# Summer Lectures in Geometry

### Topological Automorphic Forms

#### Topological Automorphic Forms II: examples, problems, and applications

I will survey some known computations of Topological Automorphic Forms. K-theory and TMF will be shown to be special cases to TAF. Certain TAF spectra have been identified with $\mathrm{BP}⟨2⟩$ by Hill and Lawson, showing these spectra admit ${E}_{\mathrm{oo}}$ ring structures. $K\left(n\right)$-local TAF gives instances of the higher real K-theories ${\mathrm{EO}}_{n}$, one of which shows up in the solution of the Kervaire invariant one problem. Associated to the TAF spectra are certain approximations of the $K\left(n\right)$-local sphere, which are expected to see "Greek letter elements" in the same manner that TMF sees the divided beta family. Finally, I will discuss some partial results and questions concerning an automorphic forms valued genus which is supposed to generalize the Witten genus.

#### References

Doug Ravenel's web page for a seminar on topological automorphic forms contains a comprehensive list of references.