W. V. D. Hodge
Encyclopedia
William Vallance Douglas Hodge FRS  (17 June 1903 – 7 July 1975) was a Scottish mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....

, specifically a geometer.
His discovery of far-reaching topological relations between algebraic geometry
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...

 and differential geometry—an area now called Hodge theory
Hodge theory
In mathematics, Hodge theory, named after W. V. D. Hodge, is one aspect of the study of the algebraic topology of a smooth manifold M. More specifically, it works out the consequences for the cohomology groups of M, with real coefficients, of the partial differential equation theory of generalised...

 and pertaining more generally to Kähler manifold
Kähler manifold
In mathematics, a Kähler manifold is a manifold with unitary structure satisfying an integrability condition.In particular, it is a Riemannian manifold, a complex manifold, and a symplectic manifold, with these three structures all mutually compatible.This threefold structure corresponds to the...

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

Life and career

He was born in Edinburgh
Edinburgh
Edinburgh is the capital city of Scotland, the second largest city in Scotland, and the eighth most populous in the United Kingdom. The City of Edinburgh Council governs one of Scotland's 32 local government council areas. The council area includes urban Edinburgh and a rural area...

, attended George Watson's College
George Watson's College
George Watson's College, known informally as Watson's, 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 merged with its sister school...

, and studied at Edinburgh University, graduating in 1923. With help from E. T. Whittaker
E. T. Whittaker
Edmund Taylor Whittaker FRS FRSE was an English mathematician who contributed widely to applied mathematics, mathematical physics and the theory of special functions. He had a particular interest in numerical analysis, but also worked on celestial mechanics and the history of physics...

 whose son J. M. Whittaker was a college friend, he then took the Cambridge Mathematical Tripos
Cambridge Mathematical Tripos
The Mathematical Tripos is the taught mathematics course at the University of Cambridge. It is the oldest Tripos that is examined in Cambridge.-Origin:...

. At Cambridge he fell under the influence of the geometer H. F. Baker
H. F. Baker
Henry Frederick Baker was a British mathematician, working mainly in algebraic geometry, but also remembered for contributions to partial differential equations , and Lie groups....

.

In 1926 he took up a teaching position at the University of Bristol
University of Bristol
The University of Bristol is a public research university located in Bristol, United Kingdom. One of the so-called "red brick" universities, it received its Royal Charter in 1909, although its predecessor institution, University College, Bristol, had been in existence since 1876.The University is...

, and began work on the interface between the Italian school of algebraic geometry
Italian school of algebraic geometry
In relation with the history of mathematics, the Italian school of algebraic geometry refers to the work over half a century or more done internationally in birational geometry, particularly on algebraic surfaces. There were in the region of 30 to 40 leading mathematicians who made major...

, particularly problems posed by Francesco Severi
Francesco Severi
Francesco Severi was an Italian mathematician.Severi was born in Arezzo, Italy. He is famous for his contributions to algebraic geometry and the theory of functions of several complex variables. He became the effective leader of the Italian school of algebraic geometry...

, and the topological methods of Solomon Lefschetz
Solomon Lefschetz
Solomon Lefschetz was an American mathematician who did fundamental work on algebraic topology, its applications to algebraic geometry, and the theory of non-linear ordinary differential equations.-Life:...

. 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
Max Newman
Maxwell Herman Alexander "Max" Newman, FRS was a British mathematician and codebreaker.-Pre–World War II:Max Newman was born Maxwell Neumann in Chelsea, London, England, on 7 February 1897...

'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
Princeton University
Princeton University is a private research university located in Princeton, New Jersey, United States. The school is one of the eight universities of the Ivy League, and is one of the nine Colonial Colleges founded before the American Revolution....

, where Lefschetz was, visiting also Oscar Zariski
Oscar Zariski
Oscar Zariski was a Russian mathematician and one of the most influential algebraic geometers of the 20th century.-Education:...

 at Johns Hopkins University
Johns Hopkins University
The Johns Hopkins University, commonly referred to as Johns Hopkins, JHU, or simply Hopkins, is a private research university based in Baltimore, Maryland, United States...

. 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
The city of Cambridge is a university town and the administrative centre of the county of Cambridgeshire, England. It lies in East Anglia about north of London. Cambridge is at the heart of the high-technology centre known as Silicon Fen – a play on Silicon Valley and the fens surrounding the...

, a position he held from 1936 to 1970. He was the first head of DPMMS
Faculty of Mathematics, University of Cambridge
The Faculty of Mathematics at the University of Cambridge comprises the Department of Pure Mathematics and Mathematical Statistics and the Department of Applied Mathematics and Theoretical Physics . It is housed in the Centre for Mathematical Sciences site in West Cambridge, alongside the Isaac...

.

He was the Master of Pembroke College, Cambridge
Pembroke College, Cambridge
Pembroke College is a constituent college of the University of Cambridge, England.The college has over seven hundred students and fellows, and is the third oldest college of the university. Physically, it is one of the university's larger colleges, with buildings from almost every century since its...

 from 1958 to 1970, and vice-president of the Royal Society
Royal Society
The Royal Society of London for Improving Natural Knowledge, known simply as the Royal Society, is a learned society for science, and is possibly the oldest such society in existence. Founded in November 1660, it was granted a Royal Charter by King Charles II as the "Royal Society of London"...

 from 1959 to 1965. He was knighted in 1959. Amongst other honours, he received the Adams Prize
Adams Prize
The Adams Prize is awarded each year by the Faculty of Mathematics at the University of Cambridge and St John's College to a young, UK based mathematician for first-class international research in the Mathematical Sciences....

 in 1937 and the Copley Medal
Copley Medal
The Copley Medal is an award given by the Royal Society of London for "outstanding achievements in research in any branch of science, and alternates between the physical sciences and the biological sciences"...

 of the Royal Society
Royal Society
The Royal Society of London for Improving Natural Knowledge, known simply as the Royal Society, is a learned society for science, and is possibly the oldest such society in existence. Founded in November 1660, it was granted a Royal Charter by King Charles II as the "Royal Society of London"...

 in 1974.

Work

The Hodge index theorem
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...

 was a result on the intersection number
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 curves, and accounting properly for tangency...

 theory for curves on an algebraic surface
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 and so of dimension four as a smooth manifold.The theory of algebraic surfaces is much more complicated than that...

: it determines the signature
Signature
A signature is a handwritten depiction of someone's name, nickname, or even a simple "X" that a person writes on documents as a proof of identity and intent. The writer of a signature is a signatory. Similar to a handwritten signature, a signature work describes the work as readily identifying...

 of the corresponding quadratic form
Quadratic form
In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. For example,4x^2 + 2xy - 3y^2\,\!is a quadratic form in the variables x and y....

. This result was sought by the Italian school of algebraic geometry
Italian school of algebraic geometry
In relation with the history of mathematics, the Italian school of algebraic geometry refers to the work over half a century or more done internationally in birational geometry, particularly on algebraic surfaces. There were in the region of 30 to 40 leading mathematicians who made major...

, 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
Algebraic variety
In mathematics, an algebraic variety is the set of solutions of a system of polynomial equations. Algebraic varieties are one of the central objects of study in algebraic geometry...

 V (assumed complex, projective and non-singular) because projective space
Projective space
In mathematics a projective space is a set of elements similar to the set P of lines through the origin of a vector space V. The cases when V=R2 or V=R3 are the projective line and the projective plane, respectively....

 itself carries such a metric. In de Rham cohomology
De Rham cohomology
In mathematics, de Rham cohomology 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 adapted to computation and the concrete representation of cohomology classes...

 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
Laplace's equation
In mathematics, 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:where ∆ = ∇² is the Laplace operator and \varphi is a scalar function...

, one can get unique α. This has the important, immediate consequence of splitting up
Hk(V(C), C)


into subspaces
Hp,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 hp,q refine the Betti number
Betti number
In algebraic topology, a mathematical discipline, the Betti numbers can be used to distinguish topological spaces. Intuitively, the first Betti number of a space counts the maximum number of cuts that can be made without dividing the space into two pieces....

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 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
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 conjectures...

.

Hodge conjecture

The Hodge conjecture
Hodge conjecture
The Hodge conjecture is a major unsolved problem in algebraic geometry which relates the algebraic topology of a non-singular complex algebraic variety and the subvarieties of that variety. More specifically, the conjecture says that certain de Rham cohomology classes are algebraic, that is, they...

 on the 'middle' spaces Hp,p is still unsolved, in general. It is one of the seven Millennium Prize Problems
Millennium Prize Problems
The Millennium Prize Problems are seven problems in mathematics that were stated by the Clay Mathematics Institute in 2000. As of September 2011, six of the problems remain unsolved. A correct solution to any of the problems results in a US$1,000,000 prize being awarded by the institute...

 set up by the Clay Mathematics Institute
Clay Mathematics Institute
The Clay Mathematics Institute is a private, non-profit foundation, based in Cambridge, Massachusetts. The Institute is dedicated to increasing and disseminating mathematical knowledge. It gives out various awards and sponsorships to promising mathematicians. The institute was founded in 1998...

.

Exposition

Hodge also wrote, with Daniel Pedoe
Daniel Pedoe
Dan Pedoe 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 expository papers in geometry. He is also the author of various core books on mathematics and geometry some of which have...

, a three-volume work Methods of Algebraic Geometry, on classical algebraic geometry, with much concrete content — illustrating though what Élie Cartan
Élie Cartan
Élie Joseph Cartan was an influential French mathematician, who did fundamental work in the theory of Lie groups and their geometric applications...

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.
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