Derived algebraic geometry, representability of derived moduli functors, virtual fundamental classes, derived group schemes and equivariant cohomology theories, elliptic cohomology, topological modular forms.

Some references:

M. Hopkins. "Topological modular forms, the Witten genus, and the theorem of the cube." Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994), 554--565, Birkhäuser, Basel, 1995.

B. Toen and G. Vezzosi, "Algebraic Geometry over model categories (a general approach to derived algebraic geometry)". On the archive: math.AG/0110109.

B. Toen. "Higher and derived stacks: a global overview." On the archive: math.AG/0604504.

J. Lurie. "Higher topos theory." Now available on my homepage at http://www.math.harvard.edu/~lurie/.

J. Lurie. "Derived algebraic geometry." (Being rewritten; old version available on my homepage at http://www.math.harvard.edu/~lurie/).

J. Lurie. "A survey of elliptic cohomology." Available on my homepage at http://www.math.harvard.edu/~lurie/).