Hilbert's fifteenth problem is one of the 23 Hilbert problems set out in a celebrated list compiled in 1900 by David Hilbert

David Hilbert

David Hilbert was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of...

. It entails a rigorous foundation of Schubert's enumerative calculus.

Splitting the question, as now it would be understood, into Schubert calculus

Schubert calculus

In mathematics, 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...

 and enumerative geometry

Enumerative geometry

In mathematics, enumerative geometry is the branch of algebraic geometry concerned with counting numbers of solutions to geometric questions, mainly by means of intersection theory.-History:...

, the former is well-founded on the basis of the topology of Grassmannian

Grassmannian

In mathematics, a Grassmannian is a space which parameterizes all linear subspaces of a vector space V of a given dimension. For example, the Grassmannian Gr is the space of lines through the origin in V, so it is the same as the projective space P. The Grassmanians are compact, topological...

s, and intersection theory

Intersection theory

In mathematics, intersection theory is a branch of algebraic geometry, where subvarieties are intersected on an algebraic variety, and of algebraic topology, where intersections are computed within the cohomology ring. The theory for varieties is older, with roots in Bézout's theorem on curves and...

. The latter has status that is less clear, if clarified with respect to the position in 1900.