James William Peter Hirschfeld (born 1940) is an Australian mathematician, resident in the United Kingdom, specializing in

combinatorial geometry Discrete geometry and combinatorial geometry are branches of geometry that study combinatorial properties and constructive methods of discrete geometric objects. Most questions in discrete geometry involve finite or discrete sets of basic geom ...

and the geometry of

finite field In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field (mathematics), field that contains a finite number of Element (mathematics), elements. As with any field, a finite field is a Set (mathematics), s ...

s. He is an

emeritus professor ''Emeritus/Emerita'' () is an honorary title granted to someone who retires from a position of distinction, most commonly an academic faculty position, but is allowed to continue using the previous title, as in "professor emeritus". In some c ...

and Tutorial Fellow at the

University of Sussex The University of Sussex is a public university, public research university, research university located in Falmer, East Sussex, England. It lies mostly within the city boundaries of Brighton and Hove. Its large campus site is surrounded by the ...

. Hirschfeld received his doctorate in 1966 from the

University of Edinburgh The University of Edinburgh (, ; abbreviated as ''Edin.'' in Post-nominal letters, post-nominals) is a Public university, public research university based in Edinburgh, Scotland. Founded by the City of Edinburgh Council, town council under th ...

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 The University of Perugia ( Italian ''Università degli Studi di Perugia'') is a public university in Perugia, Italy. It was founded in 1308, as attested by the Bull issued by Pope Clement V certifying the birth of the Studium Generale. The offi ...

and University of Rome with support from the

Royal Society The Royal Society, formally The Royal Society of London for Improving Natural Knowledge, is a learned society and the United Kingdom's national academy of sciences. The society fulfils a number of roles: promoting science and its benefits, re ...

and

Accademia nazionale dei Lincei The (; literally the "Academy of the Lynx-Eyed"), 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 Rome, Italy. Founded in ...

. He edited

Beniamino Segre Beniamino Segre (16 February 1903 – 2 October 1977) was an Italian mathematician who is remembered today as a major contributor to algebraic geometry and one of the founders of finite geometry. Life and career He was born and studied in Turin. ...

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

Galois geometry Galois geometry (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 In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ( ...

and

linear algebra Linear algebra is the branch of mathematics concerning linear equations such as :a_1x_1+\cdots +a_nx_n=b, linear maps such as :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrix (mathemat ...

taught in a first degree course, as well as a little

projective geometry In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting (''p ...

, and a very little

algebraic geometry Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...

." When ''q'' is a

prime power In mathematics, a prime power is a positive integer which is a positive integer power of a single prime number. For example: , and are prime powers, while , and are not. The sequence of prime powers begins: 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 1 ...

then there is a

GF(''q'') with ''q'' elements called a Galois field. A

vector space In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...

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 Coding theory is the study of the properties of codes and their respective fitness for specific applications. Codes are used for data compression, cryptography, error detection and correction, data transmission and computer data storage, data sto ...

, such as algebraic geometry codes, so the field is supported by

computer science Computer science is the study of computation, information, and automation. Computer science spans Theoretical computer science, theoretical disciplines (such as algorithms, theory of computation, and information theory) to Applied 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 the American mathematician Richard Collom Singleton (1928–2007), 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 ...

mathematics journal In academic publishing, a scientific journal is a periodical publication designed to further the progress of science by disseminating new research findings to the scientific community. These journals serve as a platform for researchers, schola ...

s publishing finite geometry, and conferences on

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 ...

featuring Galois geometry. Colleague Joseph A. Thas 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 The Institute of Combinatorics and its Applications (ICA) is an international scientific organization formed in 1990 to increase the visibility and influence of the combinatorial community. In pursuit of this goal, the ICA sponsors conferences, ...

Selected publications

* 1979: ''Projective Geometries over Finite Fields'',

Oxford University Press Oxford University Press (OUP) is the publishing house of the University of Oxford. It is the largest university press in the world. Its first book was printed in Oxford in 1478, with the Press officially granted the legal right to print books ...

2nd ed., Oxford, Clarendon Press 1998 * 1985: ''Finite Projective Spaces of Three Dimensions'', Oxford University Press * 1991: (with Joseph A. Thas) ''General Galois Geometries'', Oxford University Pres
2016 paperback reprint
* 2008: (with

Gábor Korchmáros Gábor Korchmáros (born 1948) is a Hungarian mathematician, who works on finite geometry. Biography Korchmáros received in 1972 from the University of Budapest a Ph.D. in mathematics. In 1973 on a postdoc grant, he studied at the Research Center ...

& Fernando Torres
Algebraic Curves over a Finite Field

Princeton University Press Princeton University Press is an independent publisher with close connections to Princeton University. Its mission is to disseminate scholarship within academia and society at large. The press was founded by Whitney Darrow, with the financial ...

References

External links

Prof James Hirschfeld
at

* {{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