W. V. D. Hodge | |
|---|---|
| Born | (1903-06-17)17 June 1903 Edinburgh, UK |
| Died | 7 July 1975(1975-07-07) (aged 72) Cambridge, UK |
| Nationality | British |
| Education | George Watson's College |
| Alma mater | University of Edinburgh St John's College, Cambridge[1] |
| Known for | Hodge conjecture Hodge dual Hodge bundle Hodge theory |
| Awards | Adams Prize(1936) Senior Berwick Prize(1952) Royal Medal(1957) De Morgan Medal(1959) Copley Medal(1974) |
| Scientific career | |
| Fields | Mathematics |
| Institutions | Pembroke College, Cambridge |
| Academic advisors | E. T. Whittaker |
| Doctoral students | Michael Atiyah Ian R. Porteous David J. Simms |
SirWilliam Vallance Douglas HodgeFRS FRSE[2] (/hɒdʒ/; 17 June 1903 – 7 July 1975) was a British mathematician, specifically ageometer.[3][4]
His discovery of far-reachingtopological relations betweenalgebraic geometry anddifferential geometry—an area now calledHodge theory and pertaining more generally toKähler manifolds—has been a major influence on subsequent work in geometry.
Hodge was born inEdinburgh in 1903, the younger son and second of three children of Jane (born 1875) and Archibald James Hodge (1869–1938) 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.[5][6][7] They lived at 1 Church Hill Place in theMorningside district.[8]
He attendedGeorge Watson's College, and studied at theUniversity of Edinburgh graduating with an MA in 1923. With help fromE. T. Whittaker, whose sonJ. M. Whittaker was a college friend, he then enrolled as an affiliated student atSt John's College, Cambridge, in order to study theMathematical Tripos. At Cambridge he fell under the influence of the geometerH. F. Baker. He gained a Cambridge BA degree in 1925, receiving the MA in 1930 and the Doctor of Science (ScD) degree in 1950.[9]
In 1926 he took up a teaching position at theUniversity of Bristol, and began work on the interface between theItalian school of algebraic geometry, particularly problems posed byFrancesco Severi, and the topological methods ofSolomon Lefschetz. This made his reputation, but led to some initial scepticism on the part of Lefschetz. According toAtiyah's memoir, Lefschetz and Hodge in 1931 had a meeting inMax Newman's rooms in Cambridge, to try to resolve issues. In the end Lefschetz was convinced.[2] In 1928 he was elected a Fellow of theRoyal Society of Edinburgh. His proposers were SirEdmund Taylor Whittaker,Ralph Allan Sampson,Charles Glover Barkla, and SirCharles Galton Darwin. He was awarded the Society'sGunning Victoria Jubilee Prize for the period 1964 to 1968.[10]
In 1930 Hodge was awarded a Research Fellowship at St John's College, Cambridge. He spent the year 1931–2 atPrinceton University, where Lefschetz was, visiting alsoOscar Zariski atJohns Hopkins University. At this time he was also assimilatingde Rham's theorem, and defining theHodge star operation. It would allow him to defineharmonic forms and so refine the de Rham theory.
On his return to Cambridge, he was offered a University Lecturer position in 1933. He became theLowndean Professor of Astronomy and Geometry atCambridge, a position he held from 1936 to 1970. He was the first head ofDPMMS.
He was the Master ofPembroke College, Cambridge from 1958 to 1970, and vice-president of theRoyal Society from 1959 to 1965. He was knighted in 1959. Amongst other honours, he received theAdams Prize in 1937 and theCopley Medal of theRoyal Society in 1974.
He died inCambridge on 7 July 1975.
TheHodge index theorem was a result on theintersection number theory for curves on analgebraic surface: it determines thesignature of the correspondingquadratic form. This result was sought by theItalian school of algebraic geometry, but was proved by the topological methods ofLefschetz.
The Theory and Applications of Harmonic Integrals[11] summed up Hodge's development during the 1930s of his general theory. This starts with the existence for anyKähler metric of a theory ofLaplacians – it applies to analgebraic varietyV (assumedcomplex,projective andnon-singular) becauseprojective space itself carries such a metric. Inde Rham cohomology terms, a cohomology class of degreek is represented by ak-formα onV(C). There is no unique representative; but by introducing the idea ofharmonic form (Hodge still called them 'integrals'), which are solutions ofLaplace's equation, one can get uniqueα. This has the important, immediate consequence of splitting up
into subspaces
according to the numberp ofholomorphic differentialsdzi wedged to make upα (the cotangent space being spanned by thedzi and their complex conjugates). The dimensions of the subspaces are theHodge numbers.
ThisHodge decomposition has become a fundamental tool. Not only do the dimensions hp,q refine theBetti numbers, 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 withmoduli problems. 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 ofmixed Hodge structure on singular varieties, and to deep analogies withétale cohomology.
TheHodge conjecture on the 'middle' spaces Hp,p is still unsolved, in general. It is one of the sevenMillennium Prize Problems set up by theClay Mathematics Institute.
Hodge also wrote, withDaniel Pedoe, a three-volume workMethods of Algebraic Geometry, on classical algebraic geometry, with much concrete content – illustrating though whatÉlie Cartan called 'the debauch of indices' in its component notation. According toAtiyah, this was intended to update and replaceH. F. Baker'sPrinciples of Geometry.
In 1929 he married Kathleen Anne Cameron.[12]
| Academic offices | ||
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| Preceded by | Master of Pembroke College, Cambridge 1958–1970 | Succeeded by |
