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 applications ranging from logic to statistical physics and from evolutionary biology to computer science.
In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group defined over a finite set of symbols consists of the permutations that can be performed on the symbols. Since there are such permutation operations, the order of the symmetric group is .
Béla Bollobás FRS is a Hungarian-born British mathematician who has worked in various areas of mathematics, including functional analysis, combinatorics, graph theory, and percolation. He was strongly influenced by Paul Erdős from the age of 14.
Endre Szemerédi is a Hungarian-American mathematician and computer scientist, working in the field of combinatorics and theoretical computer science. He has been the State of New Jersey Professor of computer science at Rutgers University since 1986. He also holds a professor emeritus status at the Alfréd Rényi Institute of Mathematics of the Hungarian Academy of Sciences.
John Griggs Thompson is an American mathematician at the University of Florida noted for his work in the field of finite groups. He was awarded the Fields Medal in 1970, the Wolf Prize in 1992, and the Abel Prize in 2008.
Sir Vaughan Frederick Randal Jones was a New Zealand mathematician known for his work on von Neumann algebras and knot polynomials. He was awarded a Fields Medal in 1990.
Olga Taussky-Todd was an Austrian and later Czech-American mathematician. She published more than 300 research papers on algebraic number theory, integral matrices, and matrices in algebra and analysis.
Peter Jephson Cameron FRSE is an Australian mathematician who works in group theory, combinatorics, coding theory, and model theory. He is currently half-time Professor of Mathematics at the University of St Andrews, and Emeritus Professor at Queen Mary University of London.
Alan Jerome Hoffman was an American mathematician and IBM Fellow emeritus, T. J. Watson Research Center, IBM, in Yorktown Heights, New York. He was the founding editor of the journal Linear Algebra and its Applications, and held several patents. He contributed to combinatorial optimization and the eigenvalue theory of graphs. Hoffman and Robert Singleton constructed the Hoffman–Singleton graph, which is the unique Moore graph of degree 7 and diameter 2.
Vera Turán Sós was a Hungarian mathematician who specialized in number theory and combinatorics. She was a student and close collaborator of both Paul Erdős and Alfréd Rényi. She also collaborated frequently with her husband Pál Turán, an analyst, number theorist, and combinatorist. Until 1987, she worked at the Department of Analysis at the Eötvös Loránd University, Budapest. Afterwards, she was employed by the Alfréd Rényi Institute of Mathematics. She was elected a corresponding member (1985) and member (1990) of the Hungarian Academy of Sciences. In 1997, Sós was awarded the Széchenyi Prize.
Jan Mycielski is a Polish-American mathematician, logician and philosopher, a professor emeritus of mathematics at the University of Colorado at Boulder. He is known for contributions to graph theory, combinatorics, set theory, topology and the philosophy of mathematics.
In mathematical finite group theory, a rank 3 permutation group acts transitively on a set such that the stabilizer of a point has 3 orbits. The study of these groups was started by Higman. Several of the sporadic simple groups were discovered as rank 3 permutation groups.
In mathematics, the O'Nan–Scott theorem is one of the most influential theorems of permutation group theory; the classification of finite simple groups is what makes it so useful. Originally the theorem was about maximal subgroups of the symmetric group. It appeared as an appendix to a paper by Leonard Scott written for The Santa Cruz Conference on Finite Groups in 1979, with a footnote that Michael O'Nan had independently proved the same result. Michael Aschbacher and Scott later gave a corrected version of the statement of the theorem.
Anne Penfold Street (1932–2016) was one of Australia's leading mathematicians, specialising in combinatorics. She was the third woman to become a mathematics professor in Australia, following Hanna Neumann and Cheryl Praeger. She was the author of several textbooks, and her work on sum-free sets became a standard reference for its subject matter. She helped found several important organizations in combinatorics, developed a researcher network, and supported young students with interest in mathematics.
Catherine Greenhill is an Australian mathematician known for her research on random graphs, combinatorial enumeration and Markov chains. She is a professor of mathematics in the School of Mathematics and Statistics at the University of New South Wales, and an editor-in-chief of the Electronic Journal of Combinatorics.
In mathematics, the Sims conjecture is a result in group theory, originally proposed by Charles Sims. He conjectured that if is a primitive permutation group on a finite set and denotes the stabilizer of the point in , then there exists an integer-valued function such that for the length of any orbit of in the set .
Colva Mary Roney-Dougal is a British mathematician specializing in group theory and computational algebra. She is Professor of Pure Mathematics at the University of St Andrews, and the Director of the Centre for Interdisciplinary Research in Computational Algebra at St Andrews. She is also known for her popularization of mathematics on BBC radio shows, including appearances on In Our Time about the mathematics of Emmy Noether and Pierre-Simon Laplace and on The Infinite Monkey Cage about the nature of infinity and numbers in the real world.
Jan Saxl was a Czech-British mathematician, and a professor at the University of Cambridge. He was known for his work in finite group theory, particularly on consequences of the classification of finite simple groups.
Martin Liebeck is a professor of Pure Mathematics at Imperial College London whose research interests include group theory and algebraic combinatorics.
Sheila Oates Williams is a British and Australian mathematician specializing in abstract algebra. She is the namesake of the Oates–Powell theorem in group theory, and a winner of the B. H. Neumann Award.
