Sir William Vallance Douglas Hodge (; 17 June 1903 – 7 July 1975) was a British mathematician, specifically a

geometer A geometer is a mathematician whose area of study is geometry. Some notable geometers and their main fields of work, chronologically listed, are: 1000 BCE to 1 BCE * Baudhayana (fl. c. 800 BC) – Euclidean geometry, geometric algebra * ...

. His discovery of far-reaching

topological In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...

relations between algebraic geometry and differential geometry—an area now called

Hodge theory In mathematics, Hodge theory, named after W. V. D. Hodge, is a method for studying the cohomology groups of a smooth manifold ''M'' using partial differential equations. The key observation is that, given a Riemannian metric on ''M'', every coh ...

and pertaining more generally to

Kähler manifold In mathematics and especially differential geometry, a Kähler manifold is a manifold with three mutually compatible structures: a complex structure, a Riemannian structure, and a symplectic structure. The concept was first studied by Jan Ar ...

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

Life and career

Hodge was born in

Edinburgh Edinburgh ( ; gd, Dùn Èideann ) is the capital city of Scotland and one of its 32 Council areas of Scotland, council areas. Historically part of the county of Midlothian (interchangeably Edinburghshire before 1921), it is located in Lothian ...

in 1903, the younger son and second of three children of Archibald James Hodge (1869-1938), a searcher of records in the property market and a partner in the firm of Douglas and Company, and his wife, Jane (born 1875), daughter of confectionery business owner William Vallance. They lived at 1 Church Hill Place in the Morningside district. He attended

George Watson's College George Watson's College is a co-educational independent day school in Scotland, situated on Colinton Road, in the Merchiston area of Edinburgh. It was first established as a hospital school in 1741, became a day school in 1871, and was merge ...

, and studied at

Edinburgh University The University of Edinburgh ( sco, University o Edinburgh, gd, Oilthigh Dhùn Èideann; abbreviated as ''Edin.'' in post-nominals) is a public research university based in Edinburgh, Scotland. Granted a royal charter by King James VI i ...

, graduating MA in 1923. With help from E. T. Whittaker, whose son

J. M. Whittaker John Macnaghten Whittaker FRS FRSE LLD (7 March 1905 – 29 January 1984) was a British mathematician and Vice-Chancellor of the University of Sheffield from 1953 to 1965. Life Whittaker was born 7 March 1905 in Cambridge, the son of math ...

was a college friend, he then took the

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

. At Cambridge he fell under the influence of the geometer

H. F. Baker Henry Frederick Baker FRS FRSE (3 July 1866 – 17 March 1956) was a British mathematician, working mainly in algebraic geometry, but also remembered for contributions to partial differential equations (related to what would become known as ...

. He gained a second MA in 1925. In 1926 he took up a teaching position at the

University of Bristol The University of Bristol is a Red brick university, red brick Russell Group research university in Bristol, England. It received its royal charter in 1909, although it can trace its roots to a Society of Merchant Venturers, Merchant Venturers' sc ...

, and began work on the interface between the

Italian school of algebraic geometry In relation to the history of mathematics, the Italian school of algebraic geometry refers to mathematicians and their work in birational geometry, particularly on algebraic surfaces, centered around Rome roughly from 1885 to 1935. There were 30 ...

, 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 on 1936, at the first delivery. Severi was born in Arezzo, Italy. He is famous for his contributions to algebr ...

, and the topological methods of

Solomon Lefschetz Solomon Lefschetz (russian: Соломо́н Ле́фшец; 3 September 1884 – 5 October 1972) was an American mathematician who did fundamental work on algebraic topology, its applications to algebraic geometry, and the theory of non-linear ...

. This made his reputation, but led to some initial scepticism on the part of Lefschetz. According to

Atiyah Atiyyah ( ar, عطية ''‘aṭiyyah''), which generally implies "something (money or goods given as regarded) received as a gift" or also means "present, gift, benefit, boon, favor, granting, giving"''.'' The name is also spelt Ateah, Atiyeh, ...

's memoir, Lefschetz and Hodge in 1931 had a meeting in

Max Newman Maxwell Herman Alexander Newman, FRS, (7 February 1897 – 22 February 1984), generally known as Max Newman, was a British mathematician and codebreaker. His work in World War II led to the construction of Colossus, the world's first operatio ...

'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 The Royal Society of Edinburgh is Scotland's national academy of science and letters. It is a registered charity that operates on a wholly independent and non-partisan basis and provides public benefit throughout Scotland. It was established i ...

. His proposers were Sir

Edmund Taylor Whittaker Sir Edmund Taylor Whittaker (24 October 1873 – 24 March 1956) was a British mathematician, physicist, and historian of science. Whittaker was a leading mathematical scholar of the early 20th-century who contributed widely to applied mathema ...

Ralph Allan Sampson Ralph Allan (or Allen) Sampson FRS FRSE LLD (25 June 1866 – 7 November 1939) was a British astronomer. Life Sampson was born in Schull, County Cork in Ireland, then part of the UK. He was the fourth of five children to James Sampson, a Cor ...

Charles Glover Barkla Charles Glover Barkla FRS FRSE (7 June 1877 – 23 October 1944) was a British physicist, and the winner of the Nobel Prize in Physics in 1917 for his work in X-ray spectroscopy and related areas in the study of X-rays ( Roentgen rays). Life ...

, and Sir

Charles Galton Darwin Sir Charles Galton Darwin (19 December 1887 – 31 December 1962) was an English physicist who served as director of the National Physical Laboratory (NPL) during the Second World War. He was a son of the mathematician George Howard Darwin ...

. He was awarded the Society's

Gunning Victoria Jubilee Prize The Gunning Victoria Jubilee Prize Lectureship is a quadrennial award made by the Royal Society of Edinburgh to recognise original work done by scientists resident in or connected with Scotland. The award was founded in 1887 by Dr Robert Hallida ...

for the period 1964 to 1968. In 1930 Hodge was awarded a Research Fellowship at

St. John's College, Cambridge St John's College is a constituent college of the University of Cambridge founded by the Tudor matriarch Lady Margaret Beaufort. In constitutional terms, the college is a charitable corporation established by a charter dated 9 April 1511. The ...

. He spent the year 1931–2 at

Princeton University Princeton University is a private research university in Princeton, New Jersey. Founded in 1746 in Elizabeth as the College of New Jersey, Princeton is the fourth-oldest institution of higher education in the United States and one of the n ...

, where Lefschetz was, visiting also

Oscar Zariski , birth_date = , birth_place = Kobrin, Russian Empire , death_date = , death_place = Brookline, Massachusetts, United States , nationality = American , field = Mathematics , work_institutions ...

Johns Hopkins University Johns Hopkins University (Johns Hopkins, Hopkins, or JHU) is a private research university in Baltimore, Maryland. Founded in 1876, Johns Hopkins is the oldest research university in the United States and in the western hemisphere. It consiste ...

. At this time he was also assimilating de Rham's theorem, and defining the

Hodge star In mathematics, the Hodge star operator or Hodge star is a linear map defined on the exterior algebra of a finite-dimensional oriented vector space endowed with a nondegenerate symmetric bilinear form. Applying the operator to an element of the ...

operation. It would allow him to define

harmonic form In mathematics, Hodge theory, named after W. V. D. Hodge, is a method for studying the cohomology groups of a smooth manifold ''M'' using partial differential equations. The key observation is that, given a Riemannian metric on ''M'', every coho ...

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 The Lowndean chair of Astronomy and Geometry is one of the two major Professorships in Astronomy (alongside the Plumian Professorship) and a major Professorship in Mathematics at Cambridge University. It was founded in 1749 by Thomas Lowndes, an a ...

of Astronomy and Geometry at

Cambridge Cambridge ( ) is a university city and the county town in Cambridgeshire, England. It is located on the River Cam approximately north of London. As of the 2021 United Kingdom census, the population of Cambridge was 145,700. Cambridge beca ...

, a position he held from 1936 to 1970. He was the first head of

DPMMS The Faculty of Mathematics at the University of Cambridge comprises the Department of Pure Mathematics and Mathematical Statistics (DPMMS) and the Department of Applied Mathematics and Theoretical Physics (DAMTP). It is housed in the Centre for ...

. 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, r ...

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

Adams Prize The Adams Prize is one of the most prestigious prizes awarded by the University of Cambridge. It is awarded each year by the Faculty of Mathematics at the University of Cambridge and St John's College to a UK-based mathematician for distingu ...

in 1937 and the

Copley Medal The Copley Medal is an award given by the Royal Society, for "outstanding achievements in research in any branch of science". It alternates between the physical sciences or mathematics and the biological sciences. Given every year, the medal is t ...

of the

in 1974. He died in

on 7 July 1975.

Work

The

Hodge index theorem In mathematics, the Hodge index theorem for an algebraic surface ''V'' determines the signature of the intersection pairing on the algebraic curves ''C'' on ''V''. It says, roughly speaking, that the space spanned by such curves (up to linear e ...

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 ...

theory for curves on an algebraic surface: it determines the

signature A signature (; from la, signare, "to sign") is a Handwriting, handwritten (and often Stylization, stylized) 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 ...

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 to ...

. This result was sought by the

, but was proved by the topological methods of

Lefschetz Solomon Lefschetz (russian: Соломо́н Ле́фшец; 3 September 1884 – 5 October 1972) was an American mathematician who did fundamental work on algebraic topology, its applications to algebraic geometry, and the theory of non-linear ...

. ''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 Kähler may refer to: ;People * Alexander Kähler (born 1960), German television journalist * Birgit Kähler (born 1970), German high jumper * Erich Kähler (1906–2000), German mathematician * Heinz Kähler (1905–1974), German art historian and ...

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 ...

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 set of solutions of a system of polynomial equations over the real or complex numbers ...

''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 In the mathematical field of algebraic geometry, a singular point of an algebraic variety is a point that is 'special' (so, singular), in the geometric sense that at this point the tangent space at the variety may not be regularly defined. In ...

) because projective space itself carries such a metric. In

de Rham cohomology In mathematics, de Rham cohomology (named after Georges de Rham) is a tool belonging both to algebraic topology and to differential topology, capable of expressing basic topological information about smooth manifolds in a form particularly adap ...

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 In mathematics and physics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace, who first studied its properties. This is often written as \nabla^2\! f = 0 or \Delta f = 0, where \Delta = \na ...

, 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 In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex de ...

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 number In mathematics, Hodge theory, named after W. V. D. Hodge, is a method for studying the cohomology groups of a smooth manifold ''M'' using partial differential equations. The key observation is that, given a Riemannian metric on ''M'', every cohomo ...

s. This ''Hodge decomposition'' has become a fundamental tool. Not only do the dimensions h^''p'',''q'' refine the

Betti number In algebraic topology, the Betti numbers are used to distinguish topological spaces based on the connectivity of ''n''-dimensional simplicial complexes. For the most reasonable finite-dimensional spaces (such as compact manifolds, finite simplicia ...

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 In mathematics, in particular algebraic geometry, a moduli space is a geometric space (usually a scheme or an algebraic stack) whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects. Such spac ...

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 In algebraic geometry, a mixed Hodge structure is an algebraic structure containing information about the Cohomology#Cohomology of varieties, cohomology of general algebraic variety, algebraic varieties. It is a generalization of a Hodge structure, ...

on singular varieties, and to deep analogies with

étale cohomology In mathematics, the étale cohomology groups of an algebraic variety or scheme are algebraic analogues of the usual cohomology groups with finite coefficients of a topological space, introduced by Grothendieck in order to prove the Weil conject ...

Hodge conjecture

The

Hodge conjecture In mathematics, the Hodge conjecture is a major unsolved problem in algebraic geometry and complex geometry that relates the algebraic topology of a non-singular complex algebraic variety to its subvarieties. In simple terms, the Hodge conjec ...

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 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. According ...

set up by the

Clay Mathematics Institute The Clay Mathematics Institute (CMI) is a private, non-profit foundation dedicated to increasing and disseminating mathematical knowledge. Formerly based in Peterborough, New Hampshire, the corporate address is now in Denver, Colorado. CMI's sc ...

Exposition

Hodge also wrote, with

Daniel Pedoe Dan Pedoe (29 October 1910, London – 27 October 1998, St Paul, Minnesota, USA) was an English-born mathematician and geometer with a career spanning more than sixty years. In the course of his life he wrote approximately fifty research and expos ...

, a three-volume work ''Methods of Algebraic Geometry'', on classical algebraic geometry, with much concrete content – illustrating though what

Élie Cartan Élie Joseph Cartan (; 9 April 1869 – 6 May 1951) was an influential French mathematician who did fundamental work in the theory of Lie groups, differential systems (coordinate-free geometric formulation of PDEs), and differential geometry ...

called 'the debauch of indices' in its component notation. According to

, this was intended to update and replace

'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