A conceptual breakthrough in sphere packing

The sphere packing problem asks for the densest packing of ${\mathbb{R}}^{n}$ with congruent balls. In other words, what is the largest fraction of ${\mathbb{R}}^{n}$ that can be covered by congruent balls with disjoint interiors?

Pathological packings may not have well-defined densities, but we can handle the technicalities as follows. A sphere packing $\mathcal{P}$ is a nonempty subset of ${\mathbb{R}}^{n}$ consisting of congruent balls with disjoint interiors. The upper density of $\mathcal{P}$ is

where $B_{r}^{n}(x)$ denotes the closed ball of radius $r$ about $x$, and the sphere packing density $\Delta_{{\mathbb{R}}^{n}}$ in ${\mathbb{R}}^{n}$ is the supremum of all the upper densities of sphere packings. In other words, we avoid technicalities by using a generous definition of the packing density. This generosity does not cause any harm, as shown by the theorem of Groemer that there exists a sphere packing $\mathcal{P}$ for which

uniformly for all $x\in{\mathbb{R}}^{n}$. Thus, the supremum of the upper densities is in fact achieved as the density of some packing, in the nicest possible way. Of course the densest packing is not unique, since there are any number of ways to perturb a packing without changing its overall density.

Why should we care about the sphere packing problem? Two obvious reasons are that it’s a natural geometric problem in its own right and a toy model for granular materials. A more surprising application is that sphere packings are error-correcting codes for a continuous communication channel. Real-world communication channels can be modeled using high-dimensional vector spaces, and thus high-dimensional sphere packings have practical importance.

Instead of justifying sphere packing by aspects of the problem or its applications, we’ll justify it by its solutions: a question is good if it has good answers. Sphere packing turns out to be a far richer and more beautiful topic than the bare problem statement suggests. From this perspective, the point of the subject is the remarkable structures that arise as dense sphere packings.

To begin, let’s examine the familiar cases of one, two, and three dimensions. The one-dimensional sphere packing problem is the interval packing problem on the line, which is of course trivial: the optimal density is $1$. The two- and three-dimensional problems are far from trivial, but the optimal packings, shown in Figure 3, are exactly what one would expect. In particular, the sphere packing density is $\pi/\sqrt{12}=0.9068\dots$ in ${\mathbb{R}}^{2}$ and $\pi/\sqrt{18}=0.7404\dots$ in ${\mathbb{R}}^{3}$. The two-dimensional problem was solved by Thue. Giving a rigorous proof requires a genuine idea, but there exist short, elementary proofs [8]. The three-dimensional problem was solved by Hales [9] via a lengthy and complex computer-assisted proof, which was extraordinarily difficult to check but has since been completely verified using formal logic [10].

In both two and three dimensions, one can obtain an optimal packing by stacking layers that are packed optimally in the previous dimension, with the layers nestled together as closely as possible. Guessing this answer is not difficult, nor is computing the density of such a packing. Instead, the difficulty lies in proving that no other construction could achieve a greater density.

Unfortunately, our low-dimensional experience is poor preparation for understanding high-dimensional sphere packing. Based on the first three dimensions, it appears that guessing the optimal packing is easy, but this expectation turns out to be completely false in high dimensions. In particular, stacking optimal layers from the previous dimension does not always yield an optimal packing. (One can recursively determine the best packings in successive dimensions under such a hypothesis [6], and this procedure yields a suboptimal packing by the time it reaches ${\mathbb{R}}^{10}$.)

The sphere packing problem seems to have no simple, systematic solution that works across all dimensions. Instead, each dimension has its own idiosyncracies and charm. Understanding the densest sphere packing in ${\mathbb{R}}^{8}$ tells us only a little about ${\mathbb{R}}^{7}$ or ${\mathbb{R}}^{9}$, and hardly anything about ${\mathbb{R}}^{10}$.

Aside from ${\mathbb{R}}^{8}$ and ${\mathbb{R}}^{24}$, our ignorance grows as the dimension increases. In high dimensions, we have absolutely no idea how the densest sphere packings behave. We do not know even the most basic facts, such as whether the densest packings should be crystalline or disordered. Here “do not know” does not merely mean “cannot prove,” but rather the much stronger “cannot predict.”

A simple greedy argument shows that the optimal density in ${\mathbb{R}}^{n}$ is at least $2^{-n}$. To see why, consider any sphere packing in which there is no room to add even one more sphere. If we double the radius of each sphere, then the enlarged spheres must cover space completely, because any uncovered point could serve as the center of a new sphere that would fit in the original packing. Doubling the radius multiplies volume by $2^{n}$, and so the original packing must cover at least a $2^{-n}$ fraction of ${\mathbb{R}}^{n}$.

That may sound appallingly low, but it is very nearly the best lower bound known. Even the most recent bounds, obtained by Venkatesh [14] in 2013, have been unable to improve on $2^{-n}$ by more than a linear factor in general and an $n\log\log n$ factor in special cases. As for upper bounds, in 1978 Kabatyanskii and Levenshtein [11] proved an upper bound of $2^{(-0.599\ldots+o(1))n}$, which remains essentially the best upper bound known in high dimensions. Thus, we know that the sphere packing density decreases exponentially as a function of dimension, but the best upper and lower bounds known are exponentially far apart.

Table 1 lists the best packing densities currently known in up to $36$ dimensions, and Figure 4 shows a logarithmic plot. The plot has several noteworthy features:

1. (1)

   The curve is jagged and irregular, with no obvious way to interpolate data points from their neighbors.

2. (2)

   The density is clearly decreasing exponentially, but the irregularity makes it unclear how to extrapolate to estimate the decay rate as the dimension tends to infinity.

3. (3)

   There seem to be parity effects. Even dimensions look slightly better than odd dimensions, multiples of four are better yet, and multiples of eight are the best of all.

4. (4)

   Certain dimensions, most notably $24$, have packings so good that they seem to pull the entire curve in their direction. The fact that this occurs is not so surprising, since one expects cross sections and stackings of great packings to be at least good, but the effect is surprisingly large.

How can we describe sphere packings? Random or pathological packings can be infinitely complicated, but the most important packings can generally be given a finite description via periodicity.

Recall that a lattice in ${\mathbb{R}}^{n}$ is a discrete subgroup of rank $n$. In other words, it consists of the integral span of a basis of ${\mathbb{R}}^{n}$. Equivalently, a lattice is the image of ${\mathbb{Z}}^{n}$ under an invertible linear operator.

A sphere packing $\mathcal{P}$ is periodic if there exists a lattice $\Lambda$ such that $\mathcal{P}$ is invariant under translation by every element of $\Lambda$. In that case, the translational symmetry group of $\mathcal{P}$ must be a lattice, since it is clearly a discrete group, and $\mathcal{P}$ consists of finitely many orbits of this group. A lattice packing is a periodic packing in which the spheres form a single orbit under the translational symmetry group (i.e., their centers form a lattice, up to translation). See Figure 5 for an illustration.

It is not known whether periodic packings attain the optimal sphere packing density in each dimension, aside from the five cases in which the sphere packing problem has been solved. They certainly come arbitrarily close to the optimal density: given an optimal packing, one can approximate it by taking the spheres contained in a large box and repeating them periodically throughout space, and the density loss is negligible if the box is large enough. However, there seems to be no reason why periodic packings should reach the exact optimum, and perhaps they don’t in high dimensions.

By contrast, lattices probably do not even come arbitrarily close to the optimal packing density in high dimensions. For example, the best periodic packing known in ${\mathbb{R}}^{10}$ is more than 8% denser than the best lattice packing known. Seen in this light, the optimality of lattices in ${\mathbb{R}}^{8}$ and ${\mathbb{R}}^{24}$ is not a foregone conclusion, but rather an indication that sphere packing in these dimensions is particularly simple.

To compute the density of a lattice packing, it’s convenient to view the lattice as a tiling of space with parallelotopes (the $n$-dimensional analogue of parallelograms). Given a basis $v_{1},\dots,v_{n}$ for a lattice $\Lambda$, the parallelotope

is called the fundamental cell of $\Lambda$ with respect to this basis. Translating the fundamental cell by elements of $\Lambda$ tiles ${\mathbb{R}}^{n}$, as in Figure 5. From this perspective, a lattice sphere packing amounts to placing spheres at the vertices of such a tiling. On a global scale, there is one sphere for each copy of the fundamental cell. Thus, if the packing uses spheres of radius $r$ and has fundamental cell $C$, then its density is the ratio

Both factors in this ratio are easily computed if we are given $r$ and $C$. The volume of a fundamental cell is just the absolute value of the determinant of the corresponding lattice basis; we will write it as $\mathop{\textup{vol}}\mathopen{}({\mathbb{R}}^{n}/\Lambda)\mathclose{}$, the volume of the quotient torus, to avoid having to specify a basis. Computing the volume of a ball of radius $r$ in ${\mathbb{R}}^{n}$ is a multivariate calculus exercise, whose answer is

where of course $(n/2)!$ means $\Gamma(n/2+1)$ when $n$ is odd. We can therefore compute the density of any lattice packing explicitly. The density of a periodic packing is equally easy to compute: if the packing consists of $N$ translates of a lattice $\Lambda$ in ${\mathbb{R}}^{n}$ and uses spheres of radius $r$, then its density is

Of course the density of a packing depends on the radius of the spheres. Given a lattice with no radius specified, it is standard to use the largest radius that does not lead to overlap. The minimal vector length of a lattice $\Lambda$ is the length of the shortest nonzero vector in $\Lambda$, or equivalently the shortest distance between two distinct points in $\Lambda$. If the minimal vector length is $r$, then $r/2$ is the largest radius that yields a packing, since that is the radius at which neighboring spheres become tangent.

Many dimensions feature noteworthy sphere packings, but the $E_{8}$ root lattice in ${\mathbb{R}}^{8}$ and the Leech lattice in ${\mathbb{R}}^{24}$ are perhaps the most remarkable of all, with connections to exceptional structures across mathematics. In this section, we’ll construct $E_{8}$ and prove some of its basic properties. It was discovered by Korkine and Zolotareff in 1873, in the guise of a quadratic form they called $W_{8}$. We’ll give a construction much like Korkine and Zolotareff’s but more modern. The Leech lattice $\Lambda_{24}$, discovered by Leech in 1967, is similar in spirit, but more complicated. In lieu of constructing it, we will briefly summarize its properties.

To specify $E_{8}$, we just need to describe a lattice basis $v_{1},\dots,v_{8}$ in ${\mathbb{R}}^{8}$. Furthermore, only the relative positions of the basis vectors matter, so all we need to specify is their inner products with each other. All this information will be encoded by the Dynkin diagram

of $E_{8}$. In this diagram, the eight nodes correspond to the basis vectors, each of squared length $2$. The inner product between distinct vectors is $-1$ if the nodes are joined by an edge, and $0$ otherwise. Thus, if we number the nodes

then the Gram matrix of inner products for this basis is given by

Before we go further, we must address a fundamental question: how do we know there really are vectors $v_{1},\dots,v_{8}$ with these inner products? All we need is for the matrix in (3.1) to be symmetric and positive definite, and indeed it is, although it’s not obviously positive definite. That can be checked in several ways. We’ll take the pedestrian approach of observing that the characteristic polynomial of this matrix is

which clearly has no roots when $t<0$ because every term is then positive.

We can now define the $E_{8}$ root lattice to be the integral span of $v_{1},\dots,v_{8}$. We will use this definition to derive several fundamental properties of $E_{8}$. These properties will let us determine its packing density, and they will also be essential for Viazovska’s proof.

The $E_{8}$ lattice is an integral lattice, which means all the inner products between vectors in $E_{8}$ are integers. This follows immediately from the integrality of the inner products of the basis vectors $v_{1},\dots,v_{8}$. Even more importantly, $E_{8}$ is an even lattice, which means the squared length of every vector is an even integer. Specifically, for $m_{1},\dots,m_{8}\in{\mathbb{Z}}$ the vector $m_{1}v_{1}+\dots+m_{8}v_{8}$ has squared length

which is visibly even. Thus, the distances between distinct points in $E_{8}$ are all of the form $\sqrt{2k}$ with $k=1,2,\dots$, and in fact each of those distances does occur.

In particular, the distance between neighboring points in $E_{8}$ is $\sqrt{2}$, so we can form a packing with spheres of radius $\sqrt{2}/2$ and density

To compute the density of the $E_{8}$ packing, all we need to compute is $\mathop{\textup{vol}}\mathopen{}({\mathbb{R}}^{8}/E_{8})\mathclose{}$.

To compute this volume, recall that it’s the absolute value of the determinant of the basis matrix:

However, we can write the Gram matrix $\big{(}\langle v_{i},v_{j}\rangle\big{)}_{1\leq i,j\leq 8}$ as the product

of the basis matrix with its transpose, and thus

Computing the determinant of the matrix in (3.1) then shows that $\mathop{\textup{vol}}\mathopen{}({\mathbb{R}}^{8}/E_{8})\mathclose{}=1$. In other words, $E_{8}$ is a unimodular lattice.

Putting together our calculations, we have proved the following proposition:

Our calculations so far have led us to what turns out to be the densest sphere packing in ${\mathbb{R}}^{8}$, but it’s not obvious from this construction that $E_{8}$ is an especially interesting lattice. The $E_{8}$ lattice is in fact magnificently symmetrical, far more so than one might naively guess based on its lopsided Dynkin diagram. Its symmetry group is the $E_{8}$ Weyl group, which is generated by reflections in the hyperplanes orthogonal to each of $v_{1},\dots,v_{8}$. We will not make use of this group, but it’s important to keep in mind that the lattice itself is far more symmetrical than its definition. This is a common pattern when defining highly symmetrical objects.

Our density calculation for $E_{8}$ was based on its being an even unimodular lattice. In fact, $E_{8}$ is the unique even unimodular lattice in ${\mathbb{R}}^{8}$, up to orthogonal transformations. Even unimodular lattices exist only when the dimension is a multiple of eight, and they play a surprisingly large role in the theory of sphere packing.

The last property of $E_{8}$ we will need for Viazovska’s proof is that it is its own dual lattice, a concept we will define shortly. Given a lattice $\Lambda$ with basis $v_{1},\dots,v_{n}$, let $v_{1}^{*},\dots,v_{n}^{*}$ be the dual basis with respect to the usual inner product. In other words,

Then the dual lattice $\Lambda^{*}$ of $\Lambda$ is the lattice with basis $v_{1}^{*},\dots,v_{n}^{*}$. It is not difficult to check that $\Lambda^{*}$ is independent of the choice of basis for $\Lambda$; one basis-free way to characterize it is that

The self-duality $E_{8}^{*}=E_{8}$ is a consequence of the following lemma:

As mentioned above, the Leech lattice $\Lambda_{24}$ is similar to $E_{8}$ but more elaborate. It’s an even unimodular lattice in ${\mathbb{R}}^{24}$, but this time with no vectors of length $\sqrt{2}$, and it’s the unique lattice with these properties, up to orthogonal transformations. The nonzero vectors in $\Lambda_{24}$ have lengths $\sqrt{2k}$ for $k=2,3,\dots$, and of course $\Lambda_{24}^{*}=\Lambda_{24}$ because $\Lambda_{24}$ is integral and unimodular. One noteworthy property of $\Lambda_{24}$ is that it’s chiral: all of its symmetries are orientation-preserving, and the Leech lattice therefore occurs in left-handed and right-handed variants, which are mirror images of each other. (By contrast, the symmetry group of $E_{8}$ is generated by reflections, so $E_{8}$ is certainly not chiral.)

The sphere packing density of the Leech lattice is

which looks awfully low, but keep in mind that the optimal density decreases exponentially as a function of dimension. In fact, the density of the Leech lattice is remarkably high, as one can see from Figure 4 and Table 1. For comparison, the best density known in ${\mathbb{R}}^{23}$ is $0.001905\dots$, which is lower than the density of the Leech lattice, and this is the only case in which the density increases from one dimension to the next in Table 1.

The underlying technique used in Viazovska’s proof is linear programming bounds for the sphere packing density in ${\mathbb{R}}^{n}$. These upper bounds were developed by Cohn and Elkies [2], based on several decades of previous work initiated by Delsarte and extended by numerous mathematicians. In this approach to sphere packing, one uses auxiliary functions with certain properties to obtain density bounds. Viazovska’s breakthrough consists of a new technique for constructing these auxiliary functions, but before we turn to her proof let’s examine the general theory and review how the bounds work. We will see that the general bounds do not refer to special dimensions such as eight and twenty-four, which makes it all the more remarkable that they can be used to solve the sphere packing problem in these dimensions.

Linear programming bounds are based on harmonic analysis. That may sound surprising, since sphere packing is a problem in discrete geometry, which at first glance seems to have little to do with the continuous problems studied in harmonic analysis. However, there is a deep connection between these fields, because the Fourier transform is essential for understanding the action of the additive group ${\mathbb{R}}^{n}$ on itself by translation, so much so that one can’t truly understand lattices without harmonic analysis.

Define the Fourier transform $\widehat{f}$ of an integrable function $f\colon{\mathbb{R}}^{n}\to{\mathbb{R}}$ by

Fourier inversion tells us that if $\widehat{f}$ is integrable as well, then one can similarly recover $f$ from $\widehat{f}$:

almost everywhere. In other words, the Fourier transform gives the unique coefficients needed to express $f$ in terms of complex exponentials.

To avoid analytic technicalities, we will focus on Schwartz functions. Recall that $f\colon{\mathbb{R}}^{n}\to{\mathbb{R}}$ is a Schwartz function if $f$ is infinitely differentiable, $f(x)=O\big{(}(1+|x|)^{-k}\big{)}$ for all $k=1,2,\dots$, and the same holds for all the partial derivatives of $f$ (of every order). Schwartz functions behave particularly well, well enough to justify everything we’d like to do with them, and they are closed under the Fourier transform. We could get by with weaker hypotheses, but in fact Viazovska’s construction produces Schwartz functions, so we might as well focus on that case.

The significance of the Fourier transform in sphere packing is that it diagonalizes the operation of translation by any vector. Specifically, (4.1) implies that

which means that translating the input to the function $f$ by $t$ amounts to multiplying its Fourier transform $\widehat{f}(y)$ by $e^{2\pi i\langle t,y\rangle}$. Simultaneously diagonalizing all these translation operators makes the Fourier transform an ideal tool for studying periodic structures.

The key technical tool behind linear programming bounds is the Poisson summation formula, which expresses a duality between summing a function over a lattice and summing the Fourier transform over the dual lattice, as defined in (3.2). Poisson summation says that if $f$ is a Schwartz function, then

In other words, summing $\widehat{f}$ over $\Lambda^{*}$ is almost the same as summing $f$ over $\Lambda$, with the only difference being a factor of $\mathop{\textup{vol}}\mathopen{}({\mathbb{R}}^{n}/\Lambda)\mathclose{}$. When expressed in this form, Poisson summation looks mysterious, but it becomes far more transparent when written in the translated form

This equation reduces to (4.2) when $t=0$, and it has a simple proof. As a function of $t$, the left side of (4.3) is periodic modulo $\Lambda$, while the right side is its Fourier series. In particular, the right side uses exactly the complex exponentials $t\mapsto e^{2\pi i\langle y,t\rangle}$ that are periodic modulo $\Lambda$, namely those with $y\in\Lambda^{*}$ (as follows easily from (3.2)). Orthogonality let us compute the coefficient of such an exponential, and some manipulation yields $\widehat{f}(y)/\!\mathop{\textup{vol}}\mathopen{}({\mathbb{R}}^{n}/\Lambda)\mathclose{}$.

Now we can state and prove the linear programming bounds, which show how to convert a certain sort of auxiliary function into a sphere packing bound. Specifically, we will use functions $f\colon{\mathbb{R}}^{n}\to{\mathbb{R}}$ such that $f$ is eventually nonpositive (i.e., there exists a radius $r$ such that $f(x)\leq r$ for $|x|\geq r$) while $\widehat{f}$ is nonnegative everywhere.

The name “linear programming” refers to optimizing a linear function subject to linear constraints. The optimization problem of choosing $f$ so as to minimize $r$ can be rephrased as an infinite-dimensional linear program after a change of variables, but we will not adopt that perspective here.

This proof technique may look absurdly inefficient. We start with Poisson summation, which expresses a deep duality, and then we recklessly throw away all the nontrivial terms, leaving only the contributions from the origin. One practical justification is that we have little choice in the matter, since we don’t know what the other terms are (they all depend on the lattice). A deeper justification is that the omitted terms are generally small, and sometimes zero, so omitting them is not as bad as it sounds.

To apply Theorem 4.1, we must choose an auxiliary function $f$. The theorem then shows how to obtain a density bound from $f$, but it says nothing about how to choose $f$ so as to minimize $r$ and hence minimize the density bound. Sadly, optimizing the auxiliary function remains an unsolved problem, and the best possible choice of $f$ is known only when $n=1$, $8$, or $24$.

As a first step towards solving this problem, note that we can radially symmetrize $f$, so that $f(x)$ depends only on $|x|$, because all the constraints on $f$ are linear and rotationally invariant. Then $f$ is really a function of one radial variable, as is $\widehat{f}$. Functions of one variable feel like they should be tractable, but this optimization problem turns out to be impressively subtle.

If we can’t fully optimize the choice of $f$, then what can we do? Several explicit constructions are known, but in general we must resort to numerical computation. For this purpose, it’s convenient to use auxiliary functions of the form $f(x)=p(|x|^{2})e^{-\pi|x|^{2}}$, where $p$ is a polynomial. These functions are flexible enough to approximate arbitrary radial Schwartz functions, but simple enough to be tractable. Numerical optimization then yields a high-precision approximation to the linear programming bound, which is shown in Figure 8 and Table 2.

The most striking property of Figure 8 is that the upper and lower bounds in ${\mathbb{R}}^{n}$ seem to touch when $n=8$ or $24$. In other words, there should be magic auxiliary functions that solve the sphere packing problem in these dimensions, by achieving $r=\sqrt{2}$ in Theorem 4.1 when $n=8$ and $r=2$ when $n=24$. (These values of $r$ are the minimal vector lengths in $E_{8}$ and $\Lambda_{24}$, respectively.) This is exactly what has now been proved, and the proof simply amounts to constructing an appropriate auxiliary function. Linear programming bounds do not seem to be sharp for any other $n>2$, which makes these two cases truly remarkable.

The existence of these magic functions was conjectured by Cohn and Elkies [2] on the basis of numerical evidence and analogies with other problems in coding theory. Further evidence was obtained by Cohn and Kumar [3] in the course of proving that the Leech lattice is the densest lattice in ${\mathbb{R}}^{24}$, while Cohn and Miller [5] carried out an even more detailed study of the magic functions. These calculations left no doubt that the magic functions existed: one could compute them to fifty decimal places, plot them, approximate their roots and power series coefficients, etc. They were perfectly concrete and accessible functions, amenable to exploration and experimentation, which indeed uncovered various intriguing patterns. All that was missing was an existence proof.

However, proving existence was no easy matter. There was no sign of an explicit formula, or any other characterization that could lead to a proof. Instead, the magic functions seemed to come out of nowhere.

The fundamental difficulty is explaining where the magic comes from. One can optimize the auxiliary function in any dimension, but that will generally not produce a sharp bound for the packing density. Why should eight and twenty-four dimensions be any different? The numerical results show that the bound is nearly sharp in those dimensions, but why couldn’t it be exact for a hundred decimal places, followed by random noise? That’s not a plausible scenario for anyone with faith in the beauty of mathematics, but faith does not amount to a proof, and any proof must take advantage of special properties of these dimensions.

For comparison, the answer is far less nice in sixteen dimensions. By analogy with $r=\sqrt{2}$ when $n=8$ and $r=2$ when $n=24$, one might guess that $r=\sqrt{3}$ when $n=16$, but that bound cannot be achieved. Instead, numerical optimization seems to converge to $r^{2}=3.0252593116828820\dots$, which is close to $3$ but not equal to it. This number has not yet been identified exactly.

Despite the lack of an existence proof, the proof of Theorem 4.1 implicitly describes what the magic functions must look like:

As discussed in the previous section, without loss of generality we can assume that $f$ is a radial function, as is $\widehat{f}$. We know exactly where the roots of $f$ and $\widehat{f}$ should be, since $E_{8}=E_{8}^{*}$ with vector lengths $\sqrt{2k}$ for $k=1,2,\dots$, while $\Lambda_{24}=\Lambda_{24}^{*}$ with vector lengths $\sqrt{2k}$ for $k=2,3,\dots$. These roots should have order two, to avoid sign changes, except that the first root of $f$ should be a single root. See Figure 9 for a diagram.

Thus, our problem is simple to state: how can we construct a radial Schwartz function $f$ such that $f$ and $\widehat{f}$ have the desired roots and no others? Note that Poisson summation over $E_{8}$ or $\Lambda_{24}$ then implies that $f(0)=\widehat{f}(0)$, and flipping the sign of $f$ if necessary ensures that all the necessary inequalities hold.

Unfortunately it’s difficult to take advantage of this characterization. The problem is that it’s hard to control a function and its Fourier transform simultaneously: it’s easy to produce the desired roots in either one separately, but not at the same time. Our inability to control $f$ without losing control of $\widehat{f}$ is at the root of the Heisenberg uncertainty principle, and it’s a truly fundamental obstacle.

One natural way to approach this problem is to carry out numerical experiments. Cohn and Miller used functions of the form $f(x)=p(|x|^{2})e^{-\pi|x|^{2}}$ to approximate the magic functions, where $p$ is a polynomial chosen to force $f$ and $\widehat{f}$ to have many of the desired roots. Such an approximation can never be exact, since it has only finitely many roots, but it can come arbitrarily close to the truth. This investigation uncovered several noteworthy properties of the magic functions, which showed that they had unexpected structure. For example, if we normalize the magic functions $f_{8}$ and $f_{24}$ in $8$ and $24$ dimensions so that $f_{8}(0)=f_{24}(0)=1$, then Cohn and Miller conjectured that their second Taylor coefficients are rational:

If all the higher-order coefficients had been rational as well, then it would have opened the door to determining these functions exactly, but frustratingly it seems that the other coefficients are far more subtle and presumably irrational. The magic functions retained their mystery, and this Taylor series behavior went unexplained until the exact formulas for the magic functions were discovered.

Given the difficulty of controlling $f$ and $\widehat{f}$ simultaneously, one natural approach is to split them into eigenfunctions of the Fourier transform. By Fourier inversion, every radial function $f$ satisfies $\widehat{\widehat{f\,}}=f$. Thus, if we set $f_{+}=\big{(}f+\widehat{f}\,\big{)}/2$ and $f_{-}=\big{(}f-\widehat{f}\,\big{)}/2$, then $f=f_{+}+f_{-}$ with $\widehat{f}_{+}=f_{+}$ and $\widehat{f}_{-}=-f_{-}$. Because $f$ and $\widehat{f}$ vanish at the same points, they share these roots with $f_{+}$ and $f_{-}$. Our goal is therefore to construct radial eigenfunctions of the Fourier transform with prescribed roots. The advantage of this approach is that it conveniently separates into two distinct problems, namely constructing the $+1$ and $-1$ eigenfunctions, but these problems remain difficult.

Ever since the Cohn-Elkies paper in 2003, number theorists had hoped to construct the magic functions using modular forms. The reasoning is simple: modular forms are deep and mysterious functions connected with lattices, as are the magic functions, so wouldn’t it make sense for them to be related? Unfortunately, they are entirely different sorts of functions, with no clear connection between them. That’s where matters stood until Viazovska discovered a remarkable integral transform, which enabled her to construct the magic functions using modular forms. We’ll get there shortly, but first let’s briefly review how modular forms work.

We’ll start with some examples. Every lattice $\Lambda$ has a theta series $\Theta_{\Lambda}$, defined by

This series converges when $\operatorname{Im}z>0$, and it defines an analytic function on the upper half-plane $\mathfrak{h}=\{z\in{\mathbb{C}}:\operatorname{Im}z>0\}$. To motivate the definition, think of the theta series as a generating function, where the coefficient of $e^{\pi itz}$ counts the number of $x\in\Lambda$ with $|x|^{2}=t$. However, there’s one aspect not explained by the generating function interpretation: why write this function in terms of $e^{\pi iz}$? Doing so may at first look like a gratuitous nod to Fourier series, but it leads to an elegant transformation law based on applying Poisson summation to a Gaussian:

If we set $\Lambda=E_{8}$, then $\Lambda^{*}=E_{8}$ as well, and we find that

Furthermore, $E_{8}$ is an even lattice, and hence the Fourier series (6.1) implies that

These two symmetries are the most important properties of $\Theta_{E_{8}}$. For exactly the same reasons, the theta series of the Leech lattice $\Lambda_{24}$ satisfies

The mappings $z\mapsto z+1$ and $z\mapsto-1/z$ generate a discrete group of transformations of the upper half-plane, called the modular group. It turns out to be the same as the action of the group $\mathop{\textup{SL}}_{2}({\mathbb{Z}})$ on the upper half-plane by linear fractional transformations, but we will not need this fact except for naming purposes.

A modular form of weight $k$ for $\mathop{\textup{SL}}_{2}({\mathbb{Z}})$ is a holomorphic function $\varphi\colon\mathfrak{h}\to{\mathbb{C}}$ such that $\varphi(z+1)=\varphi(z)$ and $\varphi(-1/z)=z^{k}\varphi(z)$ for all $z\in\mathfrak{h}$, while $\varphi(z)$ remains bounded as $\operatorname{Im}z\to\infty$. (The latter condition is called being holomorphic at infinity, because it means the singularity there is removable.) It’s not hard to show that the weight of a nonzero modular form must be nonnegative and even, and the only modular forms of weight zero are the constant functions.

We have seen that $\Theta_{E_{8}}$ and $\Theta_{\Lambda_{24}}$ satisfy the transformation laws for modular forms of weight $4$ and $12$, respectively, and it is easy to check that they are holomorphic at infinity. Thus, these theta series are modular forms.

There are a number of other well-known modular forms. For example, the Eisenstein series $E_{k}$ defined by

is a modular form of weight $k$ for $\mathop{\textup{SL}}_{2}({\mathbb{Z}})$ whenever $k$ is an even integer greater than $2$ (while it vanishes when $k$ is odd). The proofs of the required identities $E_{k}(z+1)=E_{k}(z)$ and $E_{k}(-1/z)=z^{k}E_{k}(z)$ simply amount to rearranging the sum. Here $\zeta$ denotes the Riemann zeta function, and $2\zeta(k)$ is a normalizing factor. The advantage of this normalization is that it leads to the Fourier expansion