My current interests are mathematical aspects of the superstring theory, in particular, algebraic geometry related to the mirror symmetry.
More precisely, I am studying homological algebras and moduli problems for categories of "D-branes" that extend derived categories of coherent sheaves on algebraic varieties.
Indeed, I am trying to construct Kyoji Saito's primitive forms and their associated Frobenius structures from triangulated categories defined via matrix factorizations attached to weighted homogeneous polynomials.
For example, I proved that the triangulated category for a polynomial of type ADE is equivalent to the derived category of finitely generated modules over the path algebra of the Dynkin quiver of the same type.
Now, I extend this result to the case when the polynomial corresponds to one of Arnold's 14 exceptional singularities and then showed the "mirror symmetry" between weighted homogeneous singularities and finite dimensional algebras, where a natural interpretation of the "Arnold's strange duality" is given.