Minicourse: Introduction to Schubert Calculus

Despite being studied for more than one and a half century, Schubert Calculus is finding a new life nowadays, thanks to new geometric and combinatorial techniques by R. Vakil, A. Knutson, T. Tao and many others. This class is intended to be a gentle introduction in the subject. We will study the cohomology of the Grassmannian, together with its "italian formulae" (Pieri and Giambelli). Then we will discuss quantum cohomology of the Grassmannian, and the kernel-span technique to get the quantum analogues of the italian formulae, due to Buch-Kresch-Tamvakis. If time permits, Knutson and Tao's puzzles, and their geometric interpetation, due to Vakil, will be discussed without proofs.

The course should be accessible to anyone with basic knowledge of Algebraic Topology (such as definition of homology/cohomology, Poincare duality). The main reference for the classical cohomology will be Fulton's "Young tableaux", I will distribute references for the quantum cohomology part.