A Survey of Elliptic Cohomology

On the spectrum bo∧tmf

“…Recently, it was shown (tmf, m tmf , u tmf ) also carries such structure (see [15][16][17]). This induces a unital E ∞ -ring structure on (bo ∧ tmf, m, u).…”

“…Fig. 6 shows how to use (17) to form the E 2 -page of bo ∧ B(2 i+1 ). The arrows represent differentials and the dotted lines subsequent extensions.…”

Local Homotopy Theory

“…The localization method which is presented here is not the most general, and has a handicrafted flavour by comparison to standard references such as [41], and more recent discussions of combinatorial model structures [5,75]. It does, however, have the benefit of being relatively simple to describe and apply in presheaf-theoretic contexts.…”

Affineness and chromatic homotopy theory

“…It is also fruitful to consider derived stacks as representing some type of moduli problem for (possibly nonconnective) structured ring spectra. This point of view was used by Lurie to give a construction of the even periodic enhancement of the moduli stack of elliptic curves in [Lur09b], producing the spectrum of topological modular forms TMF.…”

“…In this case, one has the fundamental: A construction of M ell is detailed in [Beh11], and another is sketched in [Lur09b]. In other words, the Goerss-Hopkins-Miller-Lurie theorem states that given a stable 1-pointed genus one curve C → Spec R, such that the classifying map Spec R → M ell isétale, 4 one can build an E ∞ -ring spectrum from the associated formal group; moreover, one can do this functorially in the elliptic curve.…”