Frank Grosshans is an American mathematician who works in invariant theory, where he is known for the discovery of Grosshans subgroups and Grosshans graded coefficients. [1] He is a professor of mathematics at West Chester University, Pennsylvania. [2] Grosshans has been an invited speaker at meetings of the Mathematical Association of America. [3]
He received his B.S. from the University of Illinois and his Ph.D. in mathematics from the University of Chicago. [2] He taught at University of Pennsylvania and Johns Hopkins University before joining the West Chester University.
Noncommutative geometry (NCG) is a branch of mathematics concerned with a geometric approach to noncommutative algebras, and with the construction of spaces that are locally presented by noncommutative algebras of functions. A noncommutative algebra is an associative algebra in which the multiplication is not commutative, that is, for which does not always equal ; or more generally an algebraic structure in which one of the principal binary operations is not commutative; one also allows additional structures, e.g. topology or norm, to be possibly carried by the noncommutative algebra of functions.
In the mathematical field of representation theory, a Lie algebra representation or representation of a Lie algebra is a way of writing a Lie algebra as a set of matrices in such a way that the Lie bracket is given by the commutator. In the language of physics, one looks for a vector space together with a collection of operators on satisfying some fixed set of commutation relations, such as the relations satisfied by the angular momentum operators.
Invariant theory is a branch of abstract algebra dealing with actions of groups on algebraic varieties, such as vector spaces, from the point of view of their effect on functions. Classically, the theory dealt with the question of explicit description of polynomial functions that do not change, or are invariant, under the transformations from a given linear group. For example, if we consider the action of the special linear group SLn on the space of n by n matrices by left multiplication, then the determinant is an invariant of this action because the determinant of A X equals the determinant of X, when A is in SLn.
Geometric group theory is an area in mathematics devoted to the study of finitely generated groups via exploring the connections between algebraic properties of such groups and topological and geometric properties of spaces on which these groups act.
Jean Alexandre Eugène Dieudonné was a French mathematician, notable for research in abstract algebra, algebraic geometry, and functional analysis, for close involvement with the Nicolas Bourbaki pseudonymous group and the Éléments de géométrie algébrique project of Alexander Grothendieck, and as a historian of mathematics, particularly in the fields of functional analysis and algebraic topology. His work on the classical groups, and on formal groups, introducing what now are called Dieudonné modules, had a major effect on those fields.
Grigory Aleksandrovich Margulis is a Russian-American mathematician known for his work on lattices in Lie groups, and the introduction of methods from ergodic theory into diophantine approximation. He was awarded a Fields Medal in 1978, a Wolf Prize in Mathematics in 2005, and an Abel Prize in 2020, becoming the fifth mathematician to receive the three prizes. In 1991, he joined the faculty of Yale University, where he is currently the Erastus L. De Forest Professor of Mathematics.
Eugene Borisovich Dynkin was a Soviet and American mathematician. He made contributions to the fields of probability and algebra, especially semisimple Lie groups, Lie algebras, and Markov processes. The Dynkin diagram, the Dynkin system, and Dynkin's lemma are named after him.
Leonard Eugene Dickson was an American mathematician. He was one of the first American researchers in abstract algebra, in particular the theory of finite fields and classical groups, and is also remembered for a three-volume history of number theory, History of the Theory of Numbers.
David Eisenbud is an American mathematician. He is a professor of mathematics at the University of California, Berkeley and former director of the then Mathematical Sciences Research Institute (MSRI), now known as Simons Laufer Mathematical Sciences Institute (SLMath). He served as Director of MSRI from 1997 to 2007, and then again from 2013 to 2022.
Tonny Albert Springer was a mathematician at Utrecht University who worked on linear algebraic groups, Hecke algebras, complex reflection groups, and who introduced Springer representations and the Springer resolution.
In mathematics, Harish-Chandra's regularity theorem, introduced by Harish-Chandra (1963), states that every invariant eigendistribution on a semisimple Lie group, and in particular every character of an irreducible unitary representation on a Hilbert space, is given by a locally integrable function. Harish-Chandra proved a similar theorem for semisimple p-adic groups.
In mathematics, a prehomogeneous vector space (PVS) is a finite-dimensional vector space V together with a subgroup G of the general linear group GL(V) such that G has an open dense orbit in V. Prehomogeneous vector spaces were introduced by Mikio Sato in 1970 and have many applications in geometry, number theory and analysis, as well as representation theory. The irreducible PVS were classified by Sato and Tatsuo Kimura in 1977, up to a transformation known as "castling". They are subdivided into two types, according to whether the semisimple part of G acts prehomogeneously or not. If it doesn't then there is a homogeneous polynomial on V which is invariant under the semisimple part of G.
In mathematics, a zonal spherical function or often just spherical function is a function on a locally compact group G with compact subgroup K (often a maximal compact subgroup) that arises as the matrix coefficient of a K-invariant vector in an irreducible representation of G. The key examples are the matrix coefficients of the spherical principal series, the irreducible representations appearing in the decomposition of the unitary representation of G on L2(G/K). In this case the commutant of G is generated by the algebra of biinvariant functions on G with respect to K acting by right convolution. It is commutative if in addition G/K is a symmetric space, for example when G is a connected semisimple Lie group with finite centre and K is a maximal compact subgroup. The matrix coefficients of the spherical principal series describe precisely the spectrum of the corresponding C* algebra generated by the biinvariant functions of compact support, often called a Hecke algebra. The spectrum of the commutative Banach *-algebra of biinvariant L1 functions is larger; when G is a semisimple Lie group with maximal compact subgroup K, additional characters come from matrix coefficients of the complementary series, obtained by analytic continuation of the spherical principal series.
Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essence, a representation makes an abstract algebraic object more concrete by describing its elements by matrices and their algebraic operations. The theory of matrices and linear operators is well-understood, so representations of more abstract objects in terms of familiar linear algebra objects helps glean properties and sometimes simplify calculations on more abstract theories.
Nolan Russell Wallach is a mathematician known for work in the representation theory of reductive algebraic groups. He is the author of the 2-volume treatise Real Reductive Groups.
James Edward Humphreys was an American mathematician, who worked in algebraic groups, Lie groups, and Lie algebras and applications of these mathematical structures. He is known as the author of several mathematical texts, such as Introduction to Lie Algebras and Representation Theory and Reflection Groups and Coxeter Groups.
In mathematics, semi-simplicity is a widespread concept in disciplines such as linear algebra, abstract algebra, representation theory, category theory, and algebraic geometry. A semi-simple object is one that can be decomposed into a sum of simple objects, and simple objects are those that do not contain non-trivial proper sub-objects. The precise definitions of these words depends on the context.
Howard Garland is an American mathematician, who works on algebraic groups, Lie algebras, and infinite-dimensional algebras.
Dan Burghelea is a Romanian-American mathematician, academic, and researcher. He is an Emeritus Professor of Mathematics at Ohio State University.
Arkady L'vovich Onishchik was a prominent Soviet and Russian mathematician, who worked on Lie groups and their geometrical applications. Onishchik was a student of Eugene Dynkin, under whose guidance he got his PhD at Moscow State University in 1960. In 1962 Onishchik received the Prize of the Moscow Mathematical Society for young mathematicians. In 1970 he got Habilitation. Since 1975 Onishchik was a professor of Yaroslavl State University.