Koszul duality may be viewed as machinery to find smaller replacements for the (co)bar construction of a dg (co)algebra. While this may seem hopelessly abstract, there are beautiful applications to concrete geometric situations that go back to the origins of homological algebra. For instance, the machinery gives a construction of the Serre spectral sequence resulting from a fibration of topological spaces, and a relation between a Lie algebra and the Koszul complex computing its cohomology (this complex is a replacement for the bar construction of the Lie algebra). In the first part of the talk I will give details on all of this, as well as relate it to the well-known Koszul duality of quadratic algebras. This is all expository.
I will then describe my work on generalizing Koszul duality to a relative setting. This was motivated by questions in homological commutative algebra and modular representation theory. I will describe some of the results on infinite free resolutions over commutative rings, and some future directions/conjectures in modular representation theory of algebraic groups and Lie algebras.