The tangent complex for the moduli stack of formal groups

The main purpose of this talk is to explain and explore the object in the title and to outline why it might be useful. In particular, I hope to organize the following questions: when can (chromatic-type) homology theories be realized by structured ring spectra and, if they can, what can you say about maps between them? Both problems can be formulated in terms of an Andre'-Quillen cohomology calculation, which is where the tangent complex comes in. With any luck, I will get to the point where I can talk about some of the applications to elliptic spectra arising from the moduli stack of elliptic curves. I am, of course, following closely in the footsteps of others, in particular of Mike Hopkins and Haynes Miller.