# Hazel Perfect

Hazel Perfect (circa 1927 – 8 July 2015)[1] was a British mathematician specialising in combinatorics.

## Contributions

Perfect was known for inventing gammoids,[2][3][AMG] for her work with Leon Mirsky on doubly stochastic matrices,[4][SP2] for her three books Topics in Geometry,[5][TIG] Topics in Algebra,[6][TIA] and Independence Theory in Combinatorics,[7][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).[8][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 ${\displaystyle n\times n}$ matrices this region is the union of regular polygons of up to ${\displaystyle n}$ sides, having the roots of unity of each degree up to ${\displaystyle n}$ as vertices. Perfect and Mirsky proved their conjecture for ${\displaystyle n\leq 3}$; it was subsequently shown to be true for ${\displaystyle n=4}$ and false for ${\displaystyle n=5}$, but remains open for larger values of ${\displaystyle n}$.[9][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.[10] 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.[11] 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.[12] She became a reader in mathematics at the University of Sheffield.[13]

## Selected publications

### Books

 TIA. Perfect, Hazel (1966), Topics in Algebra, Pergamon[6]
 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[7]

### 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[8]

## References

1. ^ "Obituaries" (PDF), Newsletter of the London Mathematical Society, p. 41, December 2015
2. ^ Schrijver, Alexander (2003), Combinatorial optimization: Polyhedra and efficiency, Vol. B: Matroids, trees, stable sets, Algorithms and Combinatorics, vol. 24, Berlin: Springer-Verlag, p. 659, ISBN 3-540-44389-4, MR 1956925
3. ^ Welsh, D. J. A. (1976), Matroid theory, London and New York: Academic Press, p. 219, ISBN 9780486474397, MR 0427112
4. ^
5. ^ a b Review of Topics in Geometry:
6. ^ a b Reviews of Topics in Algebra:
• Drechsel, Robert R. (November 1968), The Mathematics Teacher, 61 (7): 725–726, JSTOR 27957974{{citation}}: CS1 maint: untitled periodical (link)
• Matthews, Geoffrey (December 1969), The Mathematical Gazette, 53 (386): 431–432, doi:10.2307/3612506, JSTOR 3612506{{citation}}: CS1 maint: untitled periodical (link)
7. ^ a b Reviews of Independence Theory in Combinatorics:
8. ^ a b Reviews of An Introduction to the Theory of Groups:
9. ^ Levick, Jeremy; Pereira, Rajesh; Kribs, David W. (2015), "The four-dimensional Perfect–Mirsky Conjecture", Proceedings of the American Mathematical Society, 143 (5): 1951–1956, doi:10.1090/S0002-9939-2014-12412-9, MR 3314105
10. ^ Subjects of Dissertations, Theses and Published Works Presented by Successful Candidates at Examinations for Higher Degrees, University of London, 1937, p. 22 – via Google Books
11. ^ Burkill, H. (January 1999), "Gordon Marshall Petersen", Bulletin of the London Mathematical Society, 31 (1): 97–107, doi:10.1112/s0024609398005177
12. ^ Theses and Dissertations Accepted for Higher Degrees, University of London, 1967, p. 42 – via Google Books
13. ^ Author biography from A Mathematical Spectrum Miscellany: selections from Mathematical Spectrum, 1967–1994, Applied Probability Trust, 2000, p. 3, ISBN 9780902016057