John Arthur Todd (23 August 1908 – 22 December 1994) was an English mathematician who specialised in geometry.

Biography

He was born in

Liverpool Liverpool is a city and metropolitan borough in Merseyside, England. With a population of in 2019, it is the 10th largest English district by population and its metropolitan area is the fifth largest in the United Kingdom, with a popul ...

, and went up to

Trinity College, Cambridge Trinity College is a constituent college of the University of Cambridge. Founded in 1546 by Henry VIII, King Henry VIII, Trinity is one of the largest Cambridge colleges, with the largest financial endowment of any college at either Cambridge ...

in 1925. He did research under

H.F. Baker Henry Frederick Baker Royal Society, FRS Royal Society of Edinburgh, 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 ...

, and in 1931 took a position at the

University of Manchester , mottoeng = Knowledge, Wisdom, Humanity , established = 2004 – University of Manchester Predecessor institutions: 1956 – UMIST (as university college; university 1994) 1904 – Victoria University of Manchester 1880 – Victoria Univer ...

. He became a lecturer at Cambridge in 1937. He remained at Cambridge for the rest of his working life.

Work

The

Todd class In mathematics, the Todd class is a certain construction now considered a part of the theory in algebraic topology of characteristic classes. The Todd class of a vector bundle can be defined by means of the theory of Chern classes, and is encounter ...

in the theory of the higher-dimensional

Riemann–Roch theorem The Riemann–Roch theorem is an important theorem in mathematics, specifically in complex analysis and algebraic geometry, for the computation of the dimension of the space of meromorphic functions with prescribed zeros and allowed poles. It rel ...

is an example of a

characteristic class In mathematics, a characteristic class is a way of associating to each principal bundle of ''X'' a cohomology class of ''X''. The cohomology class measures the extent the bundle is "twisted" and whether it possesses sections. Characteristic classes ...

(or, more accurately, a reciprocal of one) that was discovered by Todd in work published in 1937. It used the methods of 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 ...

. The Todd–Coxeter process for

coset enumeration In mathematics, coset enumeration is the problem of counting the cosets of a subgroup ''H'' of a group ''G'' given in terms of a presentation. As a by-product, one obtains a permutation representation for ''G'' on the cosets of ''H''. If ''H'' has ...

is a major method of computational algebra, and dates from a collaboration with

H.S.M. Coxeter Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century. Biography Coxeter was born in Kensington t ...

in 1936. In 1953 he and Coxeter discovered the

Coxeter–Todd lattice In mathematics, the Coxeter–Todd lattice K12, discovered by , is a 12-dimensional even integral lattice of discriminant 36 with no norm-2 vectors. It is the sublattice of the Leech lattice fixed by a certain automorphism of order 3, and is an ...

. In 1954 he and G. C. Shephard classified the finite

complex reflection group In mathematics, a complex reflection group is a finite group acting on a finite-dimensional complex vector space that is generated by complex reflections: non-trivial elements that fix a complex hyperplane pointwise. Complex reflection groups arise ...

Honours

In March 1948 he was elected a

Fellow of the Royal Society Fellowship of the Royal Society (FRS, ForMemRS and HonFRS) is an award granted by the judges of the Royal Society of London to individuals who have made a "substantial contribution to the improvement of natural science, natural knowledge, incl ...

Selected publications

* 1936: "A practical method for enumerating cosets of a finite abstract group", Proc. Edin. Math. Soc. 5(1), 26-34 (with

Harold Scott MacDonald Coxeter Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century. Biography Coxeter was born in Kensington t ...

) * 1937: "Rational quartic primals and associated Cremona transformations of four-dimensional space", Proc. London Math. Soc. s2-42, 324-339 (with

Dennis Babbage Dennis William Babbage (26 April 1909 – 9 June 1991) was an English mathematician associated with Magdalene College, Cambridge, and with codebreaking at Bletchley Park during World War II. In 1980 Babbage was President of Magdalene College, Cam ...

), "The geometrical invariants of algebraic varieties", Proc. London Math. Soc. 43(2), 127-138, "The arithmetical invariants of algebraic loci", Proc. London Math. Soc. 43(2), 190-225 * 1939: "The geometrical invariants of algebraic loci", Proc. London Math. Soc. 45, 410-424 * 1953: "An extreme duodenary form", Can. J. Math. 5, 384-392 (with

) * 1954: "Finite unitary reflection groups", Canadian Journal of Mathematics 6, 274-304 (with

Geoffrey Colin Shephard Geoffrey Colin Shephard is a mathematician who works on convex geometry and reflection groups. He asked Shephard's problem on the volumes of projected convex bodies, posed another problem on polyhedral nets, proved the Shephard–Todd theorem in ...

) * 1960: "On complex Stiefel manifolds", Mathematical Proc. Camb. Phil. Soc. 56, 342-353 (with

Michael Atiyah Sir Michael Francis Atiyah (; 22 April 1929 – 11 January 2019) was a British-Lebanese mathematician specialising in geometry. His contributions include the Atiyah–Singer index theorem and co-founding topological K-theory. He was awarded the ...

) * 1966: "A representation of the Mathieu group M₂₄ as a collineation group", Ann. Mat. Pura Appl. 71(4), 199-238

References

External links

Todd's Mactutor biography
{{DEFAULTSORT:Todd, J. A. 1908 births 1994 deaths Alumni of Trinity College, Cambridge 20th-century English mathematicians Scientists from Liverpool Fellows of Downing College, Cambridge Fellows of the Royal Society