In 1930 Brenner earned a B.A. degree with major in chemistry from Harvard University. In graduate study there he was influenced by Hans Brinkmann, Garrett Birkhoff, and Marshall Stone. He was granted the Ph.D. in February 1936. Brenner later described some of his reminiscences of his student days at Harvard and of the state of American mathematics in the 1930s in an article for American Mathematical Monthly. In 1951 Brenner published his findings about matrices with quaternion entries. He developed the idea of a characteristic root of a quaternion matrix and shows that they must exist. He also shows that a quaternion matrix is unitarily-equivalent to a triangular matrix. In 1956 he became a Senior Mathematician at Stanford Research Institute. Brenner, in collaboration with Donald W. Bushaw and S. Evanusa, assisted in the translation and revision of Felix Gantmacher's Applications of the Theory of Matrices. Brenner translated Nikolaj Nikolaevič Krasovskii's book Stability of motion: applications of Lyapunov's second method to differential systems and equations with delay. He also translated and edited the book Problems in differential equations by Aleksei Fedorovich Filippov. Brenner translated Problems in Higher Algebra by D. K. Faddeev and I.S. Sominiski. The exercises in this book covered complex numbers, roots of unity, as well as some linear algebra and abstract algebra. In 1959 Brenner generalized propositions by Alexander Ostrowski and G. B. Price on minors of a diagonally dominant matrix. His work is credited with stimulating a reawakening of interest in the permanent of a matrix. One of the challenges in linear algebra is to find the eigenvalues and eigenvectors of a square matrix of complex numbers. In 1931 S. A. Gershgorin described geometric bounds on the eigenvectors in terms of the matrix elements. This result known as the Gershgorin circle theorem has been used as a basis for extension. In 1964 Brenner reported on Theorems of Gersgorin Type. In 1967 at University of Wisconsin—Madison, working in the Mathematics Research Center, he produced a technical reportNew root-location theorems for partitioned matrices. In 1968 Brenner, following Alston Householder, published "Gersgorin theorems by Householder’s proof". In 1970 he published the survey article "Gersgorin theorems, regularity theorems, and bounds for determinants of partitioned matrices". The article was extended with "Some determinantal identities". In 1971 Brenner extended his geometry of the spectrum of a square complex matrix deeper into abstract algebra with his paper "Regularity theorems and Gersgorin theorems for matrices over rings with valuation". He writes, "Theorems can be extended to non-commutative domains, in particular to quaternion matrices. Secondly, the ring of polynomials has a valuation... a different type of regularity..."
Collaborations
Joel Lee Brenner was a member of the American Mathematical Society from 1936. Beasley relates that he In 1981 Brenner and Roger Lyndon collaborated to polish an idea due to H. W. Kuhn for proving the fundamental theorem of algebra. In the solution by Eric S. Rosenthal to a problem in the American Mathematical Monthly posted by Harry D. Ruderman, Kuhn's work from 1974 was cited. A query was made and prompted an article by Brenner and Lyndon. The version of the fundamental theorem stated was as follows: Brenner ultimately acquired 35 coauthors in his publications.
Alternating group
Given an ordered set Ω with n elements, the even permutations on it determine the alternating group An. In 1960 Brenner proposed the following research problem in group theory: For which An does there exist an element an such that every element g is similar to a commutator of an? Brenner states that the property is true for 4 < n < 10; in symbols it may be expressed The alternating groups are simple groups, and in 1971 Brenner began a series of articles titled "Covering theorems for finite simple groups". He was interested in the cycle type of cyclic permutations, and when An ⊂ C C, where C is a conjugacy class of a certain type. In 1977 he posed the question, "What permutations in An can be expressed as a product of permutations of periods k and l" ?
