Finitistic dimensions of differential graded rings

Finitistic dimensions are important homological numerical invariants associated to a ring. In this talk we explain how to define these invariants over differential graded rings. We then explain how to extend previous results about finitistic dimensions from commutative noetherian rings to commutative noetherian differential graded rings. Finally, we discuss the noncommutative case and its relation to the finitistic dimension conjecture.