I'm trying to gain a practical understanding of Koszul duality in different areas of mathematics. Searching the internet, there's lots of homological characterisations and explanations one finds, but I would like concrete examples of where Koszul duality is either a nice way of looking at a result, or where it provides us with new results. If someone were to be able to explain for example:
Why is the Ext-algebra (Yoneda-algebra) (and the way it characterizes Koszulity) useful?
Is there a more elementary way of explaining some of the results and applications in the Beilinson-Ginzburg-Soergel paper Koszul duality patterns in representation theory?
I find it hard to grasp Manin's motivations for viewing quadratic algebras as noncommutative spaces, cfr. "Quantum groups and noncommutative geometry". Can anyone comment on this?
What are the applications of Koszul algebras? Example: I know that by a result of Fröberg every projective variety has a homogeneous coordinate ring that is Koszul, but what does this do for us really?
Those are some of the examples that spring to mind, but feel free to add other phenomena. Also, let me know if you think there's a way to improve this question.