Quasihereditary algebras are the 'finite' version of a highest weight category, and they classically occur as blocks of the category O and as Schur algebras.

They also occur as endomorphism algebras associated to modules endowed with special filtrations. The quasihereditary algebras produced in these cases are very often strongly quasihereditary (i.e. their standard modules have projective dimension at most 1).

In this talk I will define (strongly) quasihereditary algebras, give some motivation for their study, and mention some nice strongly quasihereditary algebras found in nature.

Hall algebras are a deformation of the K-group (Grothendieck group) of an abelian category, which encode some information about non-trivial extensions in the category.
A main feature of Hall algebras is that in addition to the product (which deforms the product in the K-group) there is a natural coproduct, which in certain cases makes the Hall algebra a (braided) bi-algebra. This is the content of Green's theorem and supplies the main ingredient in a construction of quantum groups.

We give a noncommutative analogue of Castelnuovo's classic theorem that (-1) lines on a smooth surface can be contracted, and show how this may be used to construct an explicit birational map between a noncommutative P^2 and a noncommutative quadric surface. This has applications to the classification of noncommutative projective surfaces, one of the major open problems in noncommutative algebraic geometry. We will not assume a background in noncommutative ring theory.  The talk is based on joint work with Rogalski and Staffor