The computation of multidimensional persistent homology is one of the major open problems in topological data analysis. 

One can define r-dimensional persistent homology to be a functor from the poset category N^r, where N is the poset of natural numbers, to the category of modules over a commutative ring with identity. While 1-dimensional persistent homology is theoretically well-understood and has been successfully applied to many real-world problems, the theory of r-dimensional persistent homology is much harder, as it amounts to understanding representations of quivers of wild type. 

In this talk I will introduce persistent homology, give some motivation for how it is related to the study of data, and present recent results related to the classification of multidimensional persistent homology.

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.