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29/06/2011, 14:00 — 15:00 — Room P3.10, Mathematics Building

Mark Behrens, *MIT*

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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 $\mathrm{BP}\u27e82\u27e9$
by Hill and Lawson, showing these spectra admit ${E}_{\mathrm{oo}}$ ring
structures. $K(n)$-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 $K(n)$-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

- Mark Behrens, Notes
on the construction of TMF (2007).
- Mark Behrens and Tyler Lawson,
Topological Automorphic Forms, Memoirs of the AMS 958
(2010).
- Paul Goerss,
Topological modular forms (after Hopkins, Miller and Lurie),
Séminaire Bourbaki, 2009.
- Mike Hopkins,
Topological modular forms, the Witten genus and the Theorem of the
cube, Proceedings of the 1994 ICM.
- Mike Hopkins,
Algebraic Topology and Modular Forms, Proceedings of the 2002
ICM.
- Tyler Lawson, An overview of
abelian varieties in homotopy theory (2008).

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

#### See also

http://www.math.ist.utl.pt/~ggranja/SummerLect11_files/Behrenstalk3.pdf