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