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Walker took up a post as Lecturer at [[Imperial College]] in 1935; the following year he was appointed as Lecturer in Pure Mathematics at the [[University of Liverpool]], a post he held until 1947, when he moved to the [[University of Sheffield]] as Professor of Pure Mathematics.<ref name=TimesObit /><ref name=MCreg />
In 1946 he was elected a Fellow of the [[Royal Society of Edinburgh]]. His proposers were [[Harold Stanley Ruse]], Sir [[Edmund Taylor Whittaker]], [[David Gibb (mathematician)|David Gibb]] and [[William Edge]].
In 1952 he returned to Liverpool University, in 1962 becoming Dean of its Faculty of Science.<ref name=MCreg /> Having been elected a Fellow of the [[Royal Society]] in 1955, he served as a member of the organisation's council from 1961-2.<ref name=MCreg /> He served as President of the [[London Mathematical Society]] from 1962-3.<ref name=KM /> Walker retired from Liverpool University in 1974.<ref name=TimesObit />
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