In progress:

• F_p-synthetic bo-resolutions, in progress.

Publications:

• Periodic phenomena in equivariant stable homotopy theory (with Jack Carlisle). Submitted, last updated, 8/20/24.
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• A deformation of Borel equivariant homotopy (with Gabriel Angelini-Knoll, Eva Belmont, and Hana Jia Kong). Submitted, last updated 9/27/23.
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• The structure of the v_2-local algebraic tmf resolution (with Prasit Bhattacharya and Dominic Culver). Submitted, last updated 1/26/23.
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• The 2-primary Hurewicz image of tmf (with Mark Mahowald and J.D. Quigley). Geometry and Topology 27 (2023) 2763-2831.
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• The telescope conjecture at height 2 and the tmf resolution (with Agnes Beaudry, Prasit Bhattacharya, Dominic Culver, and Zhouli Xu). J. Topology 14 (2021) 1243-1320.
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• C_2-equivariant stable homotopy from real motivic stable homotopy (with Jay Shah). Annals of K-theory 5 (2020) 411-464.
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• The Bousfield-Kuhn functor and topological Andre-Quillen cohomology
  (with Charles Rezk). Inventiones 220 (2020) 949-1022.
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• Detecting exotic spheres in low dimensions using coker J
  (with Mike Hill, Mike Hopkins, and Mark Mahowald). J. London Math. Soc.101 (2020) 1173-1218.
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• On the E2-term of the bo-Adams spectral sequence
  (with Agnes Beaudry, Prasit Bhattacharya, Dominic Culver, and Zhouli Xu). J. Topology 13 (2020) 356-415.
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• Topological modular and automorphic forms
  Handbook of Homotopy Theory, edited by H.~Miller, Chapman and Hall/CRC Handbooks in Mathematics Series (2019).
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• On the ring of tmf cooperations at the prime 2
  (with Kyle Ormsby, Nat Stapleton, and Vesna Stojanoska), Journal of Topology 12 (2019) 577-657.
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• Spectral algebra models of unstable v_n-periodic homotopy theory
  (with Charles Rezk). Springer Proc. in Math. and Stat. 309 (2019).
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• A C_2-equivariant analog of Mahowald's Thom spectrum theorem
  (with Dylan Wilson). Proceedings of the AMS 146 (2018), 5003-5012.
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• On the homotopy of Q(3) and Q(5) at the prime 2
  (with Kyle Ormsby) Alg. Geom. Topol. 16-5 (2016), 2459-2534.
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• The construction of tmf
  Topological Modular Forms, AMS Mathematical Surveys and Monographs 201 (2014) 131-188.
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• The Goodwillie tower and the EHP sequence
  Memoirs of the AMS. 1026 (2012) i-xi, 1-90
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• The homotopy groups of the E(2)-local sphere at p > 3, revisited
  Adv. Math. 230 (2012) 458-492.
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• The Goodwillie tower for S^1 and Kuhn's theorem
  Alg. Geom. Topol. 11 (2011), 2453-2475
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• Higher real K-theories and topological automorphic forms
  (with Mike Hopkins) J. Topology 4 (2011), 39-72.
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• Topological automorphic forms on U(1,1)
  (with Tyler Lawson) Math. Zeit. 267 (2011), 497-522.
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• Topological automorphic forms
  (with Tyler Lawson) Mem. Amer. Math. Soc. 958 (2010), i-xxiii, 1-132.
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• The homotopy fixed point spectra of profinite Galois extensions
  (with Daniel G. Davis) Trans. Amer. Math. Soc. 362 (2010) 4983-5042.
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• beta-family congruences and the f-invariant
  (with Gerd Laures) Geom. Topol. Monographs 16 (2009) 9-29.
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• Congruences between modular forms given by the divided beta family in homotopy theory
  Geometry and Topology 13 (2009) 319-357.
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• 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.
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• Buildings, elliptic curves, and the K(2)-local sphere
  American Journal of Mathematics 129 (2007) 1513-1563.
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• Some root invariants at the prime 2
  Geometry and Topology Monographs 10 (2007), 1-40.
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• Isogenies of elliptic curves and the Morava stabilizer group
  (With Tyler Lawson) Journal of Pure and Applied Algebra 207 (2006) 37-49.
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• A modular description of the K(2)-local sphere at the prime 3
  Topology 45 (2006), 343-402.
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• Root invariants in the Adams spectral sequence
  Trans. Amer. Math. Soc. 358 (2006), 4279-4341
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• 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.
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• Addendum to "A new proof of the Bott periodicity theorem"
  Topology and its Applications 143 (2004), 281-290.
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• A new proof of the Bott periodicity theorem
  Topology and its Applications 119 (2002), 167-183.
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