An algebraic variety is a geometric object defined by polynomial equations, such as a parabola or a circle, and has been studied for a long time. Among them, Calabi-Yau manifolds are an important class in the classification theory of algebraic varieties, and also play important roles in string theory in physics. In particular, an interesting conjecture called mirror symmetry is proposed by string theory. It predicts a curious relationship between two different Calabi-Yau manifolds, and is now formulated in terms of abstract notion of derived categories. On the other hand, curve counting theory on Calabi-Yau 3-folds is an interesting research subject both in mathematics and physics. There exist several invariants such as Gromov-Witten invariants, Donaldson-Thomas invariants and Gopakumar-Vafa invariants, and all of them are expected to be equivalent.

In my research, I used the notion of stability conditions on derived categories to give applications to the study of Donaldson-Thomas invariants and Gopakumar-Vafa invariants. It is fascinating to see the interplay between explicit mathematical objects such as invariants and abstract ones such as categories in this research subject. In recent years, I am also interested in categorifications of Donaldson-Thomas invariants.