Is the Schubert Calculus Effective?


Discrete Mathematics Seminar
University of California at Berkeley


Frank Sottile
MSRI & The University of Toronto

The Schubert calculus, a basic tool in enumerative geometry, gives algorithms for solving certain problems of enumerating linear subspaces of projective space, such as: How many planes in projective 5-space meet 9 given planes?1 Its core component is the intersection form, also known as the Littlewood-Richardson rule, for the cohomology (or Chow) ring of a Grassmann manifold in terms of its basis of Schubert classes.

An effective version of the Schubert calculus would, for example, explicitly find the 42 planes meeting 9 given planes in P5. It may also explicitly describe how to deform an intersection of two Schubert varieties into a sum of distinct Schubert varieties, that is, a geometric manifestation of the Littlewood-Richardson rule.

This talk will assume little prior knowledge, and could serve as an introduction to the geometry of Grassmann manifolds. It will describe some results along these lines and some of the geometry involved, including a weak solution to some problems like the 9 planes, and an effective version of Pieri's formula.
1Answer: 42