In progress:
- Exotic spheres detected by topological modular forms
(with Mike Hill, Mike Hopkins, and Mark Mahowald)
We completely determine the image of the Hurewicz
homomorphism for tmf. We draw certain conclusions on which dimensions can
contain exotic spheres. In particular, the only dimensions below 126 which
contain no exotic spheres are 1,2,3,5, 6, 12, 61, and perhaps 4.
Preprints:
- On the ring of tmf cooperations at the prime 2
(with Kyle Ormsby, Nat Stapleton, and Vesna Stojanoska) We relate
three different schemes for understanding tmf^tmf at the prime 2:
bo-Brown-Gitler modules, 2-variable modular forms, and modular forms
with level structure. draft
- On the homotopy of Q(3) and Q(5) at the prime 2
(with Kyle Ormsby) We perform computations in the spectrum Q(5). We
also revisit and extend some computations in Q(3) studied by Mahowald
and Rezk. Detection of divided beta elements is studied for each of
these. Submitted.pdf (revised 11/21/12)
- The Bousfield-Kuhn functor and topological Andre-Quillen
cohomology
(with Charles Rezk). We show that there is a natural transformation
from the Bousfield-Kuhn functor to topological Andre-Quillen
cohomology, which is an equivalence on spheres. This allows us to
compute the E-homology of the v_n-periodic Goodwillie tower of the
identity on odd spheres. draft (revised 9/1/12)
Publications:
- The Goodwillie tower for S^1 and Kuhn's theorem
Alg. Geom. Topol. 11 (2011), 2453-2475
pdf
- The Goodwillie tower and the EHP sequence
Memoirs of the AMS. 1026 (2012) i-xi, 1-90
pdf
- The homotopy groups of the E(2)-local sphere at p > 3,
revisited
Adv. Math. 230 (2012) 458-492.
pdf
- Higher real K-theories and topological automorphic forms
(with Mike Hopkins) J. Topology 4 (2011), 39-72.
pdf
- Topological automorphic forms on U(1,1)
(with Tyler Lawson) Math. Zeit. 267 (2011), 497-522.
pdf
- beta-family congruences and the f-invariant
(with Gerd Laures) Geom. Topol. Monographs 16 (2009) 9-29.
pdf
- The homotopy fixed point spectra of profinite Galois
extensions
(with Daniel G. Davis) Trans. Amer. Math. Soc. 362 (2010) 4983-5042.
pdf
- Congruences between modular forms given by the divided beta family
in homotopy theory
Geometry and Topology 13 (2009) 319-357.
pdf
- Notes on the contruction of tmf
To appear in proceedings of 2007 Talbot Workshop.
(revised: 7/26/13)
pdf
- On the existence of a v_2^32 self map on M(1,4) at the prime
2
(With Mike Hill, Mike Hopkins, and Mark Mahowald)
Homology, Homotopy, and Applications 10 (2008) 45-84.
pdf
- Topological automorphic forms
(with Tyler Lawson) Mem. Amer. Math. Soc. 958 (2010), i-xxiii, 1-132.
pdf
- Buildings, elliptic curves, and the K(2)-local sphere
American Journal of Mathematics 129 (2007) 1513-1563.
pdf
-
Isogenies of elliptic curves and the Morava stabilizer group
(With Tyler Lawson)
Journal of Pure and Applied Algebra 207 (2006) 37-49.
pdf
- Some root invariants at the prime 2
Geometry and Topology
Monographs 10 (2007), 1-40.
pdf
- A modular description of the K(2)-local sphere at the prime
3
Topology 45 (2006), 343-402.
pdf
- Root invariants in the Adams spectral sequence
Trans. Amer. Math. Soc. 358 (2006), 4279-4341
pdf
-
On the existence of the self map v_2^9 on the
Smith-Toda complex V(1) at the prime 3
(With Satya Pemmaraju)
Contemp. Math. 346 (2004), 9-49.
pdf
- A new proof of the Bott periodicity theorem
Topology and its Applications 119 (2002), 167-183.
pdf
- Addendum to "A new proof of the Bott periodicity theorem"
Topology and its Applications 143 (2004), 281-290.
pdf
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