Four Entries for Kluwer Encyclopaedia of Mathematics

The Schubert Calculus is a formal calculus of symbols representing geometric conditions used to solve problems in enumerative geometry. This originated in work of Chasles [9] on conics and was systematized and used to great effect by Schubert in his treatise “Kalkül der abzählenden Geometrie” [33]. The justification of Schubert’s enumerative calculus and the verification of the numbers he obtained was the 15th problem of Hilbert. Justifying Schubert’s enumerative calculus was a major theme of 20th century algebraic geometry and Intersection Theory provides a satisfactory modern framework. Enumerative Geometry deals with the second part of Hilbert’s problem. Fulton’s book [19] is a complete reference for Intersection Theory; for historical surveys and a discussion of Enumerative Geometry, see the surveys [24, 25]. The Schubert calculus also refers to mathematics arising from the following class of enumerative geometric problems: Determine the number of linear subspaces of projective space that satisfy incidence conditions imposed by other linear subspaces. For a survey, see [26]. For example, how many lines in projective 3-space meet 4 given lines? These problems are solved by studying both the geometry and the cohomology or Chow rings of Grassmann varieties. This field of Schubert calculus enjoys important connections not only to algebraic geometry and algebraic topology, but also to algebraic combinatorics, representation theory, differential geometry, linear algebraic groups, and symbolic computation, and has found applications in numerical homotopy continuation [22], linear algebra [20] and systems theory [8]. The Grassmannian Gm,n of m-dimensional subspaces (m-planes) in P n over a field k has distinguished Schubert varieties