Non-ordinary conjectures in Iwasawa Theory

The Iwasawa main conjecture, first developed in the 1960s and later generalised to a modular forms setting, is the prediction that algebraic and analytic constructions of a p-adic L-function agree. This has applications towards the Birch—Swinnerton-Dyer conjecture and many similar problems. This was proved by Kato (’04) and Skinner—Urban (’06) for ordinary modular forms. Progress in the non-ordinary setting is much more recent, requiring tools from p-adic Hodge theory and rigid analytic geometry. I aim to give an overview of this and discuss a new approach in the setting of unitary groups where even more things go wrong.