In progress:

• The structure of the $v_2$-local algebraic tmf resolution (joint with Mark Behrens and Prasit Bhattacharya). Under Review
  We give a complete description of the $E_1$-page of the $v_2$-local algebraic tmf resolution.

Publications & Preprints:

• Algebraic K-theory of Elliptic Cohomology (with Gabe Angelini-Knoll, Christian Ausoni, Eva Höning, and John Rognes). Accepted, to appear in Geometry and Topology
  We compute the $V(2)$-homotopy of the algebraic K-theory of the second truncated Brown-Peterson spectrum.

• Exotic K(h)-local Picard groups when 2p-1=h^2 and the Vanishing Conjecture (joint with Ningchuan Zhang). Accepted.
  

• The Motivic Lambda algebra (joint with J.D. Quigley and W. Balderrama). Accepted, to appear in Geometry and Topology
  We give a description of the motivic Lambda algebra over general base fields. We recover various calculations of Ext over the motivic Steenrod algebra and use these to determine an infinite family of $d_2$-differentials out of the 1-line in the R-motivic ASS.

• Topological Hochschild Homology of the second truncated Brown-Peterson spectrum, part 1 (Joint with Gabe Angelini-Knoll and Eva Höning)
  

• Algebraic Slice Spectral Sequences (Joint with H. Kong and J.D. Quigley). Accepted. To appear in Documenta Mathematica
  

• The telescope conjecture at height 2 and the tmf resolution (Joint with Behrens, Beaudry, Bhattacharya, and Xu). Accepted. To appear in the Journal of Topology
  We analyze the tmf-based Adams spectral sequence for the type 2 spectrum Z. This is used to deduce that the K(2)-local Adams-Novikov spectral sequence, whose E_2-term was computed by Bhattacharya-Egger, collapses. We also explore how this spectral sequence detects the telescopic homotopy of Z.

• K(1)-local tmf co-operations (Joint with Paul VanKoughnett). Accepted. To appear in JHRS
  Using Hopkins' construction of the spectrum tmf in the K(1)-local category, we compute the K(1)-local tmf cooperations.

• kq-resolutions I (Joint with JD Quigley). Accepted. To appear in Trans. AMS
  We compute the cooperations of the very effective cover of Hermitian K-theory over the complex numbers. We use this to analyze the kq-resolution and prove complex motivic analogs of Mahowald's results using bo-resolutions, such as the order of the 2-torsion in the image of J. We also formulate and provide evidence for a motivic Telescope Conjecture.

• The Adams spectral sequence for 3-local tmf. Accepted. To appear in JHRS.
  Last updated 10/08/2020

• The BP<2>-cooperations algebra at odd primes . JPAA 224 (May 2020)
  Last updated 9/20/19

• On BP<2>-cooperations.
  Algebraic & Geometric Topology 19 (2019) 807-862

• On the E2-term of the bo-Adams spectral sequence. J. Topology 13 (2020) 356-415.
  (with Mark Behrens, Agnes Beaudry, Prasit Bhattacharya, and Zhouli Xu).
  Last updated 2/1/17.

• A new basis for complex K-theory cooperations algebra.
  Topology and its Applications 234 (2018) 7-25