The affine grassmannian of a reductive group $G$ is the quotient $G(K)/G(O)$ where $O$ is the ring of formal power series and $K$ is its field of fractions, the ring of formal Laurent series.
The affine grassmannian is farmed by representation theorists for its rich geometry: it may be realized as an inreasing union of finite-dimensional varieties whose intersection cohomologies are irreducible representations of $G$.
In this talk you'll learn how to grow and use your very own heirloom variety, the affine grassmannian of $GL_n$.
J.E. Anderson and M. Kogan, The algebra of Mikovic-Vilonen Cycles in type A, ArXiv Mathematics e-print (May 2005), available at math/0505100