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Foundations for computing with infinite linear groups

Periodic Reporting for period 1 - InfGroups (Foundations for computing with infinite linear groups)

Reporting period: 2016-05-01 to 2018-04-30

This project is in a frontier area at the interface of algebra and computer science. The overall research objective is to build up a new domain of computational group theory:
computing with finitely generated linear groups over infinite fields. To achieve that goal, the following problems were solved.
We developed innovative methodology to compute in this class of groups, based on a computer realization of properties of linear algebraic groups.
We applied the methodology to design algorithms for all major classes of finitely generated linear groups. Our results also justify decidability of related algorithmic problems;
an achievement of interest in its own right.
The algorithms have been implemented in the main computer algebra systems, and made available publicly.
We have used this software to solve open problems in group theory and its applications by means of computer-aided experimentation.
The project will contribute to the international competitiveness of the EU in fundamental science,
specifically by strengthening its leadership in the development of major computer algebra systems.The project catalyzed collaboration between EU universities and academic institutions world-wide.
This, together with dissemination activities of the project, contributed to EU plans to increase numbers of postdoctoral researchers by stimulating interest in STEM research.
a. Overview of results of the project
a.1 Methodology. We developed novel techniques for computing with finitely generated linear groups over infinite domains based on methods for linear algebraic groups and Lie algebras,
and results for discrete subgroups of Lie groups.
New methods and algorithms for practical computing with matrix groups over finite rings are of independent interest.
a.2 Algorithms. The following algorithms designed in the project are milestones: testing Zariski density; computing congruence quotients of dense subgroups (computer realization of the strong
approximation theorem); computing arithmetic closure of dense subgroups of semisimple algebraic groups satisfying congruence subgroup property; algorithms for linear groups of finite rank over number fields.
a.3 Software and applications. Our algorithms have been implemented in computer algebra systems GAP, Magma.
The software designed in the project was used to investigate properties and structure of groups which arise in applications of group theory in a number of areas.

b. Exploitation and dissemination of results.
b.1 The results obtained in the project available as peer-reviewed papers, preprints, and conference proceedings
(eight published or accepted for publication, and three further to be submitted in 2018).
Open access to publications provided via online repositories.
b.2 The dissemination activities include participation in eight international conferences and workshops, and ten talks delivered at conferences and research seminars world-wide.
b.3 As a part of dissemination activities we visited twelve universities and research centers around the world; supported in part by Oberwolfach Research Institute for Mathematics
(Germany, Research-in-Pairs scheme) and International Center for Mathematical Sciences (UK, Research-in-Groups scheme).
This project has led to significant advances in computational group theory, establishing a new sub-domain within the discipline. In many cases, the results obtained are the first-ever of their kind.
They have opened up avenues for practical computing in hitherto untreatable classes of groups and other algebraic structures. We envisage the potential impact of the project as follows.

- Impact on Computational Algebra. No computer algebra systems currently provide the functionality for infinite linear groups obtained in this project.
Due to the important applications of matrix groups and their computational efficiency, such functionality is in high demand. All users of computer algebra systems will benefit from this output of the project.
- Impact on Mathematical Sciences. The role of matrix groups in mathematics and other sciences is long-standing, going as far back as the late 19th century.
This background is a launching point for the project to utilise recently discovered applications of Zariski dense subgroups and the strong approximation properties in many mathematical disciplines.
- Impact on other sciences and engineering. Methods based on group-theoretical modeling developed during the project can be adapted to answer new problems emerging across disciplines,
especially those interfacing computational algebra.
- Societal impacts came from education and public engagement activities aimed at improving enthusiasm and aspiration for learning mathematics, promoting careers in STEM.
Diagrammatic Scheme of the Project