Hazel Perfect (circa 1927 – 8 July 2015) was a British mathematician specialising in

combinatorics Combinatorics is an area of mathematics primarily concerned with counting, both as a means and as an end to obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many ...

Contributions

Perfect was known for inventing

gammoid In matroid theory, a field within mathematics, a gammoid is a certain kind of matroid, describing sets of Vertices (graph theory), vertices that can be reached by vertex-disjoint Path (graph theory), paths in a directed graph. The concept of a g ...

s, for her work with

Leon Mirsky Leonid Mirsky (19 December 1918 – 1 December 1983) was a Russian-British mathematician who worked in number theory, linear algebra, and combinatorics.... Mirsky's theorem is named after him. Biography Mirsky was born in Russia on 19 December ...

on doubly stochastic matrices, for her three books ''Topics in Geometry'', ''Topics in Algebra'', and '' Independence Theory in Combinatorics'', and for her work as a translator (from an earlier German translation) of

Pavel Alexandrov Pavel Sergeyevich Alexandrov (), sometimes romanized ''Paul Alexandroff'' (7 May 1896 – 16 November 1982), was a Soviet mathematician. He wrote roughly three hundred papers, making important contributions to set theory and topology. In topol ...

's book ''An Introduction to the Theory of Groups'' (Hafner, 1959). The Perfect–Mirsky conjecture, named after Perfect and

, concerns the region of the

complex plane In mathematics, the complex plane is the plane (geometry), plane formed by the complex numbers, with a Cartesian coordinate system such that the horizontal -axis, called the real axis, is formed by the real numbers, and the vertical -axis, call ...

formed by the

eigenvalue In linear algebra, an eigenvector ( ) or characteristic vector is a vector that has its direction unchanged (or reversed) by a given linear transformation. More precisely, an eigenvector \mathbf v of a linear transformation T is scaled by a ...

s of doubly stochastic matrices. Perfect and Mirsky conjectured that for

n\times n

matrices this region is the union of regular polygons of up to

n

sides, having the

roots of unity In mathematics, a root of unity is any complex number that yields 1 when raised to some positive integer power . Roots of unity are used in many branches of mathematics, and are especially important in number theory, the theory of group char ...

of each degree up to

n

as vertices. Perfect and Mirsky proved their conjecture for

n\le 3

; it was subsequently shown to be true for

n=4

and false for

n=5

, but remains open for larger values

Education and career

Perfect earned a master's degree through

Westfield College Westfield College was a small college situated in Hampstead, London, from 1882 to 1989. It was the first college to aim to educate women for University of London degrees from its opening. The college originally admitted only women as students and ...

(a constituent college for women in the

University of London The University of London (UoL; abbreviated as Lond or more rarely Londin in Post-nominal letters, post-nominals) is a collegiate university, federal Public university, public research university located in London, England, United Kingdom. The ...

) in 1949, with a thesis on ''The Reduction of Matrices to Canonical Form''. In the 1950s, Perfect was a lecturer at University College of Swansea; she collaborated with Gordon Petersen, a visitor to Swansea at that time, on their translation of Alexandrov's book. She completed her Ph.D. at the University of London in 1969; her dissertation was ''Studies in Transversal Theory with Particular Reference to Independence Structures and Graphs''. She became a reader in mathematics at the

University of Sheffield The University of Sheffield (informally Sheffield University or TUOS) is a public university, public research university in Sheffield, South Yorkshire, England. Its history traces back to the foundation of Sheffield Medical School in 1828, Fir ...

Selected publications

Books

Research papers

Translation

References

{{DEFAULTSORT:Perfect, Hazel 2015 deaths British mathematicians British women mathematicians Alumni of Westfield College Academics of Swansea University Academics of the University of Sheffield German–English translators Technical translators Combinatorialists Year of birth uncertain