Browsing by Author "Flannery, Dane"

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    Algebra, matrices, and computers
    (Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2019) Detinko, Alla; Flannery, Dane; Hulpke, Alexander
    What part does algebra play in representing the real world abstractly? How can algebra be used to solve hard mathematical problems with the aid of modern computing technology? We provide answers to these questions that rely on the theory of matrix groups and new methods for handling matrix groups in a computer.
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    Computing Congruence Quotients of Zariski Dense Subgroups
    (Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2018) Detinko, Alla; Flannery, Dane; Hulpke, Alexander
    We obtain a computational realization of the strong approximation theorem. That is, we develop algorithms to compute all congruence quotients modulo rational primes of a finitely generated Zariski dense group H≤SL(n,Z) for n≥2. More generally, we are able to compute all congruence quotients of a finitely generated Zariski dense subgroup of SL(n,Q) for n>2.
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    Deciding Non-Freeness of Rational Möbius Groups
    (Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2022) Detinko, Alla; Flannery, Dane; Hulpke, Alexander
    We explore a new computational approach to a classical problem: certifying non-freeness of (2-generator, parabolic) Möbius subgroups of SL(2, Q). The main tools used are algorithms for Zariski dense groups and algorithms to compute a presentation of SL(2, R) for a localization R = Z[1/b] of Z. We prove that a Möbius subgroup G is not free by showing that it has finite index in the relevant SL(2, R). Further information about the structure of G is obtained; for example, we compute the minimal subgroup of finite index in SL(2, R) that contains G.
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    Experimenting with Symplectic Hypergeometric Monodromy Groups
    (Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2019) Detinko, Alla; Flannery, Dane; Hulpke, Alexander
    We present new computational results for symplectic monodromy groups of hypergeometric differential equations. In particular, we compute the arithmetic closure of each group, sometimes justifying arithmeticity. The results are obtained by extending our previous algorithms for Zariski dense groups, based on the strong approximation and congruence subgroup properties.
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    Experimenting with Zariski dense subgroups
    (Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2017) Detinko, Alla; Flannery, Dane; Hulpke, Alexander
    We give a method to describe all congruence images of a finitely generated Zariski dense group H< SL(n; Z). The method is applied to obtain efficient algorithms for solving this problem in odd prime degree n; if n = 2 then we compute all congruence images only modulo primes. We propose a separate method that works for all n as long as H contains a known transvection. The algorithms have been implemented in GAP, enabling computer experiments with important classes of linear groups that have recently emerged.
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    GAP functionality for Zariski dense groups
    (Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2017) Detinko, Alla; Flannery, Dane; Hulpke, Alexander
    In this document we describe the functionality of GAP [4] routines for Zariski dense or arithmetic groups that are developed in [1, 2, 3]. The research underlying the software was supported through the programme "Research in Pairs", at the Mathematisches Forschungsinstitut Oberwolfach, and part of the software was written during a stay in June 2017. The hospitality we received has been greatly appreciated. Our research was also supported by a Marie Sklodowska-Curie Individual Fellowship grant under Horizon 2020 (EU Framework Programme for Research and Innovation), and a Simons Foundation Collaboration Grant Nr. 244502. All support is acknowledged with gratitude.