Hilbert's fifteenth problem

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Hilbert's fifteenth problem is one of the 23 Hilbert problems set out in a celebrated list compiled in 1900 by David Hilbert. It entails a rigorous foundation of Schubert's enumerative calculus.

Splitting the question, as now it would be understood, into Schubert calculus and enumerative geometry, the former is well-founded on the basis of the topology of Grassmannians, and intersection theory. The latter has status that is less clear, if clarified with respect to the position in 1900.


  • Kleiman, Steven L. (1976), "Problem 15: rigorous foundation of Schubert's enumerative calculus", Mathematical developments arising from Hilbert problems (Proc. Sympos. Pure Math., Northern Illinois Univ., De Kalb, Ill., 1974), Proc. Sympos. Pure Math., XXVIII, Providence, R. I.: American Mathematical Society, pp. 445–482, MR 0429938 .
  • Manin, Ju. I. (1969), "On Hilbert's fifteenth problem", Hilbert's problems (Russian), Izdat. “Nauka”, Moscow, pp. 175–181, MR 0254047 .
  • Pragacz, Piotr (1997), "The status of Hilbert's Fifteenth Problem in 1993", Hilbert's Problems (Polish) (Międzyzdroje, 1993), Warsaw: Polsk. Akad. Nauk, pp. 175–184, MR 1632447 .