Marshall Hall Jr. (17 September 1910 – 4 July 1990) was an American

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...

who made significant contributions to

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

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 ...

Education and career

Hall studied mathematics at

Yale University Yale University is a Private university, private Ivy League research university in New Haven, Connecticut, United States. Founded in 1701, Yale is the List of Colonial Colleges, third-oldest institution of higher education in the United Stat ...

, graduating in 1932. He studied for a year at

Cambridge University The University of Cambridge is a Public university, public collegiate university, collegiate research university in Cambridge, England. Founded in 1209, the University of Cambridge is the List of oldest universities in continuous operation, wo ...

under a Henry Fellowship working with

G. H. Hardy Godfrey Harold Hardy (7 February 1877 – 1 December 1947) was an English mathematician, known for his achievements in number theory and mathematical analysis. In biology, he is known for the Hardy–Weinberg principle, a basic principle of pop ...

. He returned to Yale to take his Ph.D. in 1936 under the supervision of

Øystein Ore Øystein Ore (7 October 1899 – 13 August 1968) was a Norwegian mathematician known for his work in ring theory, Galois connections, graph theory, and the history of mathematics. Life Ore graduated from the University of Oslo in 1922, with a ...

. He worked in

Naval Intelligence Military intelligence is a military discipline that uses information collection and analysis approaches to provide guidance and direction to assist commanders in their decisions. This aim is achieved by providing an assessment of data from a ...

during

World War II World War II or the Second World War (1 September 1939 – 2 September 1945) was a World war, global conflict between two coalitions: the Allies of World War II, Allies and the Axis powers. World War II by country, Nearly all of the wo ...

, including six months in 1944 at

Bletchley Park Bletchley Park is an English country house and Bletchley Park estate, estate in Bletchley, Milton Keynes (Buckinghamshire), that became the principal centre of Allies of World War II, Allied World War II cryptography, code-breaking during the S ...

, the center of British wartime code breaking. In 1946 he took a position at

Ohio State University The Ohio State University (Ohio State or OSU) is a public university, public Land-grant university, land-grant research university in Columbus, Ohio, United States. A member of the University System of Ohio, it was founded in 1870. It is one ...

. In 1959 he moved to the

California Institute of Technology The California Institute of Technology (branded as Caltech) is a private research university in Pasadena, California, United States. The university is responsible for many modern scientific advancements and is among a small group of institutes ...

where, in 1973, he was named the first IBM Professor at Caltech, the first named chair in mathematics. After retiring from Caltech in 1981, he accepted a post at

Emory University Emory University is a private university, private research university in Atlanta, Georgia, United States. It was founded in 1836 as Emory College by the Methodist Episcopal Church and named in honor of Methodist bishop John Emory. Its main campu ...

in 1985. Hall died in 1990 in

London London is the Capital city, capital and List of urban areas in the United Kingdom, largest city of both England and the United Kingdom, with a population of in . London metropolitan area, Its wider metropolitan area is the largest in Wester ...

on his way to a conference to mark his 80th birthday.

Contributions

He wrote a number of papers of fundamental importance in group theory, including his solution of

Burnside's problem The Burnside problem asks whether a finitely generated group in which every element has finite order must necessarily be a finite group. It was posed by William Burnside in 1902, making it one of the oldest questions in group theory, and was inf ...

for groups of exponent 6, showing that a finitely generated group in which the order of every element divides 6 must be finite. His work in combinatorics includes an important paper of 1943 on

projective plane In mathematics, a projective plane is a geometric structure that extends the concept of a plane (geometry), plane. In the ordinary Euclidean plane, two lines typically intersect at a single point, but there are some pairs of lines (namely, paral ...

s, which for many years was one of the most cited mathematics research papers. In this paper he constructed a family of

non-Desarguesian plane In mathematics, a non-Desarguesian plane is a projective plane that does not satisfy Desargues' theorem (named after Girard Desargues), or in other words a plane that is not a Desarguesian plane. The theorem of Desargues is true in all projective ...

s which are known today as

Hall plane In mathematics, a Hall plane is a non-Desarguesian projective plane constructed by Marshall Hall Jr. (1943). There are examples of order ''p''2''n'' for every prime ''p'' and every positive integer ''n'' provided . Algebraic construction via Hal ...

s. He also worked on

block design In combinatorial mathematics, a block design is an incidence structure consisting of a set together with a family of subsets known as ''blocks'', chosen such that number of occurrences of each element satisfies certain conditions making the co ...

s and

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 ...

. His classic book on group theory was well received when it came out and is still useful today. His book ''Combinatorial Theory'' came out in a second edition in 1986, published by

John Wiley & Sons John Wiley & Sons, Inc., commonly known as Wiley (), is an American Multinational corporation, multinational Publishing, publishing company that focuses on academic publishing and instructional materials. The company was founded in 1807 and pr ...

. He proposed Hall's conjecture on the differences between perfect squares and perfect cubes, which remains an open problem as of 2015. Hall's work on continued fractions showed that the

Lagrange spectrum In mathematics, the Markov spectrum, devised by Andrey Markov, is a complicated set of real numbers arising in Markov Diophantine equations and also in the theory of Diophantine approximation. Quadratic form characterization Consider a quadratic ...

includes all numbers greater than 6. This interval is known as Hall's Ray. The lower limit of Hall's ray was established by Freiman in 1975.

Publications

* 1943: "Projective Planes",

Transactions of the American Mathematical Society The ''Transactions of the American Mathematical Society'' is a monthly peer-reviewed scientific journal of pure and applied mathematics published by the American Mathematical Society. It was established in 1900. As a requirement, all articles must ...

54(2): 229–77 * 1959: ''The Theory of Groups'', Macmillan **

Wilhelm Magnus Hans Heinrich Wilhelm Magnus, known as Wilhelm Magnus (5 February 1907 in Berlin, Germany – 15 October 1990 in New Rochelle, New York), was a German-American mathematician. He made important contributions in combinatorial group theory, Lie algeb ...

(1960
Review: Marshall Hall, Jr. ''Theory of Groups''

Bulletin of the American Mathematical Society The ''Bulletin of the American Mathematical Society'' is a quarterly mathematical journal published by the American Mathematical Society. Scope It publishes surveys on contemporary research topics, written at a level accessible to non-experts. ...

66(3): 144–6. ** * 1964: (with James K. Senior) ''The Groups of Order 2ⁿ n ≤ 6)'', Macmillan ** Preface: "An exhaustive catalog of the 340 groups of order dividing 64 with detailed tables of defining relations, constants, and

lattice Lattice may refer to: Arts and design * Latticework, an ornamental criss-crossed framework, an arrangement of crossing laths or other thin strips of material * Lattice (music), an organized grid model of pitch ratios * Lattice (pastry), an or ...

presentations of each group in the notation the text defines. "Of enduring value to those interested in

finite groups In abstract algebra, a finite group is a group (mathematics), group whose underlying set is finite set, finite. Finite groups often arise when considering symmetry of Symmetry in mathematics, mathematical or Symmetry (physics), physical objects, ...

". * 1967: ''Combinatorial Theory'',

Blaisdell Blaisdell is a surname. Notable people with the surname include: *Alfred Blaisdell, politician from North Dakota *Daniel Blaisdell, politician from New Hampshire *Frances Blaisdell, musician from New Jersey *Frank Ellsworth Blaisdell (1862–1946), ...

Notes

References

* *

External links

* {{DEFAULTSORT:Hall, Marshall Jr 1910 births 1990 deaths 20th-century American mathematicians Mathematicians from Missouri Scientists from St. Louis American algebraists Group theorists Combinatorialists Yale Graduate School of Arts and Sciences alumni Yale College alumni Emory University faculty California Institute of Technology faculty Ohio State University faculty