Schubert calculus

English

Alternative forms

• Schubert enumerative calculus

Etymology

Named after German mathematician Hermann Schubert (1848–1911), who introduced the theory in the nineteenth century.

Noun

Schubert calculus (uncountable)

1. (mathematics) A branch of algebraic geometry concerned with solving certain types of counting problem in projective geometry; a symbolic calculus used to represent and solve such problems;
   (by generalisation) the enumerative geometry of linear subspaces; the study of analogous questions in generalised cohomology theories.
   • 1986, Christopher I. Byrnes, Anders Lindquist, Frequency Domain and State Space methods for Linear Systems, North-Holland, page 77:

     Hall appears to have been first with this observation, too, for in a lecture he gave at a 1959 Canadian Mathematical Congress conference in Banff on the algebra of symmetric polynomials he noted that the Schubert calculus has combinatorics similar to that of the symmetric polynomials [9].

   • 2014, Thomas Lam, Luc Lapointe, Jennifer Morse, Anne Schilling, Mark Shimozono, Mike Zabrocki, k-Schur Functions and Affine Schubert Calculus, Springer, Fields Institute for Research in the Mathematical Sciences, page 2,

     The rich combinatorial backbone of the theory of Schur functions, including the Robinson–Schensted algorithm, jeu-de-taquin, the plactic monoid (see for example [139]), crystal bases [127], and puzzles [74], now underlies Schubert calculus and in particular produces a direct formula for the Littlewood-Richardson coefficients.

   • 2016, Letterio Gatto, Parham Salehyan, Hasse-Schmidt Derivations on Grassmann Algebras, IMPA, Springer, page 117,

     This point of view was extensively developed by Laksov–Thorup [96–98] and Laksov [93, 94] in the case of equivariant Schubert calculus.

Translations

branch of algebraic geometry; calculus used to solve certain counting problems

See also

• Schubert cell
• Schubert variety

Further reading

• Enumerative geometry on Wikipedia.Wikipedia
• Hilbert's fifteenth problem on Wikipedia.Wikipedia
• Intersection theory on Wikipedia.Wikipedia
• Schubert calculus on Encyclopedia of Mathematics