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H2020Individual fellowship2016–2018

MODFIN · Model theory of finite and pseudofinite structures

Horizon 2020 — Marie Skłodowska-Curie Actions

Duration
2016-06-01 → 2018-05-31
EU contribution
€183,455
Participants
1
Scheme
MSCA-IF-EF-ST

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Results in brief

Model theory of finite and pseudofinite structures

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.

Data: CORDIS, © European Union

Project objective

The project is in model theory (mathematical logic), which concerns the expressibility in logical languages of properties of mathematical structures (e.g. graphs, groups, rings). Model theory aims to identify borders between `tame' and `wild' objects in mathematics, and to pin down abstract notions of independence and dimension and understand the geometry of `definable sets' in a structure, often with wide-ranging applications. This project focusses on classes of finite structures (e.g. the class of all finite fields), and on the `ultraproduct' construction which converts a class of finite structures to an infinite `pseudofinite' structure' which inherits properties of the class and is amenable to model-theoretic methods, with applications for the finite structures. Key objectives include:(i) proving a trichotomy for pseudofinite geometries -- they should be `trivial', `group-like', or `field-like'; (ii) developing current concepts from abstract model theory for pseudofinite structures;(iii) identifying first order properties of pseudofinite groups, and constraints on their possible quotients;(iv) finding links between the model-theoretic `independence theorem', Gowers' notion of `quasi-random groups', and the Szemeredi regularity theorem in graph theory. (v) model theory of finite ordered structures, and links to finite model theory.To support his future academic research career, the Fellow, Garcia, will receive training through research in the model theory groups in Leeds and (through a secondment) Lyon. There will be knowledge transfer to Garcia of expertise in model-theoretic algebra of Leeds and Lyon, and Garcia will also build knowledge of finite model theory and its computer science applications. He will receive complementary training in many research skills (including outreach), and will transfer to Leeds expertise he has gained in the excellent model theory groups in Berkeley and Bogota, while deepening EU-Colombia mathematics links.

Original text from CORDIS.

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Data: CORDIS, © European Union