This proposal aims to develop further a novel domain of computational group theory: the design of algorithms and software for finitely generated linear groups over infinite fields. Our research is motivated by the applications of linear groups in many areas of mathematics, and the high demand for algorithms to carry out practical computations. We plan to advance our recently obtained computational techniques, based on exploring Zariski density and the strong approximation property. Specifically, we will complete the design of algorithms for isometry groups over infinite domains, and apply implementations of these algorithms to solve hard open problems via computer-aided experimentation. The latter includes experiments with symplectic hypergeometric monodromy groups. All results will be new. Moreover, the algorithms will be the first-ever of their kind, satisfying the demand for practical computational methods in the areas concerned.
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