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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.

Sphere Packing:

Abstract PDF New upper bounds on sphere packings I (with Noam Elkies)
Annals of Mathematics 157 (2003), 689-714, arXiv:math.MG/0110009
Abstract PDF New upper bounds on sphere packings II
Geometry and Topology 6 (2002), 329-353, arXiv:math.MG/0110010
Abstract PDF 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
Abstract PDF 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
Abstract PDF Optimality and uniqueness of the Leech lattice among lattices (with Abhinav Kumar)
Annals of Mathematics 170 (2009), 1003-1050, arXiv:math.MG/0403263
Abstract PDF Rigidity of spherical codes (with Yang Jiao, Abhinav Kumar, and Sal Torquato)
Geometry and Topology 15 (2011), 2235-2273, arXiv:1102.5060
Abstract PDF Sphere packing bounds via spherical codes (with Yufei Zhao)
Duke Mathematical Journal 163 (2014), 1965-2002, arXiv:1212.5966
Abstract PDF Optimal simplices and codes in projective spaces (with Abhinav Kumar and Gregory Minton)
Geometry and Topology 20 (2016), 1289-1357, arXiv:1308.3188
Abstract PDF Some properties of optimal functions for sphere packing in dimensions 8 and 24 (with Stephen D. Miller)
preprint, arXiv:1603.04759
Abstract PDF Packing, coding, and ground states
PCMI 2014 lecture notes, arXiv:1603.05202
Abstract PDF 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
Abstract PDF A conceptual breakthrough in sphere packing
Notices of the American Mathematical Society 64 (2017), 102-115, arXiv:1611.01685

Uncertainty Principles:

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

Potential Energy Minimization:

Abstract PDF Universally optimal distribution of points on spheres (with Abhinav Kumar)
Journal of the American Mathematical Society 20 (2007), 99-148, arXiv:math.MG/0607446
Abstract PDF 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
Abstract PDF Counterintuitive ground states in soft-core models (with Abhinav Kumar)
Physical Review E 78 (2008), 061113, arXiv:0811.1236
Abstract PDF 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
Abstract PDF 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
Abstract PDF 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
Abstract PDF 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
Abstract PDF Three-point bounds for energy minimization (with Jeechul Woo)
Journal of the American Mathematical Society 25 (2012), 929-958, arXiv:1103.0485
Abstract PDF 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
Abstract PDF The Gaussian core model in high dimensions (with Matthew de Courcy-Ireland)
preprint, 2016 arXiv:1603.09684

Graph Limits:

Abstract PDF An Lp theory of sparse graph convergence I: limits, sparse random graph models, and power law distributions (with Christian Borgs, Jennifer Chayes, and Yufei Zhao)
preprint, 2014, arXiv:1401.2906
Abstract PDF An Lp theory of sparse graph convergence II: LD convergence, quotients, and right convergence (with Christian Borgs, Jennifer Chayes, and Yufei Zhao)
to appear in Annals of Probability, arXiv:1408.0744
Abstract PDF Consistent nonparametric estimation for heavy-tailed sparse graphs (with Christian Borgs, Jennifer Chayes, and Shirshendu Ganguly)
preprint, 2015, arXiv:1508.06675
Abstract PDF Sparse exchangeable graphs and their limits via graphon processes (with Christian Borgs, Jennifer Chayes, and Nina Holden)
to appear in Journal of Machine Learning Research, arXiv:1601.07134
Abstract PDF Sampling perspectives on sparse exchangeable graphs (with Christian Borgs, Jennifer Chayes, and Victor Veitch)
preprint, 2017, arXiv:1708.03237


Abstract PDF 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)
Abstract PDF 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
Abstract PDF 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

Coding Theory:

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

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

Self Assembly:

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

Fast Matrix Multiplication:

Abstract PDF 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
Abstract PDF 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
Abstract PDF 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
Abstract PDF 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
Abstract PDF 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

Random Tilings and the Dimer Model:

Abstract PDF 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
Abstract PDF 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
Abstract PDF 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


Abstract PDF Projective geometry over F1 and the Gaussian binomial coefficients
American Mathematical Monthly 111 (2004), 487-495, arXiv:math.CO/0407093

Random Sampling:

Abstract PDF 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

Number Theory:

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

Scholarly Publishing:

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

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