[1][2] In the course of securing the foundations of intersection theory, Van der Waerden and André Weil[3][4] related the problem to the determination of the cohomology ring H*(G/P) of a flag manifold G/P, where G is a Lie group and P a parabolic subgroup of G. The additive structure of the ring H*(G/P) is given by the basis theorem of Schubert calculus[5][6][7] due to Ehresmann, Chevalley, and Bernstein-Gel'fand-Gel'fand, stating that the classical Schubert classes on G/P form a free basis of the cohomology ring H*(G/P).
[10] While enumerative geometry made no connection with physics during the first century of its development, it has since emerged as a central element of string theory.
[11] 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.
In particular, a) Schubert's characteristic problem has been solved by Haibao Duan and Xuezhi Zhao;[12] b) Special presentations of the Chow rings of flag manifolds have been worked out by Borel, Marlin, Billey-Haiman and Duan-Zhao, et al.;[12] c) Major enumerative examples of Schubert[8] have been verified by Aluffi, Harris, Kleiman, Xambó, et al.[13][12]