An algebraic group is an algebraic variety with a group structure such that the natural multiplication map and the inverse map are morphisms of algebraic varieties. The theory of algebraic groups plays a central role in several branches of mathematics, including Schubert geometry, geometric representation theory, and the theory of principal bundles over a projective variety.

Schubert varieties / Flag varieties are central objects of algebraic geometry and representation theory and algebraic combinatorics.  There is a beautiful connection with the Borel–Bott–Weil theorem on representation theory of algebraic groups via coherent sheaves and their cohomology groups.  Schubert varieties have singularities in general. These varieties form one of the most important and best studied classes of singular algebraic varieties.  There is a resolution of singularities of a Schubert variety, known as  Bott–Samelson resolution. 

 The classes of Schubert varieties, or Schubert cycles form a basis of the singular (co)homology ring of the Grassmannian, and more generally, for general flag varieties. The study of the intersection theory on the Grassmannian was initiated by Hermann Schubert and continued by Zeuthen in the 19th century under the heading of enumerative geometry.