H2020Индивидуална стипендия2017–2019

GROUPNIP · Model theory of groups in NIP theories

„Хоризонт 2020“ — Действия „Мария Склодовска-Кюри“

Период: 2017-03-01 → 2019-02-28
Финансиране от ЕС: 183 455 €
Участници: 1
Схема: MSCA-IF-EF-ST

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Накратко на български

Математическата логика и теорията на групите изследват как се описват свойствата на структури, като например графите. Работата по този проект помага за по-доброто разбиране на класификацията на логическите теории в съвременната математика.

Този кратък обзор е генериран от изкуствен интелект

Кратко обяснение, генерирано от езиков модел по текста на CORDIS. Оригиналът е по-долу.

Резултати накратко

Model theory of groups in NIP theories

The project was in model theory (mathematical logic), with close connections to algebra, especially group theory. Model theory concerns expressibility in logical languages of properties of mathematical structures (e.g. graphs or groups). A key notion is that of a `definable set' (generalising algebraic varieties). Model theory identifies `tame' classes of structures/first order theories such as stable theories, or the much richer class of NIP theories in which definable sets are well-understood, and finds/applies generalisations of geometric notions such as algebraic independence and dimension. This project focusses on groups in theories satisfying the NIP property, and also some other related tameness properties, such as NTP2 (being a generalization of NIP), and NSOP1 (being another generalization of stability). The project deals with natural questions regarding neostability of algebraic structures. Answering such questions improves the understanding of the "neostability map" (a map of the universe of all first order theories, clustered according to the neostability properties), which is one of the main tasks of modern model theory. The connections of the undertaken research to notions from classical mathematics (such as homology groups or Galois groups) is also likely to result in interest in the subject-matter of the project among researches from other branches of mathematics. There were three main Workpackages, each with concrete objectives stated. The first concerned connections of model theory to topology obtained by applying model-theoretic ideas to so-called Polish structures, that is, to structures obtained by considering suitable actions of a Polish groups on sets. The second concerned the study of model-theoretic homology groups - an adaptation of the classical notion of homology group to the model theoretic setting, giving a new tool for analysing the structure of a type (that is, a consistent set of formulas - a fundamental notion in model theory). The third Workpackage concerned investigating the implications of the NIP property for profinite groups, i.e. inverse limits of finite groups.

Текст от CORDIS, на английски · Данни: CORDIS, © Европейски съюз

Цел на проекта

The project is in model theory (mathematical logic), with close connections to algebra, especially group theory. Model theory concerns expressibility in logical languages of properties of mathematical structures (e.g. graphs or groups). A key notion is that of a `definable set' (generalising algebraic varieties). Model theory identifies `tame' classes of structures/theories such as stable theories, or the much richer class of NIP theories in which definable sets are well-understood, and finds/applies generalisations of geometric notions such as algebraic independence. This project focusses on groups in NIP theories, both as invariants and as definable objects. The three research Workpackages (with specific objectives) concern(1) Applying methods from the recently-developed `Polish structures' to problems of topological dynamics of type spaces in NIP theories,(2) finding methods to compute homology groups (which measure `n-amalgamation') of generically stable types in NIP theories, and characterising the homology groups for algebraically closed valued fields,(3) examining the fine structure of NIP profinite groups, viewed in a 2-sorted language with open subgroups uniformly definable.The Fellow, Dobrowolski, will receive training through research in the model theory groups in Leeds and (on a 3-month secondment) in Lyon. There will be knowledge transfer to Dobrowolski of expertise in model theory, group theory, and topological dynamics in Leeds and Lyon, and Dobrowolski will transfer to Leeds and Lyon the understanding he has built up in Wroclaw and Seoul, on Polish structures and homology groups of first order theories. He will be supervised by Macpherson in Leeds and Wagner in Lyon, but also interact with the large model theory groups in both centres. He will receive complementary training in Leeds on a range of professional academic skills, including outreach, and will take advantage of opportunities for outreach activities in Leeds related to his research.

Оригинален текст от CORDIS (на английски).

Участници

UNIVERSITY OF LEEDS · LeedsКоординаторОбединеното кралство