I have been studying properties of algebraic varieties, especially those related to Geometric Invariant Theory (GIT) and birational geometry.
GIT is a method for constructing quotients of algebraic varieties by algebraic group actions, and birational geometry, very roughly, deals with operations on algebraic varieties which only changes small parts of them. Constructions of GIT quotients require a choice of extra data, so called stability conditions. Different choices of stability conditions yield different quotients, and in good situations they are birationally equivalent.
There are special cases in which the birational geometry of the quotients can be completely described in terms of GIT, and as a class of such quotient varieties the notion of Mori dream spaces was defined. In past researches I proved that images of morphisms from Mori dream spaces are again Mori dream spaces, and the positivity of the canonical line bundle of Mori dream spaces are closely related to the singularity of the variety on which the group acts.
Recently I am working on a slightly different subject, namely the bounded derived categories of coherent sheaves on algebraic varieties and specifically on how many semi-orghotognal decompositions they have. This research also has its motivation in birational geometry.