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| Schubert calculus | |
|---|---|
| Name | Schubert calculus |
| Field | Algebraic geometry |
| Introduced | 19th century |
| Founder | Hermann Schubert |
| Keywords | Enumerative geometry, Grassmannian, Schubert variety, cohomology |
Schubert calculus
Schubert calculus is a branch of algebraic geometry and enumerative geometry concerned with intersection numbers of subvarieties in flag manifolds and Grassmannians, originating in 19th‑century enumerative problems. It combines ideas from Hermann Schubert, classical projective geometry, and modern cohomological and representation‑theoretic techniques, with links to David Hilbert, Bernhard Riemann, André Weil, and Alexander Grothendieck.
Schubert calculus studies intersections of special cycles called Schubert cycles in homogeneous spaces like the Grassmannian and complete flag variety, encoding incidence conditions that arise in problems dating to René Descartes, Isaac Newton, and Jules-Henri Poincaré. It formalizes counting solutions to enumerative problems via classes in the cohomology ring, Chow ring, and K-theory of moduli spaces considered by Hermann Grassmann, Elie Cartan, and Élie Cartan. The subject interacts with major figures and objects such as Felix Klein, Sophus Lie, Évariste Galois, Richard Stanley, and the Atiyah–Bott localization theorem.
Hermann Schubert formulated combinatorial rules inspired by projective problems studied by Joseph-Louis Lagrange, Carl Friedrich Gauss, and Jean-Victor Poncelet; later foundational rigor was provided by David Hilbert and Federigo Enriques. The 20th century saw algebraic formalization through work of Oscar Zariski, André Weil, and Alexander Grothendieck, while intersection‑theoretic foundations were advanced by Jean-Pierre Serre, Alexander Grothendieck (via schemes), and Pierre Deligne. In the later 20th century, connections with representation theory and combinatorics were developed by George Lusztig, Bertram Kostant, Robert MacPherson, Kazuhiko Kodaira, William Fulton, and Richard Stanley, and computational tools emerged in software initiated by projects at Massachusetts Institute of Technology and Institut des Hautes Études Scientifiques.
The core objects include the Grassmannian Gr(k,n), the complete flag variety Flag_n, and parabolic homogeneous spaces for GL_n, SL_n, SO_n, and Sp_{2n}. Schubert cells give a cellular decomposition compatible with Bruhat order from Nicolas Bourbaki and Claude Chevalley; the closure of cells defines Schubert varieties tied to elements of the Weyl group studied by Elie Cartan and Hermann Weyl. Cohomology classes of these closures form a basis of the cohomology ring H^*(Flag) and have structure constants—the Littlewood–Richardson coefficients—related to representation theory of GL_n and symmetric functions of Isaac Newton and Arthur Cayley. Intersection theory uses tools from Jean-Pierre Serre and William Fulton while combinatorial models employ tableaux by Alfred Young, Donald Knuth, and Alexander Kirillov.
Schubert varieties are indexed by combinatorial data such as partitions, Young diagrams, and permutations from Augustin-Louis Cauchy and Siméon Denis Poisson frameworks; their classes in cohomology or Chow rings are central to enumerative counts studied by Hermann Schubert and later formalized by William Fulton and Robert MacPherson. Singularities of Schubert varieties link to the work of Mark Haiman, Erik Sommers, and Vladimir Drinfeld; rational smoothness and Kazhdan–Lusztig polynomials relate to David Kazhdan and George Lusztig. Equivariant cohomology classes under a torus action connect with localization techniques of Mikhail Atiyah and Raoul Bott.
Intersection numbers of Schubert classes solve classical problems like the 19th‑century questions addressed by Hermann Schubert and examples from Béla Kerékjártó and Giovanni Battista Ratti; modern formulations employ tools of Fulton–MacPherson compactification and Gromov–Witten invariants developed by Maxim Kontsevich and Yongbin Ruan. Quantum Schubert calculus, influenced by Kontsevich and Edward Witten, extends counts to stable maps and links to symplectic geometry through Floer theory by Andreas Floer and mirror symmetry conjectures by Cumrun Vafa. Applications include enumerative counts in projective space exploited in problems treated by Hodge and Weyl.
Structure constants of Schubert calculus coincide with multiplicities in tensor products and branching rules studied by Littlewood and Richard Littlewood and formalized in the Littlewood–Richardson rule connected to Alfred Young tableaux and Srinivasa Ramanujan‑era symmetric function theory. Schubert polynomials of Lascoux and Schützenberger relate to characters of GL_n and cohomology of flag varieties; connections to Hecke algebras studied by Iwahori and Hecke tie to Kazhdan–Lusztig theory by Kazhdan and Lusztig. The interplay with Macdonald polynomials, Jack polynomials, and Schur functions situates Schubert calculus within algebraic combinatorics advanced by Richard Stanley and Ian G. Macdonald.
Contemporary research expands to equivariant K‑theory and quantum K‑theory influenced by Andrei Okounkov, geometric representation theory of Geometric Langlands Program proponents like Edward Frenkel, and categorification initiatives by Mikhail Khovanov and Jacob Lurie. Extensions include positroid varieties related to Postnikov and total positivity studied by Lusztig, tropical geometry applications advanced by Grigory Mikhalkin and Diane Maclagan, and computational advances in software linked to Bernd Sturmfels and projects at Institute for Advanced Study. Intersections with string theory bring in figures such as Edward Witten and Cumrun Vafa, while arithmetic aspects connect to conjectures from André Weil and Pierre Deligne.