Current Research:
I mainly work on the following projects at the moment.
- The cooperations algebra for topological modular forms
This is part of an ongoing endeavor to develop machinery to make the tmf-based Adams spectral sequence as a viable tool for doing computations in stable homotopy theory, especially for computing the homotopy of the sphere. At the moment, I am mainly focusing on the cooperations algebra for tmf at the prime p=3.
- The telescope conjecture (Joint with Agnes Beaudry, Mark Behrens, Prasit Bhattacharya, Doug Ravenel, and Zhouli Xu)
In this project, we are attempting to find a counter example to Ravenel's (now infamous) telescope conjecture.
- Motivic kq-resolutions over general base fields (Joint with JD Quigley)
We are studying the kq-based Adams spectral sequence over general base fields, extending our previous work over the field of complex numbers.
In progress:
- The Motivic Lambda algebra (joint with J.D. Quigley and W. Balderrama)
We give a description of the motivic Lambda algebra over general base fields. We recover various calculations of Ext over the motivic Steenrod algebra.
- Topological Hochschild homology of second truncated Brown-Peterson spectrum, part 1 (Joint with Gabe Angelini-Knoll and Eva Hönig)
We calculate the homotopy of the topological Hochschild homology of the second truncated Brown-Peterson spectrum with coefficients in the connective Adams summand.
Publications & Preprints:
- On the tmf-resolution of Z (Joint with Behrens, Beaudry, Bhattacharya, and Xu). Submitted
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)
Using Hopkins' construction of the spectrum tmf in the K(1)-local category, we compute the K(1)-local tmf cooperations.
- On kq-resolutions in complex motivic homotopy theory (Joint with JD Quigley). submitted
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.