Sir William Vallance Douglas Hodge (; 17 June 1903 – 7 July 1975) was a British mathematician, specifically a geometer. His discovery of far-reaching topological relations between

algebraic geometry Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...

and

differential geometry Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of Calculus, single variable calculus, vector calculus, lin ...

—an area now called Hodge theory and pertaining more generally to Kähler manifolds—has been a major influence on subsequent work in geometry.

Life and career

Hodge was born in

Edinburgh Edinburgh is the capital city of Scotland and one of its 32 Council areas of Scotland, council areas. The city is located in southeast Scotland and is bounded to the north by the Firth of Forth and to the south by the Pentland Hills. Edinburgh ...

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. They lived at 1 Church Hill Place in the Morningside district. He attended George Watson's College, and studied at the

University of Edinburgh The University of Edinburgh (, ; abbreviated as ''Edin.'' in Post-nominal letters, post-nominals) is a Public university, public research university based in Edinburgh, Scotland. Founded by the City of Edinburgh Council, town council under th ...

graduating 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, formally the College of St John the Evangelist in the University of Cambridge, is a Colleges of the University of Cambridge, constituent college of the University of Cambridge, founded by the House of Tudor, Tudor matriarch L ...

, in order to study the

Mathematical Tripos The Mathematical Tripos is the mathematics course that is taught in the Faculty of Mathematics, University of Cambridge, Faculty of Mathematics at the University of Cambridge. Origin In its classical nineteenth-century form, the tripos was a di ...

. At Cambridge he fell under the influence of the geometer H. F. Baker. He gained a Cambridge BA degree in 1925, receiving the MA in 1930 and the Doctor of Science (ScD) degree in 1950. In 1926 he took up a teaching position at the

University of Bristol The University of Bristol is a public university, public research university in Bristol, England. It received its royal charter in 1909, although it can trace its roots to a Merchant Venturers' school founded in 1595 and University College, Br ...

, and began work on the interface between the Italian school of algebraic geometry, particularly problems posed by

Francesco Severi Francesco Severi (13 April 1879 – 8 December 1961) was an Italian mathematician. He was the chair of the committee on Fields Medal in 1936, at the first delivery. Severi was born in Arezzo, Italy. He is famous for his contributions to algebra ...

, 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 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. In 1930 Hodge was awarded a Research Fellowship at St John's College, Cambridge. He spent the year 1931–2 at

Princeton University Princeton University is a private university, private Ivy League research university in Princeton, New Jersey, United States. Founded in 1746 in Elizabeth, New Jersey, Elizabeth as the College of New Jersey, Princeton is the List of Colonial ...

, where Lefschetz was, visiting also Oscar Zariski at

Johns Hopkins University The Johns Hopkins University (often abbreviated as Johns Hopkins, Hopkins, or JHU) is a private university, private research university in Baltimore, Maryland, United States. Founded in 1876 based on the European research institution model, J ...

. 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 Cambridge ( ) is a List of cities in the United Kingdom, city and non-metropolitan district in the county of Cambridgeshire, England. It is the county town of Cambridgeshire and is located on the River Cam, north of London. As of the 2021 Unit ...

, 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 The Royal Society, formally The Royal Society of London for Improving Natural Knowledge, is a learned society and the United Kingdom's national academy of sciences. The society fulfils a number of roles: promoting science and its benefits, re ...

from 1959 to 1965. He was knighted in 1959. Amongst other honours, he received the Adams Prize in 1937 and the

Copley Medal The Copley Medal is the most prestigious award of the Royal Society of the United Kingdom, conferred "for sustained, outstanding achievements in any field of science". The award alternates between the physical sciences or mathematics and the bio ...

of the

in 1974. He died in

on 7 July 1975.

Work

The Hodge index theorem was a result on the

intersection number In mathematics, and especially in algebraic geometry, the intersection number generalizes the intuitive notion of counting the number of times two curves intersect to higher dimensions, multiple (more than 2) curves, and accounting properly for ta ...

theory for curves on an algebraic surface: it determines the

signature A signature (; from , "to sign") is a depiction of someone's name, nickname, or even a simple "X" or other mark that a person writes on documents as a proof of identity and intent. Signatures are often, but not always, Handwriting, handwritt ...

of the corresponding

quadratic form In mathematics, a quadratic form is a polynomial with terms all of degree two (" form" is another name for a homogeneous polynomial). For example, 4x^2 + 2xy - 3y^2 is a quadratic form in the variables and . The coefficients usually belong t ...

. 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

Laplacian In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is th ...

s – it applies to an

algebraic variety Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the solution set, set of solutions of a system of polynomial equations over the real number, ...

''V'' (assumed

complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...

, 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 :''H''^''k''(''V''(C), C) into subspaces :''H''^''p'',''q'' 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 h^''p'',''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.

Hodge conjecture

The Hodge conjecture on the 'middle' spaces H^''p'',''p'' is still unsolved, in general. It is one of the seven

Millennium Prize Problems The Millennium Prize Problems are seven well-known complex mathematics, mathematical problems selected by the Clay Mathematics Institute in 2000. The Clay Institute has pledged a US $1 million prize for the first correct solution to each problem ...

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

Family

In 1929 he married Kathleen Anne Cameron.

Publications

* * * *

References

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