This course will be an introduction to modern deformation theory using homotopy Lie algebras or L-infinity algebras. These tools are very important for the local study of moduli spaces. But with these techniques, one gets more than just the classical moduli space, one also gets the derived structure. This approach is therefore also relevant for the local study of derived schemes. Moreover, these techniques shed light on the obstruction theories involved in the construction of virtual fundamental classes, which are basic for many numerical invariants (Gromov-Witten or Donaldson-Thomas, for example) defined in terms of moduli spaces. We will explain the general theory and then specialize to the cyclic case, which corresponds to the case of a symmetric obstruction theory. This case is of particular importance for the Donaldson-Thomas theory of Calabi-Yau threefolds.

We hope to make this lecture series accessible to graduate students with a strong background in algebraic geometry.

Literature:
M. Manetti: Lectures on deformations of complex manifolds,
arXiv:math.AG/0507286
M. Manetti: Deformation theory via differential graded Lie algebras,
arXiv:math.AG/0507284