Homotopical uniqueness and elliptic cohomology

Using Lurie's work in spectral algebraic geometry, one can conjure up many highly coherent stable operations on the universal periodic elliptic cohomology theory TMF. For many purposes it is more desirable to work with Tmf or tmf, two cousins of TMF formed by some "compactification'' process. In this talk, we will discuss how to extend operations on TMF over these "compactifications". In particular, we quickly see that we require a certain level of homotopical coherence, then describe an obstruction theory which captures these coherences, and show that these obstructions vanish in a sufficient range. Some applications of this obstruction theory will be mentioned too, including the uniqueness of the topological q-expansion map up to 3-homotopy, and possibly others.