This article is about a mathematician. For the companion of Samuel Johnson, see Hodge (cat). Hodge is the name of one of Samuel Johnsons cats, immortalized in a characteristically whimsical passage in James Boswells Life of Johnson: 1 The latter paragraph is used as the epigraph to Vladimir Nabokovs acclaimed poem/novel Pale Fire. ...

William Vallance Douglas Hodge (17 June 1903 - 7 July 1975) was a Scottish mathematician, specifically a geometer. His discovery of topological relations between algebraic geometry and differential geometry - now called Hodge theory and pertaining more generally to Kähler manifolds - was a major influence on subsequent work. He was born in Edinburgh, and was a professor at Cambridge from 1936 to 1970. Amongst other honours, he received the Copley Medal of the Royal Society June 17 is the 168th day of the year in the Gregorian calendar (169th in leap years), with 197 days remaining. ... 1903 has the latest occurring solstices and equinoxes for 400 years, because the Gregorian calendar hasnt had a leap year for seven years or a century leap year since 1600. ... July 7 is the 188th day of the year (189th in leap years) in the Gregorian Calendar, with 177 days remaining. ... 1975 was a common year starting on Wednesday (the link is to a full 1975 calendar). ... A mathematician is a person whose area of study and research is mathematics. ... A geometer is a mathematician whose area of study is geometry. ... Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. ... In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. ... In mathematics, Hodge theory is the study of the consequences for the algebraic topology of a smooth manifold M of the partial differential equation theory of generalised Laplacian operators associated to a Riemannian metric on M. It was developed by W. V. D. Hodge in the 1930s as an extension... In mathematics, a Kähler manifold is a complex manifold which also carries a Riemannian metric and a symplectic form on the underlying real manifold in such a way that the three structures (complex, Riemannian, and symplectic) are all mutually compatible. ... Edinburghs location in Scotland Edinburgh viewed from Arthurs Seat. ... The city of Cambridge is an old English university town and the regional centre of the county of Cambridgeshire. ... 1936 was a leap year starting on Wednesday (link will take you to calendar). ... 1970 was a common year starting on Thursday. ... The Copley Medal is a scientific award for work in any field of science, the highest award granted by the Royal Society of London. ... The Royal Society of London is claimed to be the oldest learned society still in existence. ...

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. 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 equivalence) has a one-dimensional subspace on which it is positive... In mathematics, the concept of intersection number arose in algebraic geometry, where two curves intersecting at a point may be considered to meet twice if they are tangent there. ... In mathematics, an algebraic surface is an algebraic variety of dimension two. ... For use of the term in mathematics, see signature (mathematics). ... In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. ... In relation with the history of mathematics, the Italian school of algebraic geometry refers to the work over half a century or more (flourishing roughly 1885-1935) done internationally in birational geometry, particularly on algebraic surfaces. ... Solomon Lefschetz (3 September 1884-5 October 1972) was a US mathematician who did fundamental work on algebraic topology, its applications to algebraic geometry, and the theory of non-linear ordinary differential equations. ...

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 Laplacian solutions, 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). In mathematics, a Kähler manifold is a complex manifold which also carries a Riemannian metric and a symplectic form on the underlying real manifold in such a way that the three structures (complex, Riemannian, and symplectic) are all mutually compatible. ... In vector calculus, the Laplace operator or Laplacian is a differential operator equal to the sum of all the unmixed second partial derivatives of a dependent variable. ... In classical algebraic geometry (and to some extent also in modern algebraic geometry), the main objects of study are algebraic varieties. ... In mathematics, a projective space is a fundamental construction from any vector space. ... 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. ...

This Hodge decomposition is 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. In algebraic topology, the Betti numbers of a topological space X are a sequence b0, b1, ... of topological invariants. ... In algebraic geometry, the moduli problem is to describe the parameters on which algebraic varieties depend. ...

In particular the Hodge conjecture on the 'middle' spaces H^p,p is still unsolved, in general. The Hodge conjecture is a major unsolved problem of algebraic geometry. ...

Further developments by others led in particular to an idea of mixed Hodge structure on singular varieties, and to deep analogies with étale cohomology. In mathematics, the étale cohomology theory of algebraic geometry is a refined construction of homological algebra, introduced in order to attack the Weil conjectures. ...

Hodge also wrote (with Daniel Pedoe) a three-volume work on algebraic geometry with much concrete content - but illustrating what Elie Cartan called 'the debauch of indices', in its component notation. In fact a story of Hodge's lecturing style concerned his favouring not only of subscripts and superscripts, but of the letters r and s - which he wrote on a blackboard so as to be indistinguishable. Dan Pedoe (1910 to 1998) was an English-born mathematician and geometer with a career spanning more than sixty years. ... Élie Joseph Cartan (9 April 1869 - 6 May 1951) was a French mathematician, who did fundamental work in the theory of Lie groups and their geometric applications. ...

External links

• Biography (http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Hodge.html) at the MacTutor archive