic in computer science, philosophical logic), with close connections between these subgroups, joint seminars, other postdoctoral fellows, and a large group of over 20 PhD students who interact closely in joint seminars. Thus, García was able to interact with a large collection of PhD students, and see model theory within a wider context of logic and its connections to computer science.

The project led to a two-way transfer of knowledge between García and between the host institution in Leeds (and the wider model theory community). García’s deep understanding of pseudofinite structures was conveyed in the above seminars and courses, and also in his contributions to study groups, such as one on the ultraproduct approach by Elek and Szegedy to the hypergraph Szemerédi Regularity Lemma. His understanding of pure model theory (in particular, of generalisations of stability theory) was conveyed to PhD students in Leeds in a lecture course, and he gave extensive other support to Leeds PhD students. In the other direction, his understanding of group theory grew through discussions with Macpherson, and of general model theory through other seminar activities in Leeds.

García also gained wider experience relevant to his academic career. He played the leading role in managing his fellowship and gained experience organising conferences. His interactions and support for PhD students in Leeds and the two workshops he ran for school students in Leeds equip him for future PhD supervision and experience in outreach activities. García has now begun a postdoctoral fellowship in the very strong model theory group in Bogotá. He aims for a permanent academic career in Colombia, and to build on the collaborations with European researchers developed during the Fellowship. In particular, one goal is to set up future networks and links between Colombian and European model theorists.

Leeds was an ideal host for the project, since it has one of the leading logic groups in Europe, with breadth across logic (model theory set theory, proof theory and constructivism, computability theory, logic in computer science, philosophical logic), with close connections between these subgroups, joint seminars, other postdoctoral fellows, and a large group of over 20 PhD students who interact closely in joint seminars. Thus, García was able to interact with a large collection of PhD students, and see model theory within a wider context of logic and its connections to computer science.

The project led to a two-way transfer of knowledge between García and between the host institution in Leeds (and the wider model theory community). García’s deep understanding of pseudofinite structures was conveyed in the above seminars and courses, and also in his contributions to study groups, such as one on the ultraproduct approach by Elek and Szegedy to the hypergraph Szemerédi Regularity Lemma. His understanding of pure model theory (in particular, of generalisations of stability theory) was conveyed to PhD students in Leeds in a lecture course, and he gave extensive other support to Leeds PhD students. In the other direction, his understanding of group theory grew through discussions with Macpherson, and of general model theory through other seminar activities in Leeds.

García also gained wider experience relevant to his academic career. He played the leading role in managing his fellowship and gained experience organising conferences. His interactions and support for PhD students in Leeds and the two workshops he ran for school students in Leeds equip him for future PhD supervision and experience in outreach activities. García has now begun a postdoctoral fellowship in the very strong model theory group in Bogotá. He aims for a permanent academic career in Colombia, and to build on the collaborations with European researchers developed during the Fellowship. In particular, one goal is to set up future networks and links between Colombian and European model theorists.