Hilbert's fifteenth problem
Hilbert's fifteenth problem is one of the 23 Hilbert problems set out in a celebrated list compiled in 1900 by David Hilbert. The problem is to put Schubert's enumerative calculus on a rigorous foundation.
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.
The entirety of the original problem statement is as follows:
The problem consists in this: To establish rigorously and with an exact determination of the limits of their validity those geometrical numbers which Schubert especially has determined on the basis of the so-called principle of special position, or conservation of number, by means of the enumerative calculus developed by him.
Although the algebra of today guarantees, in principle, the possibility of carrying out the processes of elimination, yet for the proof of the theorems of enumerative geometry decidedly more is requisite, namely, the actual carrying out of the process of elimination in the case of equations of special form in such a way that the degree of the final equations and the multiplicity of their solutions may be foreseen.
Schubert calculus is a branch of algebraic geometry introduced in the nineteenth century by Hermann Schubert, in order to solve various counting problems of projective geometry (part of enumerative geometry). It was a precursor of several more modern theories, for example characteristic classes, and in particular its algorithmic aspects are still of current interest.
The objects introduced by Schubert are the Schubert cells, which are locally closed sets in a Grassmannian defined by conditions of incidence of a linear subspace in projective space with a given flag. For details see Schubert variety.
- Katz, Sheldon (2006), Enumerative Geometry and String Theory, Student Mathematical Library, 32, American Mathematical Society
- Hilbert, David, "Mathematische Probleme" Göttinger Nachrichten, (1900), pp. 253-297, and in Archiv der Mathematik und Physik, (3) 1 (1901), 44-63 and 213-237. Published in English translation by Dr. Maby Winton Newson, Bulletin of the American Mathematical Society 8 (1902), 437-479   doi:10.1090/S0002-9904-1902-00923-3 . [A fuller title of the journal Göttinger Nachrichten is Nachrichten von der Königl. Gesellschaft der Wiss. zu Göttingen.]
- 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.