There are several "duals" in different areas of mathematics that seem to share a number of similarities, and these include

(i) the relation between the Poincaré and Hilbert series in commutative algebra and combinatorics;
(ii) the relation between the homotopy groups of a topological space and its (co)homology groups;
(iii) the relation between the cohomology of a reductive group over C and the cohomology of its classifying space;
(iv) the relation between the ring of differential operators and the de Rham dg-algebra of differential forms; as well as between the modules over such algebras;
(v) the relation between an augmented algebra A and an associated Ext-algebra; as well as between modules over such algebras;

among many others.

At first glance, it seems hopeless to try and unify such phenomena, but surprisingly, it turns out that these can all be viewed as manifestations of a single kind of duality, called Koszul duality. We will give an introduction to Koszul duality in what I consider the simplest and most down-to-earth case: the relation between the representation theory of SL(2) and the geometry of its flag variety (the projective line), and how the interplay between the algebra and the geometry reveal some interesting symmetries that seem to be hidden when working with either aspect alone.