My mathematical interests lie in representation theory and homological algebra. At present I am thinking about quiver Hecke algebras. This family of algebras were defined in 2008 by Khovanov and Lauda, and independently by Rouquier, as such they are often called KLR algebras. These algebras are connected, via categorification and perverse sheaves, to canonical basis for quantum groups. They have also proved significant in the representation theory of the symmetric group since Brundan and Kleshchev have shown that their cyclotomic quotients are isomorphic to the cyclotomic Hecke algebra.
For more information on some of the topics mentioned above see:
Brundan – Quiver Hecke algebras and categorification – arXiv
Khovanov and Lauda – A diagrammatic approach to categorification of quantum groups I – arXiv
Kleshchev – Representation theory of the symmetric group and related Hecke algebras – arXiv
Kleshchev, Loubert and Miemietz – Affine cellularity of Khovanov-Lauda-Rouquier algebras in type A – arXiv
Rouquier – 2-Kac-Moody algebras – Preprint
Rouquier – Quiver Hecke algebras and 2-Lie algebras – Preprint