This page links to all my research papers. The icons on the left let you read the abstract or download the paper in PDF format. The abstract page also links to other formats.

I tend to put papers on the web only when they are essentially complete. If you would like to see the current version of a draft manuscript you don't see here or some work in progress, just let me know. Another thing I'm not currently posting here are lecture slides and course notes. They aren't designed to stand on their own, so they are of interest primarily to people who were there. If you were there and want a copy (or weren't there and want to see what you missed), send me e-mail and I'll send you a link.

I've organized the papers here by topic rather than date, since that seems most useful. Here is a full list in roughly chronological order.

New upper bounds on sphere packings I
(with Noam Elkies)
Annals of Mathematics 157 (2003), 689-714, arXiv:math.MG/0110009 |
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New upper bounds on sphere packings II
Geometry and Topology 6 (2002), 329-353, arXiv:math.MG/0110010 |
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The densest lattice in twenty-four dimensions
(with Abhinav Kumar)
Electronic Research Announcements of the American Mathematical Society 10 (2004), 58-67, arXiv:math.MG/0408174 |
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Uniqueness of the (22,891,1/4) spherical code
(with Abhinav Kumar)
New York Journal of Mathematics 13 (2007), 147-157, arXiv:math.MG/0607448 |
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Optimality and uniqueness of the Leech lattice among lattices
(with Abhinav Kumar)
Annals of Mathematics 170 (2009), 1003-1050, arXiv:math.MG/0403263 |
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Rigidity of spherical codes (with Yang Jiao,
Abhinav Kumar, and
Sal Torquato)
Geometry and Topology 15 (2011), 2235-2273, arXiv:1102.5060 |
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Sphere packing bounds via spherical codes
(with Yufei Zhao)
Duke Mathematical Journal 163 (2014), 1965-2002, arXiv:1212.5966 |
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Optimal simplices and codes in projective spaces
(with Abhinav Kumar and Gregory Minton)
Geometry and Topology 20 (2016), 1289-1357, arXiv:1308.3188 |
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Some properties of optimal functions for sphere packing in dimensions 8 and 24
(with Stephen D. Miller)
preprint, arXiv:1603.04759 |
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Packing, coding, and ground states
PCMI 2014 lecture notes, arXiv:1603.05202 |
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The sphere packing problem in dimension 24 (with Abhinav Kumar,
Stephen D. Miller,
Danylo Radchenko, and
Maryna Viazovska)
Annals of Mathematics 185 (2017), 1017-1033, arXiv:1603.06518 |
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A conceptual breakthrough in sphere packing
Notices of the American Mathematical Society 64 (2017), 102-115, arXiv:1611.01685 |

An optimal uncertainty principle in twelve dimensions via modular forms (with Felipe Gonçalves),
preprint, 2017, arXiv:1712.04438 |

Universally optimal distribution of points on spheres
(with Abhinav Kumar)
Journal of the American Mathematical Society 20 (2007), 99-148, arXiv:math.MG/0607446 |
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The D_4 root system is not universally optimal
(with John Conway, Noam Elkies,
and Abhinav Kumar)
Experimental Mathematics 16 (2007), 313-320, arXiv:math.MG/0607447 |
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Counterintuitive ground states in soft-core models
(with Abhinav Kumar)
Physical Review E 78 (2008), 061113, arXiv:0811.1236 |
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Experimental study of energy-minimizing point configurations on spheres
(with Brandon Ballinger, Greg Blekherman,
Noah Giansiracusa,
Elizabeth Kelly, and
Achill Schürmann)
Experimental Mathematics 18 (2009), 257-283, arXiv:math.MG/0611451 |
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Ground states and formal duality relations in the Gaussian core model
(with Abhinav Kumar and
Achill Schürmann)
Physical Review E 80 (2009), 061116, arXiv:0911.2169 |
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Point configurations that are asymmetric yet balanced
(with Noam Elkies,
Abhinav Kumar, and
Achill Schürmann)
Proceedings of the American Mathematical Society 138 (2010), 2863-2872, arXiv:0812.2579 |
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Order and disorder in energy minimization
Proceedings of the International Congress of Mathematicians, Hyderabad, August 19-27, 2010, Volume IV, pages 2416-2443, Hindustan Book Agency, New Delhi, 2010, arXiv:1003.3053 |
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Three-point bounds for energy minimization
(with Jeechul Woo)
Journal of the American Mathematical Society 25 (2012), 929-958, arXiv:1103.0485 |
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Formal duality and generalizations of the Poisson summation formula
(with Abhinav Kumar,
Christian Reiher,
and Achill Schürmann)
Discrete Geometry and Algebraic Combinatorics, vol. 625 of Contemporary Mathematics, American Mathematical Society, 2014, pp. 123-140, arXiv:1306.6796 |
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The Gaussian core model in high dimensions
(with Matthew de Courcy-Ireland)
Duke Mathematical Journal 167 (2018), 2417-2455, arXiv:1603.09684 |

An L
(with Christian Borgs,
Jennifer Chayes, and
Yufei Zhao)
^{p} theory of sparse graph convergence I: limits, sparse random graph models, and power law distributionsTransactions of the American Mathematical Society, to appear, arXiv:1401.2906 |
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An L
(with Christian Borgs,
Jennifer Chayes, and
Yufei Zhao)
^{p} theory of sparse graph convergence II: LD convergence, quotients, and right convergenceAnnals of Probability 46 (2018), 337-396, arXiv:1408.0744 |
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Consistent nonparametric estimation for heavy-tailed sparse graphs
(with Christian Borgs,
Jennifer Chayes, and
Shirshendu Ganguly)
preprint, 2015, arXiv:1508.06675 |
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Sparse exchangeable graphs and their limits via graphon processes
(with Christian Borgs,
Jennifer Chayes, and
Nina Holden)
Journal of Machine Learning Research, to appear, arXiv:1601.07134 |
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Sampling perspectives on sparse exchangeable graphs
(with Christian Borgs,
Jennifer Chayes, and
Victor Veitch)
preprint, 2017, arXiv:1708.03237 |
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Identifiability for graphexes and the weak kernel metric
(with Christian Borgs,
Jennifer Chayes, and
László Miklós Lovász)
preprint, 2018, arXiv:1804.03277 |

Ideal forms of Coppersmith's theorem and Guruswami-Sudan list decoding
(with Nadia Heninger)
Advances in Mathematics of Communications 9 (2015), 311-339, arXiv:1008.1284 (extended abstract was published in proceedings of ICS 2011) |
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Approximate common divisors via lattices
(with Nadia Heninger)
in Proceedings of the Tenth Algorithmic Number Theory Symposium, The Open Book Series 1 (2013), 271-293, arXiv:1108.2714 |
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The impossibility of obfuscation with auxiliary input or a universal simulator
(with Nir Bitansky, Ran Canetti, Shafi Goldwasser, Yael Tauman Kalai, Omer Paneth, and Alon Rosen)
Advances in Cryptology -- CRYPTO 2014, Lecture Notes in Computer Science 8617, pp. 71-89, Springer, 2014, arXiv:1401.0348 |

See also the papers with Nadia Heninger above for other work in coding theory.

Energy-minimizing error-correcting codes
(with Yufei Zhao)
IEEE Transactions on Information Theory 60 (2014), 7442-7450, arXiv:1212.1913 |

Algorithmic design of self-assembling structures
(with Abhinav Kumar)
Proc. Natl. Acad. Sci. USA 106 (2009), 9570-9575, arXiv:0906.3550 |

A group-theoretic approach to fast matrix multiplication
(with Chris Umans)
Proceedings of the 44th Annual Symposium on Foundations of Computer Science (11-14 October 2003, Cambridge, MA), IEEE Computer Society, pp. 438-449, arXiv:math.GR/0307321 |
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Group-theoretic algorithms for matrix multiplication
(with Bobby Kleinberg,
Balazs Szegedy, and
Chris Umans)
Proceedings of the 46th Annual Symposium on Foundations of Computer Science (23-25 October 2005, Pittsburgh, PA), IEEE Computer Society, pp. 379-388, arXiv:math.GR/0511460 |
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Fast matrix multiplication using coherent configurations (with Chris Umans)
Proceedings of the 24th Annual ACM-SIAM Symposium on Discrete Algorithms (6-8 January 2013, New Orleans, LA), Society for Industrial and Applied Mathematics, pp. 1074-1087, arXiv:1207.6528 |
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On cap sets and the group-theoretic approach to matrix multiplication (with
Jonah Blasiak,
Thomas Church,
Joshua A. Grochow,
Eric Naslund,
William F. Sawin,
and Chris Umans)
Discrete Analysis, 2017:3, 27 pp., arXiv:1605.06702 |
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Which groups are amenable to proving exponent two for matrix multiplication? (with
Jonah Blasiak,
Thomas Church,
Joshua A. Grochow,
and Chris Umans)
preprint, 2017, arXiv:1712.02302 |

Local statistics for random domino tilings of the Aztec diamond
(with Noam Elkies and Jim Propp)
Duke Mathematical Journal 85 (1996), 117-166, arXiv:math.CO/0008243 |
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The shape of a typical boxed plane partition
(with
Michael Larsen
and Jim Propp)
New York Journal of Mathematics 4 (1998), 137-165, arXiv:math.CO/9801059 |
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A variational principle for domino tilings
(with Rick Kenyon
and Jim Propp)
Journal of the American Mathematical Society 14 (2001), 297-346, arXiv:math.CO/0008220 |

Projective geometry over F
_{1} and the Gaussian binomial coefficients
American Mathematical Monthly 111 (2004), 487-495, arXiv:math.CO/0407093 |

Generating a random sink-free orientation in quadratic time
(with Robin Pemantle
and Jim Propp)
Electronic Journal of Combinatorics 9 (2002), #R10, arXiv:math.PR/0103189 |

Symmetry and specializability in continued fractions
Acta Arithmetica 75.4 (1996), 297-320, arXiv:math.NT/0008221 |
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2-adic behavior of numbers of domino tilings
Electronic Journal of Combinatorics 6 (1999), #R14, arXiv:math.CO/0008222 |
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A short proof of the simple continued fraction expansion of
e
American Mathematical Monthly 113 (2006), 57-62, arXiv:math.NT/0601660 |
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Metacommutation of Hurwitz primes
(with Abhinav Kumar)
Proceedings of the American Mathematical Society 143 (2015), 1459-1469, arXiv:1307.0443 |

See also the statement of purpose of the cost of knowledge boycott of Elsevier.

Mathematicians take a stand
(with Douglas N. Arnold)
Notices of the American Mathematical Society 59 (2012), 828-833, arXiv:1204.1351 |