I will discuss how various formalisms of intersection theory (Chow groups, K-theory, cobordism) can be extended to the setting of derived schemes and stacks. This gives a new approach to virtual phenomena such as the virtual fundamental class and virtual Riemann-Roch formulas.
Links:
[1] http://www.mpim-bonn.mpg.de/taxonomy/term/39
[2] http://www.mpim-bonn.mpg.de/node/3444
[3] http://www.mpim-bonn.mpg.de/node/3207