T. A. Springer

Tonny Albert Springer was a mathematician at Utrecht University who worked on linear algebraic groups, Hecke algebras, complex reflection groups, who introduced Springer representations and the Springer resolution. Springer began his undergraduate studies in 1945 at Leiden University and remained there for his graduate work in mathematics, earning his PhD in 1951 under Hendrik Kloosterman with thesis Over symplectische Transformaties; as a postdoc Springer spent the academic year 1951/1952 at the University of Nancy and returned to Leiden University, where he was employed until 1955. In 1955 he accepted a lectureship at Utrecht University, where he became professor ordinarius in 1959 and continued in that position until 1991 when he retired as professor emeritus. Springer's visiting professorships included many institutions: the University of Göttingen, the Institute for Advanced Study, IHES, Tata Institute of Fundamental Research, UCLA, the Australian National University, the University of Sydney, the University of Rome Tor Vergata, the University of Basel, the Erwin Schrödinger Institute in Vienna, the University of Paris VI.

In 1964 Springer was elected to the Royal Netherlands Academy of Sciences. In 1962 he was an invited speaker at the International Congress of Mathematicians in Stockholm and in 2006 at Madrid. Springer, Tonny A. Jordan Algebras and Algebraic Groups, Classics in Mathematics, Springer-Verlag, ISBN 3-540-63632-3 Reprint of the 1973 edition. Springer, Tonny A.. Linear algebraic groups, Birkhäuser, ISBN 978-0-8176-4021-7. 1981. Springer, Tonny A. Invariant theory, Lecture Notes in Mathematics, 585, Springer-Verlag Profile Springer's home page. T. A. Springer at the Mathematics Genealogy Project

Michel Demazure

Michel Demazure is a French mathematician. He made contributions in the fields of abstract algebra, algebraic geometry, computer vision, participated in the Nicolas Bourbaki collective, he has been president of the French Mathematical Society and directed two French science museums. In the 1960s, Demazure was a student of Alexandre Grothendieck, together with Grothendieck, he ran and edited the Séminaire de Géométrie Algébrique du Bois Marie on group schemes at the Institut des Hautes Études Scientifiques near Paris from 1962 to 1964. Demazure obtained his doctorate from the Université de Paris in 1965 under Grothendieck's supervision, with a dissertation entitled Schémas en groupes reductifs, he was maître de conférence at Strasbourg University, university professor at Paris-Sud in Orsay and the École Polytechnique in Palaiseau. From 1965 to 1985, he was one of the core members of the Bourbaki group, a group of French mathematicians writing under the collective pseudonym Nicolas Bourbaki. In 1988 Demazure was the president of the Société Mathématique de France.

From 1991 to 1998, he was the director of the Palais de la Découverte in Paris and, from 1998 to 2002, the chairman of the Cité des Sciences et de l'Industrie in La Villette, two major science museums in France. Demazure chairs the regional advisory committee of research for Languedoc-Roussillon. In SGA3, Demazure introduced the definition of a root datum, a generalization of root systems for reductive groups, central to the notion of Langlands duality. A 1970 paper of Demazure on subgroups of the Cremona group has been recognized as the beginning of the study of toric varieties; the Demazure character formula and Demazure modules and Demazure conjecture are named after Demazure, who wrote about them in 1974. Demazure modules are submodules of a finite-dimensional representation of a semisimple Lie algebra, the Demazure character formula is an extension of the Weyl character formula to these modules. Demazure's work in this area was marred by a dependence on a false lemma in an earlier paper.

In his career, Demazure's research emphasis shifted from pure mathematics to more computational problems, involving the application of algebraic geometry to image reconstruction problems in computer vision. The Kruppa–Demazure theorem, stemming from this work, shows that if a scene consisting of five points is viewed from two cameras with unknown positions but known focal lengths in general, there will be ten different scenes that could have generated the same two images. Austrian mathematician Erwin Kruppa had many years earlier narrowed the number of possible scenes to eleven, Demazure provided the first complete solution to the problem. Schémas en groupes. I: Propriétés générales des schémas en groupes. Lecture Notes in Mathematics 151, Berlin: Springer-Verlag, 1970. MR0274458. Schémas en groupes. II: Groupes de type multiplicatif, et structure des schémas en groupes généraux. Lecture Notes in Mathematics 152, Berlin: Springer-Verlag, 1970. MR0274459. Schémas en groupes. III: Structure des schémas en groupes réductifs.

Lecture Notes in Mathematics 153, Berlin: Springer-Verlag, 1970. MR0274460. Groupes algébriques. Tome I: Géométrie algébrique, généralités, groupes commutatifs. Masson, Amsterdam: North Holland, 1970. MR0302656. Translated into English by J. Bell as Introduction to Algebraic Geometry and Algebraic Groups, Volume 39 of North-Holland Mathematics Studies, Elsevier, 1980, MR0563524. Lectures on p-divisible groups. Lecture Notes in Mathematics 302, Berlin: Springer-Verlag, 1972, 1986, ISBN 3-540-06092-8. MR0344261, MR0883960. Bifurcations and catastrophes: Geometry of solutions to nonlinear problems. Universitext, Berlin: Springer-Verlag, 2000. Translated from the French by David Chillingworth. MR1739190. Cours d'Algèbre: Primalité. Divisibilité. Codes. Paris: Cassini, 1997, 2008. MR1466448

Root system

In mathematics, a root system is a configuration of vectors in a Euclidean space satisfying certain geometrical properties. The concept is fundamental in the theory of Lie groups and Lie algebras the classification and representation theory of semisimple Lie algebras. Since Lie groups and Lie algebras have become important in many parts of mathematics during the twentieth century, the special nature of root systems belies the number of areas in which they are applied. Further, the classification scheme for root systems, by Dynkin diagrams, occurs in parts of mathematics with no overt connection to Lie theory. Root systems are important for their own sake, as in spectral graph theory; as a first example, consider the six vectors in 2-dimensional Euclidean space, R2, as shown in the image at the right. These vectors span the whole space. If you consider the line perpendicular to any root, say β the reflection of R2 in that line sends any other root, say α, to another root. Moreover, the root to which it is sent equals α + nβ.

These six vectors satisfy the following definition, therefore they form a root system. Let E be a finite-dimensional Euclidean vector space, with the standard Euclidean inner product denoted by. A root system Φ in E is a finite set of non-zero vectors that satisfy the following conditions: An equivalent way of writing conditions 3 and 4 is as follows: Some authors only include conditions 1–3 in the definition of a root system. In this context, a root system that satisfies the integrality condition is known as a crystallographic root system. Other authors omit condition 2. In this article, all root systems are assumed to be crystallographic. In view of property 3, the integrality condition is equivalent to stating that β and its reflection σα differ by an integer multiple of α. Note that the operator ⟨ ⋅, ⋅ ⟩: Φ × Φ → Z defined by property 4 is not an inner product, it is not symmetric and is linear only in the first argument. The rank of a root system Φ is the dimension of E. Two root systems may be combined by regarding the Euclidean spaces they span as mutually orthogonal subspaces of a common Euclidean space.

A root system which does not arise from such a combination, such as the systems A2, B2, G2 pictured to the right, is said to be irreducible. Two root systems and are called isomorphic if there is an invertible linear transformation E1 → E2 which sends Φ1 to Φ2 such that for each pair of roots, the number ⟨ x, y ⟩ is preserved; the root lattice of a root system Φ is the Z-submodule of E generated by Φ. It is a lattice in E; the group of isometries of E generated by reflections through hyperplanes associated to the roots of Φ is called the Weyl group of Φ. As it acts faithfully on the finite set Φ, the Weyl group is always finite. In the A 2 case, the "hyperplanes" are the lines perpendicular to the roots, indicated by dashed lines in the figure; the Weyl group is the symmetry group of an equilateral triangle. In this case, the Weyl group is not the full symmetry group of the root system. There is only one root system of rank 1, consisting of two nonzero vectors; this root system is called A 1. In rank 2 there are four possibilities, corresponding to σ α = β + n α, where n = 0, 1, 2, 3.

Note that a root system is not determined by the lattice that it generates: A 1 × A 1 and B 2 both generate a square lattice while A 2 and G 2 generate a hexagonal lattice, only two of the five possible types of lattices in two dimensions. Whenever Φ is a root system in E, S is a subspace of E spanned by Ψ = Φ ∩ S Ψ is a root system in S. Thus, the exhaustive list of four root systems of rank 2 shows the geometric possibilities for any two roots chosen from a root system of arbitrary rank. In particular, two such roots must meet at an angle of 0, 30, 45, 60, 90, 120, 135, 150, or 180 degrees. If g is a complex semisimple Lie algebra and h is a Cartan subalgebra, we can construct a root system as follows. We say that α ∈ h ∗ is a root of g relative to h if α ≠ 0 {\displaystyle \al