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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 (mathematician)|William Edge]]. He won the Society's Keith Medal for the period 1947/49.<ref>{{cite book|title=Biographical Index of Former Fellows of the Royal Society of Edinburgh 1783–2002|date=July 2006|publisher=The Royal Society of Edinburgh|isbn=0 902 198 84 X|url=https://www.royalsoced.org.uk/cms/files/fellows/biographical_index/fells_indexp2.pdf|access-date=17 March 2019|archive-url=https://web.archive.org/web/20160304074135/https://www.royalsoced.org.uk/cms/files/fellows/biographical_index/fells_indexp2.pdf|archive-date=4 March 2016|url-status=dead}}</ref>
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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==Awards and honours==
*Fellow of the [[Royal Astronomical Society]], 1934<ref name=MCreg />
*Fellow of the [[Royal Society of Edinburgh]], 1946<ref name = KM>{{cite web|url=http://www.royalsoced.org.uk/cms/files/fellows/biographical_index/fells_indexp2.pdf|title
*[[Berwick Prize]], 1947<ref name=TimesObit /><ref name=MCreg />
*[[Keith Medal]], 1947-9<ref>{{cite news|url=https://news.google.com/newspapers?id=IEhAAAAAIBAJ&sjid=ZZEMAAAAIBAJ&pg=3924%2C129473|title= Royal Society of Edinburgh. Awards to Professors |work=[[The Glasgow Herald]] |date=2 May 1950 |page=3 |accessdate=1 May 2018}}</ref>
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