---
tag: FWGHMADJR4
kind: concept
name: Schubert calculus
msc: [14N15, 14M15, "05E14"]
sources:
  - url: https://en.wikipedia.org/wiki/Schubert_calculus
same_as:
  wikidata: Q7432936
  wikipedia: Schubert calculus
---

## Summary

Schubert calculus is the computation of intersections of [[Schubert varieties|Z014Q8KPDD]] in [[Grassmannians|CYPA882RB9]] and [[flag varieties|NHRWCQQKTH]].
It works in the [[cohomology ring|RDT4785M4D]], or the Chow ring, where the classes of Schubert varieties form a basis.
The products of these classes are given by the Pieri rule and by Littlewood–Richardson coefficients.
It answers classical questions of [[enumerative geometry|ZPZA1FKZZC]], such as how many lines meet four general lines in space.

## Explanations

### L1 human

Hermann Schubert, in the nineteenth century, counted geometric configurations by turning conditions into symbols and multiplying them.
For example, "a line meets a given line" is one symbol $\sigma_1$, and four such conditions give $\sigma_1^4$.
A set of multiplication rules then turns $\sigma_1^4$ into $2$ times the symbol of a single line, so the answer is $2$.
Today these symbols are cohomology classes of Schubert varieties, and the rules are theorems.