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, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...

and

differential geometry Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and mult ...

—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 co ...

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 Arn ...

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 m ...

, and studied at Edinburgh University, graduating MA 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. He gained a second MA in 1925. In 1926 he took up a teaching position at the

University of Bristol , mottoeng = earningpromotes one's innate power (from Horace, ''Ode 4.4'') , established = 1595 – Merchant Venturers School1876 – University College, Bristol1909 – received royal charter , type ...

, 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. 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'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,

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 a ...

. 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 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 ...

, where Lefschetz was, visiting also Oscar Zariski at

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 consi ...

. 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 ...

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 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 ...

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 bec ...

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

Pembroke College, Cambridge Pembroke College (officially "The Master, Fellows and Scholars of the College or Hall of Valence-Mary") is a constituent college of the University of Cambridge, England. The college is the third-oldest college of the university and has over 700 ...

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 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 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 t ...

theory for curves on an

algebraic surface In mathematics, an algebraic surface is an algebraic variety of dimension two. In the case of geometry over the field of complex numbers, an algebraic surface has complex dimension two (as a complex manifold, when it is non-singular) and so of di ...

: 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. ''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 the ...

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, 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 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 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 co ...

s. 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 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, 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 geometr ...

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

, 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