Poset filtrations of triangulated categories in algebra and geometry

For a finite poset $S$ and a triangulated category $\mathcal{D}$, I will define an (admissible, strict) $S$-filtration on $\mathcal{D}$. I will describe how to glue a t-structure via such a filtration. I will give examples of poset filtrations and the glued t-structures for quasi-hereditary algebras and birational morphisms of smooth surfaces. I will also introduce a generalisation of Ringel duality and describe it in the above examples.