Hazel Perfect

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

Contributions

Perfect was known for inventing gammoids,^[AMG] for her work with Leon Mirsky on doubly stochastic matrices,^[SP2] for her three books Topics in Geometry,^[TIG] Topics in Algebra,^[TIA] and Independence Theory in Combinatorics,^[ITC] and for her work as a translator (from an earlier German translation) of Pavel Alexandrov's book An Introduction to the Theory of Groups (Hafner, 1959).^[ITG]

The Perfect–Mirsky conjecture, named after Perfect and Leon Mirsky, concerns the region of the complex plane formed by the eigenvalues 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 of each degree up to $n$ as vertices. Perfect and Mirsky proved their conjecture for $n\leq 3$; it was subsequently shown to be true for $n=4$ and false for $n=5$, but remains open for larger values of $n$.^[SP2]

Education and career

Perfect earned a master's degree through Westfield College (a constituent college for women in the University of London) 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.

Selected publications

Books

TIG.

Perfect, Hazel (1963), Topics in Geometry, Pergamon, MR 0155210

TIA.

Perfect, Hazel (1966), Topics in Algebra, Pergamon

ITC.

Bryant, Victor; Perfect, Hazel (1980), Independence Theory in Combinatorics: An introductory account with applications to graphs and transversals, London and New York: Chapman & Hall, ISBN 0-412-16220-2, MR 0604173

Research papers

SP2.

Perfect, Hazel; Mirsky, L. (1965), "Spectral properties of doubly-stochastic matrices", Monatshefte für Mathematik, 69: 35–57, doi:10.1007/BF01313442, MR 0175917, S2CID 120466093

AMG.

Perfect, Hazel (1968), "Applications of Menger's graph theorem", Journal of Mathematical Analysis and Applications, 22: 96–111, doi:10.1016/0022-247X(68)90163-7, MR 0224494

Translation

ITG.

Alexandroff, P. S. (1959), An Introduction to the Theory of Groups, translated by Perfect, Hazel; Petersen, G. M., New York: Hafner Publishing Co., MR 0099361