Research

Recent projects

My graduate studies focused on the following projects:

• gravitational n-body choreographies and rigorous numerical existence proofs therefor;
• rigorous numerical existence proofs for codes in compact manifolds; and
• linear and semidefinite programming bounds for spherical codes.

The latter two projects are joint work with Abhinav Kumar and Henry Cohn.

Publications

• "Linear recurrence sequences satisfying congruence conditions", Proceedings of the American Mathematical Society (2014). doi:10.1090/S0002-9939-2014-12168-X
• "Improved concentration bounds for Count-Sketch" (with E. Price), awarded Best Student Paper at SODA (2014). arXiv:1207.5200
• "Unique factorization in the theory of quadratic forms", Diophantine Methods, Lattices, and Arithmetic Theory of Quadratic Forms, Contemporary Mathematics 587 (2013). doi:10.1090/conm/587/11681
• "Three approaches to a sequence problem", Mathematics Magazine 84 (2011), 33-37. doi:10.4169/math.mag.84.1.033
• "Inversions in ranking data" (with J.P. Grossman), Discrete Mathematics 309 (2009), 6149-6151. doi:10.1016/j.disc.2009.04.030
• "Voting, the symmetric group, and representation theory" (with Z. Daugherty, A. Eustis, and M. Orrison), American Mathematical Monthly 116 (2009), 667-687. doi:10.4169/193009709X460796 arXiv:0712.2837
• "Borda meets Pascal" (with M. Jameson and M. Orrison), Math Horizons 16 (2008), 8-10. jstor/25678765
• "A new mechanism for non-locality from string theory: UV-IR quantum entanglement and its imprints on the CMB" (with V. Sahakian), Physical Review D 77 (2008), no. 026008. doi:10.1103/PhysRevD.77.026008 arXiv:0707.3786
• "Problems of circle tangency", Harvard College Mathematics Review 1 (2007), 47-53. pdf

Presentations

• "Rigorous numerical methods for gravitational orbits", Joint Mathematics Meetings 2013 (San Diego, CA). abstract
• "Dancing with the Stars", Microsoft Research Special Seminar 2013 (Cambridge, MA).
• "Optimality of spherical codes subject to symmetries", Joint Mathematics Meetings 2012 (Boston, MA). abstract
• "Studying voting paradoxes through representation theory", Joint Mathematics Meetings 2009 (Washington, DC). abstract
• "Dot product representations of graphs" (poster), Joint Mathematics Meetings 2008 (San Diego, CA).
• "Algebraic insights into voting theory", Pacific Coast Undergraduate Mathematics Conference 2006 (Los Angeles, CA).
• "Comparing voting methods using representation theory" (poster), Joint Mathematics Meetings 2006 (San Antonio, TX).
• "The spinning cube" (with R. Minton), Joint Mathematics Meetings 2004 (Phoenix, AZ).
• "The stability of the L1 Lagrange point", International Space Olympics 2003 (Korelov, Russia).

Miscellaneous notes

This section contains notes that are expository, abandoned, just for fun, or otherwise not research-caliber. I am slowly posting them; items not yet uploaded are available upon request.

• "Skew fields in central simple algebras"
• "Non-cohomological proof that the Brauer group is torsion"
• "Galois embedding problems" (talk notes, 11/28/2012) abstract
• "Conway's Angel problem" (talk notes, 11/30/2011)
• "Fields in which every separable extension is Galois"
• "Prime proving algorithms" (talk notes, 11/23/2010)
• "Exact value of Fiege's game with parallel repetition"
• "Moments of random orthogonal matrices"
• "Conway's 'Look-and-say sequence'"