James Joseph Sylvester

James Joseph Sylvester was born in London on September 3, 1814, into a Jewish family. He studied at St. John's College, Cambridge, but was unable to take his degree because he refused to subscribe to the Church of England's Thirty-nine Articles, a requirement at the time. He later earned a BA from Trinity College, Dublin, in 1841. Sylvester worked as an actuary and a lawyer before becoming a professor of mathematics at University College London in 1838. In 1876, he crossed the Atlantic to become the first professor of mathematics at the newly founded Johns Hopkins University in Baltimore, where he established the American Journal of Mathematics. He returned to England in 1883 to take the Savilian Chair of Geometry at Oxford, a position he held until his death in 1897.

Sylvester's most significant work was in invariant theory, which he developed jointly with Arthur Cayley. Their collaboration, beginning in the 1850s, produced a systematic theory of invariants and covariants of binary forms. Sylvester also made important contributions to matrix theory, introducing the term "matrix" in 1850 and developing the concept of the nullity of a matrix. In number theory, he studied partitions and proved several results on the distribution of primes. He also worked on the theory of determinants, the theory of forms, and the theory of algebraic equations. His work on the theory of invariants influenced later mathematicians such as David Hilbert, who solved the finite basis problem that Sylvester had posed.

Sylvester's influence extends beyond his own research. He was a mentor to many young mathematicians, including George Bruce Halsted and Fabian Franklin, and he played a key role in establishing mathematics as a research discipline in the United States through his work at Johns Hopkins. His creation of the American Journal of Mathematics provided a vital outlet for mathematical research in America. Sylvester's terminology has become standard: "matrix," "discriminant," "totient," and "invariant" are all his coinages. His collected works, published in four volumes, remain a valuable resource for historians of mathematics. The Sylvester Medal, awarded by the Royal Society, is named in his honor.

Beyond his major achievements, Sylvester had many lesser-known facets. He was an accomplished poet and wrote verses that were published in the Johns Hopkins University Circulars. He was also interested in the theory of music and wrote a paper on the mathematical theory of musical intervals. Sylvester had a lifelong interest in the history of mathematics and wrote biographical sketches of mathematicians such as Euclid and Descartes. He was known for his eccentric personality and his tendency to work in bursts of intense activity. He also made contributions to the theory of probability, and his work on the "Sylvester-Gallai theorem" (proved later by Tibor Gallai) is a classic result in combinatorial geometry. Sylvester's early career was hampered by religious discrimination, but he overcame these obstacles to become one of the leading mathematicians of his era.

Invariant theory

The study of algebraic expressions that remain unchanged under a group of transformations.

Matrix

A rectangular array of numbers or symbols arranged in rows and columns, used to represent linear transformations.

Totient

A function that counts the positive integers up to a given integer that are relatively prime to it.

Discriminant

A polynomial expression that determines the nature of the roots of a polynomial equation.