I am a sophomore at MIT.
I'm interested in dualities in chromatic homotopy theory and (derived) algebraic geometry, as well as the numerous connections with number theory.
**Email address:**sanathd[at]mit[dot]edu**CV:**[pdf]
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. |

- Roots of unity in K(n)-local E
_{∞}-rings (2017).

We prove 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. - The derived Lubin-Tate stack and K(n)-local duality phenomena (2017).

In this paper, we use derived algebraic geometry to provide proofs and generalizations of some duality phenomena in the K(n)-local stable homotopy category. - Ekedahl-Oort types of Jacobians of hyperelliptic curves in odd characteristic (2017), with John Halliday. Draft, in preparation.
- Chromatic homotopy theory and arithmetic geometry (2017). Notes from two lectures that I gave at Emory. These are unpolished.
- My notes from Talbot 2017.
- My notes for 18.917 (topics in algebraic topology), taught by Professor Clark Barwick in Spring 2017.
- My notes for 18.905 (algebraic topology). The combined notes that include the content of 18.906 (algebraic topology ii) can be found at this link (which links to a draft of a book that I'm writing with Haynes Miller and others). This is also on github.
- My notes for 18.715 (representation theory), taught by Professor Bjorn Poonen in fall 2016.
- Coherent cospans and a combinatorial generalization of complete Segal operads (2016). Preprint available on the arXiv.

Let C be a category with pullbacks and a terminal object. Cospans in C form a bicategory. In this project, we study a simple combinatorial generalization of Barwick's complete Segal operads that incorporates coherent cospans. We describe 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.