Publications (63)
The densest lattice in twenty-four dimensions
Henry Cohn, Abhinav Kumar
In this research announcement we outline the methods used in our recent proof that the Leech lattice is the unique densest lattice in R^24. Complete details will appear elsewhere,…
Variations on five-dimensional sphere packings
Henry Cohn, Isaac Rajagopal
We analyze SzöllÅsi's recent construction of a conjecturally optimal five-dimensional kissing configuration and produce a new such configuration, the fourth to be discovered. We…
Metacommutation of Hurwitz primes
Henry Cohn, Abhinav Kumar
Conway and Smith introduced the operation of metacommutation for pairs of primes in the ring of Hurwitz integers in the quaternions. We study the permutation induced on the primes…
The shape of a typical boxed plane partition
Henry Cohn, Michael Larsen, James Propp
Using a calculus of variations approach, we determine the shape of a typical plane partition in a large box (i.e., a plane partition chosen at random according to the uniform distr…
New upper bounds on sphere packings II
Henry Cohn
We continue the study of the linear programming bounds for sphere packing introduced by Cohn and Elkies. We use theta series to give another proof of the principal theorem, and pre…
Sparse exchangeable graphs and their limits via graphon processes
Christian Borgs, Jennifer T. Chayes, Henry Cohn +1
In a recent paper, Caron and Fox suggest a probabilistic model for sparse graphs which are exchangeable when associating each vertex with a time parameter in . Here w…
Sphere packing bounds via spherical codes
Henry Cohn, Yufei Zhao
The sphere packing problem asks for the greatest density of a packing of congruent balls in Euclidean space. The current best upper bound in all sufficiently high dimensions is due…
Algorithmic design of self-assembling structures
Henry Cohn, Abhinav Kumar
We study inverse statistical mechanics: how can one design a potential function so as to produce a specified ground state? In this paper, we show that unexpectedly simple potential…
Free partition functions and an averaged holographic duality
Nima Afkhami-Jeddi, Henry Cohn, Thomas Hartman +1
We study the torus partition functions of free bosonic CFTs in two dimensions. Integrating over Narain moduli defines an ensemble-averaged free CFT. We calculate the averaged parti…
Projective geometry over F_1 and the Gaussian binomial coefficients
Henry Cohn
There is no field with only one element, yet there is a well-defined notion of what projective geometry over such a field means. This notion is familiar to experts and plays an int…
On cap sets and the group-theoretic approach to matrix multiplication
Jonah Blasiak, Thomas Church, Henry Cohn +4
In 2003, Cohn and Umans described a framework for proving upper bounds on the exponent of matrix multiplication by reducing matrix multiplication to group algebra multiplicati…
Which groups are amenable to proving exponent two for matrix multiplication?
Jonah Blasiak, Thomas Church, Henry Cohn +2
The Cohn-Umans group-theoretic approach to matrix multiplication suggests embedding matrix multiplication into group algebra multiplication, and bounding in terms of the repre…
The sphere packing problem in dimension 24
Henry Cohn, Abhinav Kumar, Stephen D. Miller +2
Building on Viazovska's recent solution of the sphere packing problem in eight dimensions, we prove that the Leech lattice is the densest packing of congruent spheres in twenty-fou…
Optimal simplices and codes in projective spaces
Henry Cohn, Abhinav Kumar, Gregory Minton
We find many tight codes in compact spaces, i.e., optimal codes whose optimality follows from linear programming bounds. In particular, we show the existence (and abundance) of sev…
Point configurations that are asymmetric yet balanced
Henry Cohn, Noam D. Elkies, Abhinav Kumar +1
A configuration of particles confined to a sphere is balanced if it is in equilibrium under all force laws (that act between pairs of points with strength given by a fixed function…
New upper bounds on sphere packings I
Henry Cohn, Noam Elkies
We develop an analogue for sphere packing of the linear programming bounds for error-correcting codes, and use it to prove upper bounds for the density of sphere packings, which ar…
A group-theoretic approach to fast matrix multiplication
Henry Cohn, Christopher Umans
We develop a new, group-theoretic approach to bounding the exponent of matrix multiplication. There are two components to this approach: (1) identifying groups G that admit a certa…
Dual linear programming bounds for sphere packing via modular forms
Henry Cohn, Nicholas Triantafillou
We obtain new restrictions on the linear programming bound for sphere packing, by optimizing over spaces of modular forms to produce feasible points in the dual linear program. In…
Identifiability for graphexes and the weak kernel metric
Christian Borgs, Jennifer T. Chayes, Henry Cohn +1
In two recent papers by Veitch and Roy and by Borgs, Chayes, Cohn, and Holden, a new class of sparse random graph processes based on the concept of graphexes over -finite measu…
An optimal uncertainty principle in twelve dimensions via modular forms
Henry Cohn, Felipe Gonçalves
We prove an optimal bound in twelve dimensions for the uncertainty principle of Bourgain, Clozel, and Kahane. Suppose is an integrable fun…
Three-point bounds for energy minimization
Henry Cohn, Jeechul Woo
Three-point semidefinite programming bounds are one of the most powerful known tools for bounding the size of spherical codes. In this paper, we use them to prove lower bounds for…
Group-theoretic algorithms for matrix multiplication
Henry Cohn, Robert Kleinberg, Balazs Szegedy +1
We further develop the group-theoretic approach to fast matrix multiplication introduced by Cohn and Umans, and for the first time use it to derive algorithms asymptotically faster…
A short proof of the simple continued fraction expansion of e
Henry Cohn
This note presents an especially short and direct variant of Hermite's proof of the simple continued fraction expansion e = [2,1,2,1,1,4,1,1,6,...] and explains some of the motivat…
Universal optimality of the and Leech lattices and interpolation formulas
Henry Cohn, Abhinav Kumar, Stephen D. Miller +2
We prove that the root lattice and the Leech lattice are universally optimal among point configurations in Euclidean spaces of dimensions and , respectively. In other…
Three-point bounds for sphere packing
Henry Cohn, David de Laat, Andrew Salmon
We define three-point bounds for sphere packing that refine the linear programming bound, and we compute these bounds numerically using semidefinite programming by choosing a trunc…
Formal duality and generalizations of the Poisson summation formula
Henry Cohn, Abhinav Kumar, Christian Reiher +1
We study the notion of formal duality introduced by Cohn, Kumar, and Schürmann in their computational study of energy-minimizing particle configurations in Euclidean space. In par…
The impossibility of obfuscation with auxiliary input or a universal simulator
Nir Bitansky, Ran Canetti, Henry Cohn +4
In this paper we show that the existence of general indistinguishability obfuscators conjectured in a few recent works implies, somewhat counterintuitively, strong impossibility re…
Rigidity of spherical codes
Henry Cohn, Yang Jiao, Abhinav Kumar +1
A packing of spherical caps on the surface of a sphere (that is, a spherical code) is called rigid or jammed if it is isolated within the space of packings. In other words, aside f…
Sign uncertainty principles and low-degree polynomials
Henry Cohn, Dingding Dong, Felipe Gonçalves
We prove an asymptotically sharp version of the Bourgain-Clozel-Kahane and Cohn-Gonçalves sign uncertainty principles for polynomials of sublinear degree times a Gaussian, as the…
Energy-minimizing error-correcting codes
Henry Cohn, Yufei Zhao
We study a discrete model of repelling particles, and we show using linear programming bounds that many familiar families of error-correcting codes minimize a broad class of potent…
Ground states and formal duality relations in the Gaussian core model
Henry Cohn, Abhinav Kumar, Achill Schuermann
We study dimensional trends in ground states for soft-matter systems. Specifically, using a high-dimensional version of Parrinello-Rahman dynamics, we investigate the behavior of t…
Finite matrix multiplication algorithms from infinite groups
Jonah Blasiak, Henry Cohn, Joshua A. Grochow +2
The Cohn-Umans (FOCS '03) group-theoretic framework for matrix multiplication produces fast matrix multiplication algorithms from three subsets of a finite group satisfying a s…
An theory of sparse graph convergence I: limits, sparse random graph models, and power law distributions
Christian Borgs, Jennifer T. Chayes, Henry Cohn +1
We introduce and develop a theory of limits for sequences of sparse graphs based on graphons, which generalizes both the existing theory of dense graph limits and…
Uniqueness of the (22,891,1/4) spherical code
Henry Cohn, Abhinav Kumar
We use techniques of Bannai and Sloane to give a new proof that there is a unique (22,891,1/4) spherical code; this result is implicit in a recent paper by Cuypers. We also correct…
Local statistics for random domino tilings of the Aztec diamond
Henry Cohn, Noam Elkies, James Propp
We prove an asymptotic formula for the probability that, if one chooses a domino tiling of a large Aztec diamond at random according to the uniform distribution on such tilings, th…
Mathematicians take a stand
Douglas N. Arnold, Henry Cohn
We survey the reasons for the ongoing boycott of the publisher Elsevier. We examine Elsevier's pricing and bundling policies, restrictions on dissemination by authors, and lapses i…
Counterintuitive ground states in soft-core models
Henry Cohn, Abhinav Kumar
It is well known that statistical mechanics systems exhibit subtle behavior in high dimensions. In this paper, we show that certain natural soft-core models, such as the Gaussian c…
A variational principle for domino tilings
Henry Cohn, Richard Kenyon, James Propp
We formulate and prove a variational principle (in the sense of thermodynamics) for random domino tilings, or equivalently for the dimer model on a square grid. This principle stat…
The Gaussian core model in high dimensions
Henry Cohn, Matthew de Courcy-Ireland
We prove lower bounds for energy in the Gaussian core model, in which point particles interact via a Gaussian potential. Under the potential function with $0…
Sampling perspectives on sparse exchangeable graphs
Christian Borgs, Jennifer T. Chayes, Henry Cohn +1
Recent work has introduced sparse exchangeable graphs and the associated graphex framework, as a generalization of dense exchangeable graphs and the associated graphon framework. T…
The D_4 root system is not universally optimal
Henry Cohn, John H. Conway, Noam D. Elkies +1
We prove that the D_4 root system (equivalently, the set of vertices of the regular 24-cell) is not a universally optimal spherical code. We further conjecture that there is no uni…
Sphere packing bounds via rescaling
Henry Cohn, Andrew Salmon
We study the relationship between local and global density for sphere packings, and in particular the convergence of packing densities in large, compact regions to the Euclidean li…
2-adic behavior of numbers of domino tilings
Henry Cohn
We study the 2-adic behavior of the number of domino tilings of a 2n-by-2n square as nvaries. It was previously known that this number was of the form 2^n f(n)^2, where f(n) is an…
From sphere packing to Fourier interpolation
Henry Cohn
Viazovska's solution of the sphere packing problem in eight dimensions is based on a remarkable construction of certain special functions using modular forms. Great mathematics has…
Universally optimal distribution of points on spheres
Henry Cohn, Abhinav Kumar
We study configurations of points on the unit sphere that minimize potential energy for a broad class of potential functions (viewed as functions of the squared Euclidean distance…
Order and disorder in energy minimization
Henry Cohn
How can we understand the origins of highly symmetrical objects? One way is to characterize them as the solutions of natural optimization problems from discrete geometry or physics…
Fast matrix multiplication using coherent configurations
Henry Cohn, Christopher Umans
We introduce a relaxation of the notion of tensor rank, called s-rank, and show that upper bounds on the s-rank of the matrix multiplication tensor imply upper bounds on the ordina…
Improved kissing numbers in seventeen through twenty-one dimensions
Henry Cohn, Anqi Li
We prove that the kissing numbers in 17, 18, 19, 20, and 21 dimensions are at least 5730, 7654, 11692, 19448, and 29768, respectively. The previous records were set by Leech in 196…
Approximate common divisors via lattices
Henry Cohn, Nadia Heninger
We analyze the multivariate generalization of Howgrave-Graham's algorithm for the approximate common divisor problem. In the m-variable case with modulus N and approximate common d…
Packing, coding, and ground states
Henry Cohn
These are the lecture notes from my 2014 PCMI graduate summer school lectures. In these lectures, we'll study simple models of materials from several different perspectives: geomet…
Optimality and uniqueness of the Leech lattice among lattices
Henry Cohn, Abhinav Kumar
We prove that the Leech lattice is the unique densest lattice in R^24. The proof combines human reasoning with computer verification of the properties of certain explicit polynomia…
Symmetry and specializability in continued fractions
Henry Cohn
We study explicit continued fraction expansions for certain series. Some of these expansions have symmetry that generalizes some remarkable examples discovered independently by Kmo…
Matrix multiplication via matrix groups
Jonah Blasiak, Henry Cohn, Joshua A. Grochow +2
In 2003, Cohn and Umans proposed a group-theoretic approach to bounding the exponent of matrix multiplication. Previous work within this approach ruled out certain families of grou…
A conceptual breakthrough in sphere packing
Henry Cohn
This expository paper describes Viazovska's breakthrough solution of the sphere packing problem in eight dimensions, as well as its extension to twenty-four dimensions by Cohn, Kum…
High-dimensional sphere packing and the modular bootstrap
Nima Afkhami-Jeddi, Henry Cohn, Thomas Hartman +2
We carry out a numerical study of the spinless modular bootstrap for conformal field theories with current algebra , or equivalently the linear programming bo…
An theory of sparse graph convergence II: LD convergence, quotients, and right convergence
Christian Borgs, Jennifer T. Chayes, Henry Cohn +1
We extend the theory of sparse graph limits, which was introduced in a companion paper, by analyzing different notions of convergence. Under suitable restrictions on node wei…
Optimality of spherical codes via exact semidefinite programming bounds
Henry Cohn, David de Laat, Nando Leijenhorst
We show that the spectral embeddings of all known triangle-free strongly regular graphs are optimal spherical codes (the new cases are points in dimensions, points i…
The work of Maryna Viazovska
Henry Cohn
On July 5th, 2022, Maryna Viazovska was awarded a Fields Medal for her solution of the sphere packing problem in eight dimensions, as well as further contributions to related extre…
Consistent nonparametric estimation for heavy-tailed sparse graphs
Christian Borgs, Jennifer T. Chayes, Henry Cohn +1
We study graphons as a non-parametric generalization of stochastic block models, and show how to obtain compactly represented estimators for sparse networks in this framework. Our…
Some properties of optimal functions for sphere packing in dimensions 8 and 24
Henry Cohn, Stephen D. Miller
We study some sequences of functions of one real variable and conjecture that they converge uniformly to functions with certain positivity and growth properties. Our conjectures im…
Generating a random sink-free orientation in quadratic time
Henry Cohn, Robin Pemantle, James Propp
A sink-free orientation of a finite undirected graph is a choice of orientation for each edge such that every vertex has out-degree at least 1. Bubley and Dyer (1997) use Markov Ch…
Ideal forms of Coppersmith's theorem and Guruswami-Sudan list decoding
Henry Cohn, Nadia Heninger
We develop a framework for solving polynomial equations with size constraints on solutions. We obtain our results by showing how to apply a technique of Coppersmith for finding sma…
Experimental study of energy-minimizing point configurations on spheres
Brandon Ballinger, Grigoriy Blekherman, Henry Cohn +3
In this paper we report on massive computer experiments aimed at finding spherical point configurations that minimize potential energy. We present experimental evidence for two new…