## Periodic Reporting for period 1 - GROUPNIP (Model theory of groups in NIP theories)

Reporting period: 2017-03-01 to 2019-02-28

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.

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.

The work on Workpackage 1 was influenced by refutation of the main conjectures by P. Simon. The main results on this Workpackage were obtained by Dobrowolski in collaboration with F.-V. Kuhlamann on generalized IFS(iterated function system)-attractors (which was a continuation of an earlier collaboration). Some natural examples of IFS-attractors (called also fractals) occur very naturally in algebra, via certain valued fields, many of them being also important objects in model theory. The goal of the work with Kuhlmann was to find a generalization of the notion of an IFS-attractor allowing more algebraic and model-theoretic examples, but being in close analogy with the original definition.

Several such possible generalizations were introduced, and for each of them topological consequences were proven, and classes of algebraic examples were found.

On Workpackage 2, in a joint work, Dobrowolski, B. Kim, A. Kolesnikov, and J. Lee studied a notion of localized Lascar-Galois group - this is a natural analogue of the notion of Lascar group - a very important invariant of a first order theory. In the work by Dobrowolski, Kim, Kolesnikov, and Lee, the questions about some desirable properties of these groups (related also to homology groups) hold were addressed. A positive answer was given under some additional assumptions, and it was proven that none of these assumptions can be removed by constructing several counterexamples to stronger statements.

Another work within this Workpackage was done by Dobrowolski in collaboration with B. Kim and N. Ramsey on Kim-independence in NSOP1 theories. Kim-independence is a variant of forking-independence - an independence relation enjoying very nice properties in so-called simple theories, thus closely resembling linear independence in linear spaces and algebraic independence in fields. In fact, simple theories are exactly those in which forking independence behaves nicely, and NSOP1 theories as those in which Kim-independence behaves nicely - the latter was established in an earlier work by I. Kaplan and N. Ramsey. However, there was a limitation in their setting - they considered only independence over a model. In the work by Dobrowolski, Kim, and Ramsey this limitation was removed, at a cost of a very mild assumption of forking existence, which is conjectured to be true in all NSOP1 theories.

On Workpackage 3, several algebraic consequences of tameness assumptions on groups were proven. Dobrowolski in collaboration with J. Goodrick proved that left-ordered groups of inp-rank equal to one (where inp rank is a cardinal-valued rank giving a hierarchy on the class of NTP2 theories) are abelian, generalizing a previous result of P. Simon on bi-ordered groups. Secondly, it was proven by Dobrowolski and F. Wagner that omega-categorical groups of finite inp-rank are finite-by-abelian-by-finite, thus very close to being commutative. Finally, Dobrowolski, D. Hoffmann, and J. Lee, generalized a well-known theorem characterizing elementary equivalence of pseudo algebraically closed (PAC) fields in terms of isomorphisms of their absolute Galois groups to the setting of pseudo algebraically closed structures, an abstract generalization of the class of PAC (perfect) fields.

Dobrowolski has established the project website http://www.math.uni.wroc.pl/~dobrowol/mc.html

where the outcomes of the project are presented. In particular, information on where to find the papers resulting from the project in open access can be found is provided there. The results were also disseminated widely through seven invited conference talks, and through several seminar talks in Leeds, Wroclaw, Jerusalem, Lyon, Paris, and Manchester.

Several such possible generalizations were introduced, and for each of them topological consequences were proven, and classes of algebraic examples were found.

On Workpackage 2, in a joint work, Dobrowolski, B. Kim, A. Kolesnikov, and J. Lee studied a notion of localized Lascar-Galois group - this is a natural analogue of the notion of Lascar group - a very important invariant of a first order theory. In the work by Dobrowolski, Kim, Kolesnikov, and Lee, the questions about some desirable properties of these groups (related also to homology groups) hold were addressed. A positive answer was given under some additional assumptions, and it was proven that none of these assumptions can be removed by constructing several counterexamples to stronger statements.

Another work within this Workpackage was done by Dobrowolski in collaboration with B. Kim and N. Ramsey on Kim-independence in NSOP1 theories. Kim-independence is a variant of forking-independence - an independence relation enjoying very nice properties in so-called simple theories, thus closely resembling linear independence in linear spaces and algebraic independence in fields. In fact, simple theories are exactly those in which forking independence behaves nicely, and NSOP1 theories as those in which Kim-independence behaves nicely - the latter was established in an earlier work by I. Kaplan and N. Ramsey. However, there was a limitation in their setting - they considered only independence over a model. In the work by Dobrowolski, Kim, and Ramsey this limitation was removed, at a cost of a very mild assumption of forking existence, which is conjectured to be true in all NSOP1 theories.

On Workpackage 3, several algebraic consequences of tameness assumptions on groups were proven. Dobrowolski in collaboration with J. Goodrick proved that left-ordered groups of inp-rank equal to one (where inp rank is a cardinal-valued rank giving a hierarchy on the class of NTP2 theories) are abelian, generalizing a previous result of P. Simon on bi-ordered groups. Secondly, it was proven by Dobrowolski and F. Wagner that omega-categorical groups of finite inp-rank are finite-by-abelian-by-finite, thus very close to being commutative. Finally, Dobrowolski, D. Hoffmann, and J. Lee, generalized a well-known theorem characterizing elementary equivalence of pseudo algebraically closed (PAC) fields in terms of isomorphisms of their absolute Galois groups to the setting of pseudo algebraically closed structures, an abstract generalization of the class of PAC (perfect) fields.

Dobrowolski has established the project website http://www.math.uni.wroc.pl/~dobrowol/mc.html

where the outcomes of the project are presented. In particular, information on where to find the papers resulting from the project in open access can be found is provided there. The results were also disseminated widely through seven invited conference talks, and through several seminar talks in Leeds, Wroclaw, Jerusalem, Lyon, Paris, and Manchester.

The project led to mutual transfer of knowledge between Dobrowolski and the model theory group in Leeds (as well as Lyon, during the secondment there). Dobrowolski received extensive training on subjects important to the project, such as the theory of profinite groups and p-groups, through discussions with Macpherson, and on various other subjects through seminars and reading groups. On the other hand, the background of Dobrowolski from Wroclaw and Seoul allowed him to convey his understanding of various problems, especially those related to connections between model theory and topology, in seminar talks, study groups, and discussions with PhD students.

Dobrowolski has also gained experience important to his planned academic career. This includes skills such as planning a budget, organizing conferences (he organised a 3-day workshop in Leeds), applying for funds, as well as teaching experience and outreach activities (he gave seven classes for high school students).

Dobrowolski aims to develop the collaborations started during the fellowship, thus strengthening the connection between various model theory groups, such as Leeds, Lyon, Wroclaw, Jerusalem and Seoul. He has now started a one-year postdoctoral fellowship at the University of Leeds. The scientific results and experience gained by him during the GROUPNIP project put him in a strong position to apply for a permanent position in one of the model theory centres as the next step in his career.

Dobrowolski has also gained experience important to his planned academic career. This includes skills such as planning a budget, organizing conferences (he organised a 3-day workshop in Leeds), applying for funds, as well as teaching experience and outreach activities (he gave seven classes for high school students).

Dobrowolski aims to develop the collaborations started during the fellowship, thus strengthening the connection between various model theory groups, such as Leeds, Lyon, Wroclaw, Jerusalem and Seoul. He has now started a one-year postdoctoral fellowship at the University of Leeds. The scientific results and experience gained by him during the GROUPNIP project put him in a strong position to apply for a permanent position in one of the model theory centres as the next step in his career.