Mathematics Colloquium University of Notre Dame Real Schubert calculus Frank Sottile University of Wisconsin 22 January 1999 Schubert's calculus was a collection of algorithms for enumerating the linear subspaces of a vector space having given positions with respect to other, fixed subspaces. Its justification was important for the development of modern algebraic geometry, and it has connection with other areas of mathematics including combinatorics and representation theory. In this talk, I will discuss recent work which shows it is possible to have all solutions be real not only for these classical enumerative problems, but also to problems of counting rational curves on Grassmannians.