The odd Littlewood-Richardson rule (November 2011): A plactic definition of odd Schur functions and a proof that these and the Schur functions of the following two papers all coincide. An odd analogue of the Littlewood-Richardson rule is given, as is an interpretation of this rule in terms of Knutson-Tao hives. arXiv:math.QA/1111.3932.
The odd nilHecke algebra and its diagrammatics, joint with Mikhail Khovanov and Aaron Lauda (November 2011): Construction of odd analogues of the nilHecke algebra and its cyclotomic quotients, its thin and thick diagrammatics, and the cohomology rings of complex Grassmannians. As in the even setting, the odd nilHecke algebra is isomorphic to a matrix algebra over the ring of odd symmetric functions (in finitely many variables). Like their even counterparts, odd nilHecke algebras categorify the positive half of quantum sl(2). arXiv:math.QA/1111.1320.
The Hopf algebra of odd symmetric functions, joint with Mikhail Khovanov (July 2011): Introduction of a (noncommutative, non-cocommutative) Z-graded Hopf superalgebra which exhibits many of the combinatorial properties as the classical symmetric functions. arXiv:math.QA/1107.5610.
Non-research
Algebraic K0 and K1 (May 2009): An essay I wrote for Dylan Thurston's Algebraic Topology class. Discusses the basics of K0 and K1 for rings, topological spaces, and exact categories.
The Grothendieck-Riemann-Roch Theorem (May 2008): My essay for Part III (advisor: Burt Totaro). Discusses the Grothendieck-Riemann-Roch theorem for non-singular varieties, largely following the paper of Borel and Serre.
Notes on the X-ray Transform in Geometry and Dynamics (March 2008): My notes on Gabriel Paternain's course, "The X-ray Transform in Geometry and Dynamics", a Part III course at Cambridge from the Lent term of 2008. (Updated June 2008.)
Dunking Donuts: Culinary Calculations of the Euler Characteristic (December 2006): A slightly expanded, written version of a talk I gave at the Harvard Math Table (undergraduate colloquium) on November 28, 2006. Morse theory and the Poincaré-Hopf Index Theorem are introduced as ways to calculate the Euler characteristic. The title refers to two culinary analogies used in the case of closed surfaces. Published in the Harvard College Mathematics Review (Vol. 1, No. 1, Spring 2007).
Notes on Topological Tic-Tac-Toe (October 2006): A section handout I wrote as the course assistant for Harvard Math 131, Topology (taught by Véronique Godin). Tic-tac-toe is considered on the torus, the projective plane, the Klein bottle, the cylinder, and the Möbius strip. This is very sparse, closer to an outline than actual notes. A much more complete exposition I have written can be found at Concrete Nonsense (direct links: 1, 2, 3).