From a magical mathematical function - one solution to rule them all

Three years ago, Marina Vyazovskaya from the Swiss Federal Institute of Technology in Lausanne amazed mathematicians by discovering the densest way to pack spheres of the same size in eight- and 24-dimensional spaces (in the second case, with the help of four co-authors). And now they and co-authors have proved something even more surprising: configurations that solve the problem of densely packing spheres in the mentioned dimensions also solve an infinite number of other problems related to the best arrangement of points trying to avoid each other.

Points, for example, can denote an infinite set of electrons that repel each other and try to settle in the configuration with the lowest energy. Or, these points may indicate the centers of long, twisted polymers in solution, trying to arrange themselves so as not to collide with neighbors. There are many options for such problems, and it is not obvious that each will have the same solution. Mathematicians believe that in most dimensions this is very unlikely to be the case.

But the spaces, consisting of 8 and 24 dimensions, contain a special, very symmetrical configuration of points, which, as we now know, simultaneously solves all these different problems. In the language of mathematics, these two configurations are called "universally optimal."

This new large-scale discovery seriously summarizes the previous work of Vyazovskaya and her colleagues. “The fireworks didn't stop,” said Thomas Hales , a mathematician at the University of Pittsburgh, who proved in 1998 that the well-known pyramidal arrangement of oranges is the densest way to pack spheres in three-dimensional space.

Eight and 24 join one dimension in a small list of dimensions containing universally optimal configurations. On the two-dimensional plane there is a candidate for universal optimality - a grid of equilateral triangles - but there is no proof. In the three-dimensional world, a full zoo reigns: different configurations of points show different results in different circumstances, and for some problems mathematicians do not even have tolerable guesses about the best configuration.

“Change the measurement, or change the task a little, and the situation becomes incomprehensible,” said Richard Schwartz , a mathematician at Brown University in Providence. “I don’t know why the mathematical universe is so arranged.”

To prove universal optimality is much more difficult than to solve the problem of packing spheres. In particular, because universal optimality includes an infinite number of different tasks at once, but also because these tasks are more complicated in themselves. In the packing of spheres, each sphere is concerned only with its closest neighbors, but in a problem such as the distribution of electrons, each of the electrons interacts with all the others, regardless of the distance between them. “Even in the light of my early work, I did not expect that this universally optimal proof could be done,” said Hales.

“It's very, very impressive,” said Sylvia Serfati , a mathematician at New York University. “This thing is on par with major mathematical breakthroughs of the 19th century.”

Magic certificate

It may seem strange that dimensions 8 and 24 should behave differently than, say, dimensions 7, 18 or 25. But mathematicians have long known that dense packing of objects in space works differently in different dimensions. For example, consider a multidimensional sphere, defined simply as a set of points located at a fixed distance from the center. If we compare the volume of the sphere with the volume of the smallest cube that describes it, then the higher the dimension, the smaller the cube takes up the sphere. If you wanted to mail an eight-dimensional soccer ball in the smallest possible box, the ball would take up less than 2% of the box’s volume - and everything else would be stray empty space.

In each dimension larger than three, it is possible to create a configuration similar to the pyramid of oranges, and with increasing dimensions, the gaps between the spheres grow. Having reached the eighth dimension, we suddenly encounter the fact that in these spaces there is enough space to squeeze the spheres there. The result is an extremely symmetrical configuration called the E ₈ grille . In the 24th dimension, a Lich lattice arises in a similar way , when additional spheres can be crammed into the gaps, thus creating another well-known construction for packing spheres.

For reasons not completely understood by mathematicians, these two lattices suddenly appear either in one area of ​​mathematics or in another, from number theory and mathematical analysis to mathematical physics. “I don’t know one reason for all this,” said Henry Cohn of the Microsoft Research New England Institute in Cambridge, Massachusetts, one of the five authors of the work.

For more than ten years, mathematicians have had convincing numerical evidence that E ₈ and the Lich lattice are universally optimal in their dimensions - but until recently they had no idea how to prove this. Then in 2016, Vyazovskaya took the first step towards this, proving that these two lattices are the best ways to pack spheres.

And if the Hales proof for the three-dimensional case stretches over hundreds of pages and requires expensive calculations on the computer, the proof from Vyazovskaya for the case of E ₈ fits on 23 pages. The essence of her arguments is connected with the definition of a “magic” function (as mathematicians now call it), which gives out what Hales called a “certificate” for E ₈ for the best packing of spheres - this proof is difficult to obtain, but after its appearance it has instant convincingness. For example, if someone asked you if there is a real number x such that the polynomial x ² - 6x + 9 becomes negative, you could think about the answer. However, realizing that this polynomial is equivalent to (x - 3) ², you would immediately understand that the answer is no, because the square of a real number cannot be negative.

The method of searching for the magic function of Vyazovskaya proved to be powerful - and almost too powerful. The task of packing spheres concerns only the interaction of nearby points, but the Vyazovskaya approach seemed to work for long-range interactions, as is the case with remote electrons.

Uncertainty in higher dimensions

To show that the configuration of points in space is universally optimal, it is first necessary to determine this universality. There is no point configuration that is optimal for any purpose: for example, when the force of attraction acts on the points, the configuration with the lowest energy is not some lattice, but a massive heap in which all the points are in one place.

Vyazovskaya, Cohn and their colleagues limited the scope of their study to the universality of repulsive forces. More specifically, they considered monotonous forces, that is, those in which the repulsion becomes stronger when the points approach each other. This vast family includes many of the common forces of the physical world. This includes the power laws of the Universe - including the Coulomb law for electrically charged particles, and Gaussians, bell-based graphing functions that describe the behavior of entities with many independent repulsive parts, such as long polymers. The task of packing the spheres is on the outer edge of this universe: the requirement that the spheres do not intersect turns into infinitely strong repulsion when the distance between their centers is less than their diameter.

For any of these monotonous forces, the question arises - what will be the configuration with the lowest energy - the "ground state" - for an infinite set of particles? In 2006, Kon and Kumar developed a method for finding a smaller energy boundary of the ground state by comparing a function that describes energy with smaller “auxiliary” functions with very convenient properties. They found an infinite supply of auxiliary functions for each dimension, but did not know how to find the best auxiliary function.


Five authors of the new work: Henry Cohn, Abkhinav Kumar, Marina Vyazovskaya, Stephen Miller and Danilo Radchenko

In most measurements, the numerical limitations discovered by Kohn and Kumar do not resemble the energy of the best configuration possible. But in dimensions 8 and 24, the boundaries came stunningly close to the energy E ₈ and the Lich lattice for each repulsive force on which Kon and Kumar tested their method. It was natural to think about whether, for any repulsive force, there exists some ideal auxiliary function that would give a boundary exactly coinciding with the energy E ₈ or the Lich lattice. For the task of packing spheres, this was exactly what Vyazovskaya did three years ago: she discovered an ideal, “magical” auxiliary function, studying a class of functions called modular functions , whose special symmetry properties centuries ago made them an object of study.

When it came to other problems with repulsive points, for example, the problem with electrons, the researchers knew what properties any magic function should satisfy: at certain points, it should take special values, and its Fourier transform , which measures the natural frequencies of the function, should take special values ​​at other points. What they did not know was whether such a function existed.

It is usually quite simple to construct a function that does what you need at your favorite points, but it is surprisingly difficult to control both the function and its Fourier image at the same time. “When you start to make something do one of them, the other does something completely different from your desires,” Cohn said.

In fact, this finicky is nothing more than a disguised principle of uncertainty in physics. The Heisenberg uncertainty principle is that it says that the more you know about the location of a particle, the less you know about its momentum, and vice versa, is a special case of this general principle, because the particle's momentum wave is the Fourier transform of its location wave.

In the case of repulsive force in dimensions 8 or 24, Vyazovskaya put forward a bold hypothesis: the restrictions that the team wanted to impose on their magic function and its Fourier image are exactly on the border between the possible and the impossible. She suspected that if you add any more restrictions, there would be no such function; if you reduce the restrictions, then there may be many such functions. She suggested that in the situation that interested the team, there should be exactly one suitable function.

“I think this is one of Marina’s great features,” Cohn said. “She is very insightful, and also very brave.”

At that time, Kon was skeptical - Vyazovskaya’s hunch seemed too good to be true - but the team eventually proved it. They not only showed that for each repulsive force there is exactly one magic function, but also gave a recipe for its manufacture. As in the case of packing spheres, this design immediately gave optimality certificates for the E ₈ and the Lich lattice. “It's kind of a monumental result,” Schwartz said.

Triangular Grid

In addition to solving the problem of universal optimality, a new proof answers the urgent question that mathematicians have faced since Vyazovskaya solved the problem of packing spheres three years ago: where did its magical function come from? “I think many were puzzled,” Vyazovskaya said. “They asked: What is the meaning of this?”

In a new work, Vyazovskaya and her colleagues showed that the magic function of packing spheres is the first in a series of building blocks of modular forms that can be used to create magic functions for each repulsive force. “Now she has many brothers and sisters,” said Vyazovskaya.

It still seems wonderful to Kon that the picture worked out so well. “In mathematics, some things have to be achieved through perseverance and brute force,” he said. “And there are times, as it is now, as if mathematics wants something to happen.”

The next natural question is whether these methods can be adapted to prove universal optimality for the only remaining candidate: lattices of equilateral triangles on a two-dimensional plane. For mathematicians, the fact that no one was able to give evidence in such simple conditions is considered “a terrible shame for the entire community,” said Edward Saff , a mathematician at Vanderbilt University in Nashville.

Unlike E ₈and Lich lattices, a two-dimensional triangular lattice appears in different places in nature, from cell structures to the location of funnels in superconductors. Physicists already imply the optimality of this lattice in a wide range of contexts based on a mountain of experiments and simulations. But, Cohn says, no one has a conceptual explanation of why a triangular lattice should be universally optimal — something that, hopefully, will provide mathematical proof.

Dimension 2 is the only one, with the exception of 8 and 24, in which the numerical lower boundary of Kohn and Kumar works well. This clearly suggests that a magic function must exist in two dimensions. However, the command method for constructing magic functions can hardly be transferred to this new area: it strongly depends on the fact that the numbers denoting the distances between points in E ₈ and the Lich lattice behave especially well, which does not happen in two dimensions. So far, this dimension “seems to be beyond human capabilities,” Cohn said.

So far, mathematicians are celebrating their new insight associated with the strange worlds of 8- and 24-dimensional spaces. This, as Schwartz said, is “one of the best things that I will most likely see in my life.”