---
tag: WPQPBXNN5F
kind: problem
name: Hilbert's fifteenth problem
msc: [14N15, 14N10]
uses: [ZPZA1FKZZC, FWGHMADJR4, 4HV4WTAX9P]
about: [FWGHMADJR4, ZPZA1FKZZC]
sources:
  - url: https://en.wikipedia.org/wiki/Hilbert%27s_fifteenth_problem
same_as:
  wikidata: Q5761149
  wikipedia: Hilbert's fifteenth problem
history:
  posed_by: "David Hilbert"
  first_stated: "1900"
---

## Problem

Give a rigorous foundation for the enumerative calculus of Hermann Schubert.
Determine exactly the range of validity of his methods, and check the numbers of [[enumerative geometry|ZPZA1FKZZC]] that he and his contemporaries found.

## Remarks

Hilbert posed this problem in his list of 23 problems presented in 1900.
Schubert had computed many enumerative numbers with symbolic rules, but these rules lacked proofs, and some classical answers were later found to need correction.
The modern foundations are [[intersection theory|4HV4WTAX9P]] and the cohomology of Grassmannians and flag varieties, in which [[Schubert calculus|FWGHMADJR4]] becomes a set of theorems.
Much of the problem is regarded as settled in this way, but checking all of Schubert's numbers is a separate task, and the problem is often listed as partially solved.

## Explanations

### L1 human

In the nineteenth century, geometers counted figures, such as lines meeting other lines or conics touching other conics, with clever but unproven rules.
The rules usually gave right answers, but sometimes wrong ones, and nobody knew why.
Hilbert asked for a solid foundation that explains when the rules work.
The answer came through the theory of intersections and the topology of Grassmannians, developed during the twentieth century.