Sir W. V. D. Hodge

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Sir William Vallance Douglas Hodge

Birthplace: Edinburgh, Edinburgh City, United Kingdom
Death: July 07, 1975 (72)
Cambridge, Cambridgeshire, United Kingdom
Immediate Family:

Son of Archibald James Hodge and Janet Hodge
Husband of Private
Father of Private and Private
Brother of Archibald Vallance Hodge and Janet Horsburgh Broadbent

Managed by: Martin Severin Eriksen
Last Updated:
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Immediate Family

About Sir W. V. D. Hodge

William Hodge's parents were Janet Vallence, whose father owned a confectionary business in Edinburgh, Scotland, and Archibald James Hodge who worked in the property market. William was the second of his parents' children, having one older brother and one younger sister.

He was educated at George Watson's College in Edinburgh, entering in 1909 and studying there until 1920. While at the College he won a mathematical bursary which allowed him to progress to become a student at Edinburgh University. He was taught there by Whittaker who advised him to continue his studies at Cambridge. After graduating with First Class Honours in mathematics from Edinburgh in 1923, he entered St John's College, Cambridge. His studies at Cambridge were financed by a van Dunlop bursary which he had won from Edinburgh and a scholarship from St John's College. After gaining distinction in the Mathematical Tripos of 1925 he went on to win a Smith's Prize and spent a further year researching at Cambridge financed by a Ferguson scholarship.

Hodge was appointed to an assistant lectureship at the University of Bristol in 1926 and spent five years there. During this period he married Kathleen Anne Cameron, the daughter of the manager of the Edinburgh branch of Oxford University Press. They had one son and one daughter.

His main mathematical interests were in algebraic geometry and differential geometry. In 1930 he applied ideas of Lefschetz to solve a problem posed by Severi. That same year he gained election to a fellowship at St John's College, Cambridge. In the following year, after winning an 1851 Exhibition Studentship, he went to Princeton so that he could work with Lefschetz who he greatly admired. While in the United States Hodge spent two months at Johns Hopkins University studying with Zariski.

After his visit to the United States, Hodge returned to Cambridge in 1932. He was appointed as a university lecturer in the following year and, in 1935, was elected to a fellowship at Pembroke College, Cambridge. During this period he developed the relationship between geometry, analysis and topology and produced some of his best remembered work on the theory of harmonic integrals. For these contributions Hodge won the Adams Prize in 1937 and Weyl described this contribution as:-

... one of the great landmarks in the history of science in the present century.

Hodge published a polished account of his important theory in 1941. This work marked an important change in direction for the Cambridge school of geometry which, under Baker's leadership, had become somewhat isolated from other areas of mathematics.

In 1936 Hodge had been appointed as Lowndean Professor of Astronomy and Geometry, succeeding Baker, and he held this chair at Cambridge until 1970. He continued to work at Cambridge during World War II, but took on extra duties to compensate for the shortage of staff who were away in the forces; in particular he acted as bursar of Pembroke. In 1958 he was appointed as Master of Pembroke, holding the post until he retired from university life in 1970.

Hodge was one of the originators of the British Mathematical Colloquium, an annual conference which visits different British universities. He also played a major role in setting up the International Mathematical Union in 1952, being elected as vice-president from 1954 to 1958. In 1959 the London Mathematical Society awarded him their De Morgan Medal. He was elected to the Royal Society of London in 1938, received the Society's Royal Medal in 1957, and was vice-president from 1959 to 1965. The Royal Medal was awarded in:-

... recognition of his distinguished work on algebraic geometry.

He was also awarded the Copley Medal by the Royal Society in 1974:-

... in recognition of his pioneering work in algebraic geometry, notably in his theory of harmonic integrals.

He received many other honours. He was elected to the Royal Society of Edinburgh, and the American National Academy of Sciences. He was awarded honorary degrees by many universities including Bristol (1957), Edinburgh (1958), Leicester (1959), Sheffield (1960), Exeter (1961), Wales (1961), and Liverpool (1961). He was knighted in 1959.

He is described in [2] as follows:-

Hodge was very unlike the conventional picture of a mathematician. Jovial, informal and down-to-earth, he could easily have passed for a businessman.

Atiyah writes that Hodge was:-

... modest and unassuming. Genial in manner and temperament, endowed with sturdy Scots common sense, he got on well with his colleagues and students.

Article by: J J O'Connor and E F Robertson

William Vallance Douglas Hodge FRS (17 June 1903 – 7 July 1975) was a Scottish mathematician, specifically a geometer.

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 manifolds—has been a major influence on subsequent work in geometry.

He was born in Edinburgh, attended George Watson's College, and studied at Edinburgh University, graduating in 1923. With help from E. T. Whittaker whose son J. M. Whittaker was a college friend, he then took the Cambridge Mathematical Tripos. At Cambridge he fell under the influence of the geometer H. F. Baker.

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 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.

In 1930 Hodge was awarded a Research Fellowship at St. John's College, Cambridge. He spent a 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 forms 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 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.

The Hodge index theorem was a result on the intersection number theory for curves on an algebraic surface: it determines the 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 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 Laplacians — it applies to an algebraic variety V (assumed complex, projective and 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 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

Hk(V(C), C) 

into subspaces


according to the number p of holomorphic differentials dzi wedged to make up α (the cotangent space being spanned by the dzi and their complex conjugates). The dimensions of the subspaces are the Hodge numbers.

This Hodge decomposition has become a fundamental tool. Not only do the dimensions hp,q refine the Betti 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 with moduli 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 of mixed Hodge structure on singular varieties, and to deep analogies with étale cohomology.

[edit] Hodge conjecture

The Hodge conjecture on the 'middle' spaces Hp,p is still unsolved, in general. It is one of the seven Millennium Prize Problems set up by the Clay Mathematics Institute.

[edit] 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 Atiyah, this was intended to update and replace H. F. Baker's Principles of Geometry.

[edit] References

Michael Atiyah, William Vallance Douglas Hodge'. Royal Society biographical memoir, reprinted in Atiyah's Collected Papers Vol.I, pp. 231–254

[edit] External links

O'Connor, John J.; Robertson, Edmund F., "W. V. D. Hodge", MacTutor History of Mathematics archive, University of St Andrews.
W. V. D. Hodge at the Mathematics Genealogy Project.

Other References

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Sir W. V. D. Hodge's Timeline

June 17, 1903
Edinburgh, Edinburgh City, United Kingdom
July 7, 1975
Age 72
Cambridge, Cambridgeshire, United Kingdom