James William Peter Hirschfeld (born 1940) is an Australian mathematician, resident in the United Kingdom, specializing in combinatorial geometry and the geometry of finite fields. He is an emeritus professor and Tutorial Fellow at the University of Sussex. Hirschfeld received his doctorate in 1966 from the University of Edinburgh with thesis advisor William Leonard Edge and thesis ''The geometry of cubic surfaces, and Grace's extension of the double-six, over finite fields''. To pursue further studies in finite geometry Hirschfeld went to University of Perugia and University of Rome with support from the Royal Society and

Accademia nazionale dei Lincei The Accademia dei Lincei (; literally the "Academy of the Lynx-Eyed", but anglicised as the Lincean Academy) is one of the oldest and most prestigious European scientific institutions, located at the Palazzo Corsini on the Via della Lungara in Rom ...

. He edited Beniamino Segre's 100-page monograph "Introduction to Galois Geometries" (1967). In 1979 Hirschfeld published the first of a trilogy on

Galois geometry Galois geometry (so named after the 19th-century French mathematician Évariste Galois) is the branch of finite geometry that is concerned with algebraic and analytic geometry over a finite field (or ''Galois field''). More narrowly, ''a'' Ga ...

, pegged at a level depending only on "the group theory and linear algebra taught in a first degree course, as well as a little projective geometry, and a very little

algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...

." When ''q'' is a prime power then there is a finite field GF(''q'') with ''q'' elements called a Galois field. A vector space over GF(''q'') of ''n'' + 1 dimensions produces an n-dimensional Galois geometry PG(''n,q'') with its subspaces: one-dimensional subspaces are the points of the Galois geometry and two-dimensional subspaces are the lines. Non-singular linear transformations of the vector space provide motions of PG(''n,q''). The first book (1979) covered PG(1,''q'') and PG(2,''q''). The second book addressed PG(3,''q'') and the third PG(''n,q''). Chapters are numbered sequentially through the trilogy: 14 in the first book, 15 to 21 in the second, and 22 to 27 in the third. Finite geometry has contributed to coding theory, such as algebraic geometry codes, so the field is supported by computer science. In the preface of the 1991 text Hirschfeld summarizes the status of Galois geometry, mentioning

maximum distance separable code In coding theory, the Singleton bound, named after Richard Collom Singleton, is a relatively crude upper bound on the size of an arbitrary block code C with block length n, size M and minimum distance d. It is also known as the Joshibound. proved b ...

mathematics journal In academic publishing, a scientific journal is a periodical publication intended to further the progress of science, usually by reporting new research. Content Articles in scientific journals are mostly written by active scientists such as s ...

s publishing finite geometry, and conferences on

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

featuring Galois geometry. Colleague

Joseph A. Thas Joseph Adolphe François Thas (born 13 October 1944, Dilbeek, Belgium) is a Belgian mathematician, who works on combinatorics, incidence geometry and finite geometries. Thas received in 1969 his PhD from Ghent University under Julien Bilo with ...

is coauthor of ''General Galois Geometries'' on PG(''n,q'') where ''n'' ≥ 4. Hirschfeld was cited as the ultimate editor of ''Design Theory'' (1986). In 2018 he received the 2016 Euler Medal.

Selected publications

* 1979: ''Projective Geometries over Finite Fields'', Oxford University Press 2nd ed., Oxford, Clarendon Press 1998 * 1985: ''Finite Projective Spaces of Three Dimensions'', Oxford University Press * 1991: (with

) ''General Galois Geometries'', Oxford University Pres
2016 paperback reprint
* 2008: (with Gábor Korchmáros & Fernando Torres
Algebraic Curves over a Finite Field
Princeton University Press

References

External links

Prof James Hirschfeld
at University of Sussex * {{DEFAULTSORT:Hirschfeld, James William Peter Alumni of the University of Edinburgh Academics of the University of Sussex 20th-century British mathematicians 21st-century British mathematicians Combinatorialists Geometers 1940 births Living people