View source for W. V. D. Hodge

{{Short description|British mathematician}}
{{About|a mathematician|other uses|Hodge (disambiguation)}}
{{EngvarB|date=August 2014}}
{{Use dmy dates|date=August 2014}}
{{Infobox scientist
| honorific_prefix  = [[Sir]]
| honorific_suffix  = {{postnominal|country=GBR|FRS|FRSE}}
| name              = W. V. D. Hodge
| image             = William Vallance Douglas Hodge (DPJ - b01dbz97ze).jpg
| caption           = Hodge {{c.|1970}}
| birth_date        = {{Birth date|df=yes|1903|06|17}}
| birth_place       = [[Edinburgh]], UK
| death_date        = {{Death date and age|df=yes|1975|07|07|1903|06|17}}
| death_place       = [[Cambridge]], UK
| nationality       = British
| fields            = Mathematics
| workplaces        = [[Pembroke College, Cambridge]]
| alma_mater        = [[University of Edinburgh]]<br>[[St John's College, Cambridge]]<ref>[http://www-history.mcs.st-andrews.ac.uk/Biographies/Hodge.html Hodge biography - University of St Andrews]</ref>
| academic_advisors = [[E. T. Whittaker]]
| doctoral_students = [[Michael Atiyah]]<br>[[Ian R. Porteous]]<br>[[David J. Simms]]
| known_for         = [[Hodge conjecture]]<br>[[Hodge dual]]<br>[[Hodge bundle]]<br>[[Hodge theory]]
| awards            = [[Adams Prize]] {{small|(1936)}}<br>[[Senior Berwick Prize]] <small>(1952)</small><br>[[Royal Medal]] {{small|(1957)}}<br>[[De Morgan Medal]] {{small|(1959)}}<br>[[Copley Medal]] {{small|(1974)}}
| education         = [[George Watson's College]]
}}
Sir '''William Vallance Douglas Hodge''' {{postnominal|country=GBR|FRS|FRSE}}<ref name="frs">{{Cite journal | last1 = Atiyah | first1 = M. F. | author-link = Michael Atiyah| doi = 10.1098/rsbm.1976.0007 | title = William Vallance Douglas Hodge. 17 June 1903 -- 7 July 1975 | journal = [[Biographical Memoirs of Fellows of the Royal Society]] | volume = 22 | pages = 169–192| year = 1976 | s2cid = 72054846 }}</ref> ({{IPAc-en|h|ɒ|dʒ}}; 17 June 1903 – 7 July 1975) was a British mathematician, specifically a [[geometer]].<ref name="mactutor">{{MacTutor Biography|id=Hodge}}</ref><ref name="mathgene">{{MathGenealogy |id=18583}}</ref>

His discovery of far-reaching [[topological]] relations between [[algebraic geometry]] and [[differential geometry]]—an area now called [[Hodge theory]] and pertaining more generally to [[Kähler manifold]]s—has been a major influence on subsequent work in geometry.

==Early life and education==
Hodge was born in [[Edinburgh]] in 1903, the younger son and second of three children of Jane (born 1875) and Archibald James Hodge (1869&ndash;1938) His father was a searcher of records in the property market and a partner in the firm of Douglas and Company and his mother was the daughter of a confectionery business owner William Vallance.<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_indexp1.pdf|archive-url=https://web.archive.org/web/20160112063055/https://www.royalsoced.org.uk/cms/files/fellows/biographical_index/fells_indexp1.pdf |archive-date=12 January 2016 }}</ref><ref>{{cite web | url=https://mathshistory.st-andrews.ac.uk/Biographies/Hodge | title=William Hodge - Biography }}</ref><ref>{{cite ODNB | url=https://www.oxforddnb.com/view/10.1093/ref:odnb/9780198614128.001.0001/odnb-9780198614128-e-31241 | isbn=978-0-19-861412-8 | doi=10.1093/ref:odnb/31241 | title=Hodge, Sir William Vallance Douglas (1903–1975), mathematician }}</ref> They lived at 1 Church Hill Place in the [[Morningside, Edinburgh|Morningside]] district.<ref>Edinburgh and Leith Post Office Directory 1903-4</ref>

[[File:1 Church Hill Place, Edinburgh.jpg|thumb|Hodge's childhood home at 1 Church Hill Place in [[Edinburgh]]]]

He attended [[George Watson's College]] and then the [[University of Edinburgh]], where he graduated with an MA in 1923. With help from [[E. T. Whittaker]], whose son [[J. M. Whittaker]] was a college friend, he then enrolled as an affiliated student at [[St John's College, Cambridge|St John's College]] at the [[University of Cambridge]] in order to study the [[Mathematical Tripos]]. At Cambridge, he was influenced by the geometer [[H. F. Baker]]. He was awarded a BA from the University of Cambridge in 1925, an MA in 1930, and a Doctor of Science (ScD) in 1950.<ref>The Annual Register of the University of Cambridge for the year 1968-69</ref>

==Career==
In 1926, he took up a teaching position at the [[University of Bristol]], and began work on the interface between the [[Italian school of algebraic geometry]], particularly problems posed by [[Francesco Severi]], and the topological methods of [[Solomon Lefschetz]]. This made his reputation, but led to some initial scepticism on the part of Lefschetz. According to [[Michael Atiyah|Atiyah]]'s memoir, Lefschetz and Hodge in 1931 had a meeting in [[Max Newman]]'s rooms in Cambridge, to try to resolve issues. In the end Lefschetz was convinced.<ref name="frs"/> In 1928 he was elected a Fellow of the [[Royal Society of Edinburgh]]. His proposers were Sir [[Edmund Taylor Whittaker]], [[Ralph Allan Sampson]], [[Charles Glover Barkla]], and Sir [[Charles Galton Darwin]]. He was awarded the Society's [[Gunning Victoria Jubilee Prize]] for the period 1964 to 1968.<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_indexp1.pdf|archive-date=24 January 2013|access-date=1 November 2016|archive-url=https://web.archive.org/web/20130124115814/http://www.royalsoced.org.uk/cms/files/fellows/biographical_index/fells_indexp1.pdf|url-status=dead}}</ref>

In 1930, Hodge was awarded a Research Fellowship at St John's College, Cambridge. He spent the year 1931–2 at [[Princeton University]], where Lefschetz was, visiting also [[Oscar Zariski]] at [[Johns Hopkins University]]. At this time he was also assimilating [[de Rham's theorem]], and defining the [[Hodge star]] operation. It would allow him to define [[harmonic form]]s and so refine the de Rham theory.

On his return to Cambridge, he was offered a University Lecturer position in 1933. He became the [[Lowndean Professor]] of Astronomy and Geometry at [[Cambridge]], a position he held from 1936 to 1970. He was the first head of [[Faculty of Mathematics, University of Cambridge|DPMMS]].

He was the Master of [[Pembroke College, Cambridge]] from 1958 to 1970, and vice-president of the [[Royal Society]] from 1959 to 1965.  He was knighted in 1959. Amongst other honours, he received the [[Adams Prize]] in 1937 and the [[Copley Medal]] of the [[Royal Society]] in 1974.

==Death==
He died in [[Cambridge]] on 7 July 1975.

==Work==
The [[Hodge index theorem]] was a result on the [[intersection number]] theory for curves on an [[algebraic surface]]: it determines the [[metric signature|signature]] of the corresponding [[quadratic form]]. This result was sought by the [[Italian school of algebraic geometry]], but was proved by the topological methods of [[Lefschetz]].

''The Theory and Applications of Harmonic Integrals''<ref>{{cite journal|author=Struik, D. J.|author-link=Dirk Jan Struik|title=Review: W. V. D. Hodge, ''The theory and applications of harmonic integrals''|journal=Bull. Amer. Math. Soc.|date=1944|volume=50|issue=1|pages=43–45|url=http://projecteuclid.org/euclid.bams/1183505582|doi=10.1090/s0002-9904-1944-08054-3|doi-access=free}}</ref> summed up Hodge's development during the 1930s of his general theory. This starts with the existence for any [[Kähler metric]] of a theory of [[Laplacian]]s – it applies to an [[algebraic variety]] ''V'' (assumed [[complex numbers|complex]], [[projective variety|projective]] and [[nonsingular variety|non-singular]]) because [[projective space]] itself carries such a metric. In [[de Rham cohomology]] terms, a cohomology class of degree ''k'' is represented by a [[differential form|''k''-form]] ''α'' on ''V''('''C'''). There is no unique representative; but by introducing the idea of ''harmonic form'' (Hodge still called them 'integrals'), which are solutions of [[Laplace's equation]], one can get unique ''α''. This has the important, immediate consequence of splitting up

:''H''<sup>''k''</sup>(''V''('''C'''), '''C''')

into subspaces

:''H''<sup>''p'',''q''</sup>

according to the number ''p'' of [[Holomorphic function|holomorphic]] differentials ''dz<sup>i</sup>'' wedged to make up ''α'' (the cotangent space being spanned by the ''dz<sup>i</sup>'' and their complex conjugates). The dimensions of the subspaces are the [[Hodge number]]s.

This ''Hodge decomposition'' has become a fundamental tool. Not only do the dimensions h<sup>''p'',''q''</sup> refine the [[Betti number]]s, by breaking them into parts with identifiable geometric meaning; but the decomposition itself, as a varying 'flag' in a complex vector space, has a meaning in relation with [[moduli problem]]s. In broad terms, Hodge theory contributes both to the discrete and the continuous classification of algebraic varieties.

Further developments by others led in particular to an idea of [[mixed Hodge structure]] on singular varieties, and to deep analogies with [[étale cohomology]].

==Hodge conjecture==
{{Further|Hodge conjecture}}
The [[Hodge conjecture]] on the 'middle' spaces H<sup>''p'',''p''</sup> is still unsolved, in general. It is one of the seven [[Millennium Prize Problems]] set up by the [[Clay Mathematics Institute]].

==Exposition==
Hodge also wrote, with [[Daniel Pedoe]], a three-volume work ''Methods of Algebraic Geometry'', on classical algebraic geometry, with much concrete content – illustrating though what [[Élie Cartan]] called 'the debauch of indices' in its component notation. According to [[Michael Atiyah|Atiyah]], this was intended to update and replace [[H. F. Baker]]'s ''Principles of Geometry''.

==Family==
In 1929 he married Kathleen Anne Cameron.<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_indexp1.pdf|archive-date=24 January 2013|access-date=1 November 2016|archive-url=https://web.archive.org/web/20130124115814/http://www.royalsoced.org.uk/cms/files/fellows/biographical_index/fells_indexp1.pdf|url-status=dead}}</ref>

==Publications==
*{{Citation | last1=Hodge | first1=W. V. D. | title=The Theory and Applications of Harmonic Integrals | url=https://books.google.com/books?id=-8k8AAAAIAAJ | publisher=[[Cambridge University Press]] | isbn= 978-0-521-35881-1 |mr=0003947 | date=1941}}
*{{Citation | last1=Hodge | first1=W. V. D. | last2=Pedoe | first2=D. | author2-link=Daniel Pedoe | title=Methods of Algebraic Geometry, Volume I (Book II) | orig-year=1947 | publisher=[[Cambridge University Press]] | isbn=978-0-521-46900-5 | date=1994}}<ref>{{cite journal|author=Coxeter, H. S. M.|author-link=H. S. M. Coxeter|title=Review: ''Methods of algebraic geometry''. By W. V. D. Hodge and D. Pedoe|journal=Bull. Amer. Math. Soc.|date=1949|volume=55|issue=3, Part 1|pages=315–316|url=http://www.ams.org/journals/bull/1949-55-03/S0002-9904-1949-09193-0/S0002-9904-1949-09193-0.pdf|doi=10.1090/s0002-9904-1949-09193-0}}</ref>
*{{Citation | last1=Hodge | first1=W. V. D. | last2=Pedoe | first2=Daniel | author2-link=Daniel Pedoe | title=Methods of Algebraic Geometry: Volume 2 Book III: General theory of algebraic varieties in projective space. Book IV: Quadrics and Grassmann varieties. | orig-year=1952 | url=https://books.google.com/books?id=bJwbn3RSWhwC | publisher=[[Cambridge University Press]] | series=Cambridge Mathematical Library | isbn=978-0-521-46901-2 |mr=0048065 | date=1994}}<ref>{{cite journal|author=Coxeter, H. S. M.|title=Review: ''Methods of algebraic geometry.'' Vol. 2. By W. V. D. Hodge and D. Pedoe|journal=Bull. Amer. Math. Soc.|date=1952| volume=58|issue=6|pages=678–679|url=http://www.ams.org/journals/bull/1952-58-06/S0002-9904-1952-09661-0/S0002-9904-1952-09661-0.pdf|doi=10.1090/s0002-9904-1952-09661-0|doi-access=free}}</ref>
*{{Citation | last1=Hodge | first1=W. V. D. | last2=Pedoe | first2=Daniel | author2-link=Daniel Pedoe | title=Methods of Algebraic Geometry: Volume 3 | orig-year= 1954 | url=https://books.google.com/books?id=XeX_p8pHCAIC | publisher=[[Cambridge University Press]] | isbn=978-0-521-46775-9 | date=1994}}<ref>{{cite journal|author=Samuel, P.|author-link=Pierre Samuel|journal=Bull. Amer. Math. Soc.|date=1955|volume=61|issue=3, Part 1|pages=254–257|title=Review: ''Methods of algebraic geometry''. Vol. III. ''Birational geometry''. By W. V. D. Hodge and D. Pedoe|url=http://www.ams.org/journals/bull/1955-61-03/S0002-9904-1955-09910-5/S0002-9904-1955-09910-5.pdf|doi=10.1090/s0002-9904-1955-09910-5|doi-access=free}}</ref>

==See also==
* [[List of things named after W. V. D. Hodge]]

==References==
{{reflist}}

{{s-start}}
{{s-aca}}
{{s-bef|before=[[Sydney Castle Roberts]]}}
{{s-ttl|title=[[List of Masters of Pembroke College, Cambridge|Master of Pembroke College, Cambridge]]|years=1958–1970}}
{{s-aft|after=[[W. A. Camps]]}}
{{s-end}}

{{De Morgan Medallists}}
{{Copley Medallists 1951–2000}}
{{Authority control}}

{{DEFAULTSORT:Hodge, William Vallance Douglas}}
[[Category:1903 births]]
[[Category:1975 deaths]]
[[Category:Algebraic geometers]]
[[Category:Cambridge mathematicians]]
[[Category:Scientists from Edinburgh]]
[[Category:People educated at George Watson's College]]
[[Category:Fellows of Pembroke College, Cambridge]]
[[Category:Alumni of St John's College, Cambridge]]
[[Category:Alumni of the University of Edinburgh]]
[[Category:Fellows of the Royal Society]]
[[Category:Foreign associates of the National Academy of Sciences]]
[[Category:Royal Medal winners]]
[[Category:Recipients of the Copley Medal]]
[[Category:Masters of Pembroke College, Cambridge]]
[[Category:Lowndean Professors of Astronomy and Geometry]]
[[Category:20th-century Scottish mathematicians]]
[[Category:Alumni of the University of St Andrews]]
[[Category:Presidents of the Cambridge Philosophical Society]]