I'm a sophomore at MIT, broadly interested in chromatic homotopy theory and (derived) algebraic geometry, as well as the numerous connections with number theory. I also do non-math stuff: I'm a drummer, I occasionally play basketball, and I'm an avid biker. ## In IAP/January 2018, I will be teaching a course/leading a seminar on chromatic homotopy theory. Here's the website for this.In the fall of 2017, I'll be organizing the Student Colloquium for Undergraduates in Mathematics ((un)fortunately abbreviated "SCUM"!). I'm also participating in/organizing a reading seminar on p-adic Hodge theory and THH, a la Nikolaus-Scholze and Bhatt-Morrow-Scholze. See there for rough notes. **Email address:**sanathd[at]mit[dot]edu**CV:**[pdf]
Here are the courses that I've taken. This is a link to some things I've been thinking about. Here are some outlines of projects that I'm interested in thinking about. This is a link to a chart depicting the things I'm trying to learn/am interested in. |

- Slides from my talk on "Roots of unity in
K(n)-local E
_{∞}-rings", at JMM 2018 (in San Diego, California). - An equivariant version of Wood's theorem. Last
update: December 2017.

Proves an equivariant version of Wood's equivalence KU = KO /\ C(eta) and discusses a generalization to equivariant TMF. - A textbook on algebraic topology, joint with
Haynes Miller. Last update: December 2017.

Lecture notes from the 18.905-906 sequence taught by Haynes Miller in the 2016-17 academic year. This links to a draft of a book that I'm writing with Haynes Miller. The TeX code is available on github. I advise caution when reading this, since the notes are in the process of being revised. - The Dieudonn'e modules and Ekedahl-Oort
types of Jacobians of hyperelliptic curves in odd characteristic, with
John Halliday. Last update: December 2017. arXiv.

Provides explicit formulae for the Frobenius and Verschiebung acting on the mod p Dieudonn'e module of the Jacobian of a hyperelliptic curve, when p is an odd prime. These formulae are used to settle some questions posed by Glass and Pries from 2004. The code used is available at this github repository. - The Lubin-Tate stack and Gross-Hopkins duality. Last
update: December 2017. arXiv.

Uses derived algebraic geometry to provide proofs and generalizations of some duality phenomena in K(n)-local stable homotopy theory. - Roots of unity in K(n)-local
E
_{∞}-rings. Last update: November 2017. arXiv.

Shows that for n≥1, any K(n)-local E_{∞}-ring R with a primitive p^k-th root of unity in pi_0 R is trivial, thus proving that the Lubin-Tate tower does not lift to a tower of derived stacks over Morava E-theory. - Talbot proceedings: Obstruction theory for
structured ring spectra, joint with Eva Belmont et. al. Last update:
September 2017.

The proceedings from the Talbot workshop on obstruction theory for structured ring spectra, which took place from May 21-27, 2017.

- A global perspective on stable homotopy theory. Last update: December 2017. Notes for the Kan seminar, which discusses Ravenel's influential conjectures (with some motivation stemming from algebraic geometry).
- The Wood cofiber sequence via algebraic geometry. Last update: November 2017. An expository account of an algebro-geometric viewpoint on the Wood cofiber sequence.
- Examples of Goodwillie Calculus, for a talk at Juvitop. Last update: October 2017.
- Milnor's exotic spheres, for a talk at the Kan seminar. Last update: September 2017.
- Chromatic homotopy theory and arithmetic geometry. Last update: August 2017. Notes from two lectures that I gave at Emory. These are unpolished.
- Coherent cospans and a combinatorial generalization of
complete Segal operads (2016). Preprint available on the arXiv.

Studies a naive combinatorial generalization of Barwick's complete Segal operads that incorporates coherent cospans, and describes explicit examples of such "2-fold complete Segal operads".

- The
cohomology of Ω
*S*.^{n} - Elliptic curve cryptography, for a class at SPLASH in Fall 2016.
- My blog at ErdosNinth, hosted on Wordpress.