From the magic math function - one solution to rule them all

Three years ago, Marina Vyazovskaya from the Swiss Federal Institute of Technology in Lausanne amazed mathematicians, finding the most dense way of packing 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 their coauthors have proved something even more amazing: the configurations that solve the problem of dense packing of spheres in the mentioned dimensions also solve an infinite number of other problems associated with the best location of points trying to avoid each other.

Dots, 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 denote the centers of long, twisted polymers in solution, trying to settle down so as not to collide with neighbors. There are many variants of such problems, and it is not obvious that for each there will be the same solution. Mathematicians believe that in most dimensions it is very unlikely to be so.

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."
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This new large-scale discovery seriously summarizes the previous work of Vyazovskaya and her colleagues. “Fireworks did not stop,” said Thomas Heyls , 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 are attached to one dimension in a small list of dimensions containing universally optimal configurations. On a two-dimensional plane there is a candidate for universal optimality - a grid of equilateral triangles - but there is no proof. A full zoo reigns in the three-dimensional world: 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.

“You change the dimension, or you change the task a little, and the situation becomes incomprehensible,” said Richard Schwartz , a mathematician at Brown University in Providence. "I do not know why the mathematical universe is so arranged."

Proving universal optimality is much more difficult than solving 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 in themselves more difficult. In the packing of spheres, each sphere is disturbed only by its nearest neighbors, but in a problem like the distribution of electrons, each of the electrons interacts with all the others, regardless of the distance between them. “Even in the light of early work, I did not expect that this universally optimal proof could be done,” said Heils.

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

Magic certificate

It may seem strange that measurements 8 and 24 should not behave like, say, measurements 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 a sphere with the volume of the smallest cube describing it, then the higher the dimension, the smaller the cube is occupied by the sphere. If you wanted to mail an eight-dimensional soccer ball in the smallest possible box, the ball would take less than 2% of the box volume - and everything else would be a parasitic empty space.

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

For reasons not completely understood by mathematicians, these two grids unexpectedly arise in one area of ​​mathematics, then in another, from number theory and mathematical analysis to mathematical physics. “I don’t know one reason for all of this,” said Henry Cohn of the Microsoft Research New England Institute in Cambridge, Mass., One of five authors of the paper.

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

And if the Heils proof for a three-dimensional case is stretched over hundreds of pages and requires expensive computations on a computer, the proof from Vyazovskaya for the case of E ₈ fits into 23 pages. The essence of her arguments is related to the definition of the “magic” function (as mathematicians now call it), which gives what Heils called the “certificate” for E ₈ for the best packaging of spheres - this proof is difficult to get, but after its appearance it has instant convincingness. For example, if someone asked you if there is a real number x, at which 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 search method for the Vyazovsk magical function turned out 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 configuration of points that is optimal for any purpose: for example, when a force acts on a point, the configuration with the lowest energy is not some kind of lattice, but a massive handful in which all points are in one place.

Vyazovskaya, Kohn, and their colleagues restricted the field of study to the universality of repulsive forces. More specifically, they considered monotonic forces, that is, those whose repulsion becomes stronger as the points approach each other. This extended family includes many common forces of the physical world. This includes the power laws of the Universe - including the Coulomb law for electrically charged particles, and Gaussians, functions with bell-shaped graphs describing the behavior of entities with many independent repelling parts, such as long polymers. The task of packing spheres is on the outer edge of this universe: the requirement that the spheres do not intersect turns into an infinitely strong repulsion in the case when the distance between their centers are 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, Kohn and Kumar developed a method for finding a smaller boundary for the energy of the ground state through comparing a function describing 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, Abhinav Kumar, Marina Vyazovskaya, Stephen Miller and Danilo Radchenko

In most measurements, the numerical constraints detected by Kohn and Kumar remind little of the energy of the best configuration possible. But in measurements 8 and 24, the boundaries came amazingly close to the energy of E ₈ and the Litch lattice for each repulsive force at which Kohn and Kumar tried their method. It was natural to think about whether for any repulsive force there is some ideal auxiliary function that would give a boundary that exactly coincides with the energy E ₈ or the Litch lattice. For the problem of packing spheres, this is exactly what Vyazovskaya did three years ago: she discovered an ideal, “magic” auxiliary function, studying a class of functions called modular functions , whose special symmetry properties a century ago made them an object of study.

When it came to other tasks 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 - measuring the natural frequencies of the function - should 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 transform at the same time. “When you start forcing one to do something, the other does something completely different from your desires,” said Cohn.

In fact, this fastidiousness is nothing more than a disguised principle of uncertainty in physics. Heisenberg's uncertainty principle - stating that the more you know about a particle's location, the less you know about its impulse, and vice versa is a special case of this general principle, since a particle's impulse wave is a Fourier transform of its positional wave.

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

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

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

Triangular grid

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

In the 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 a lot of little brothers and little sisters,” said Vyazovskaya.

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

The next natural question is whether it is possible to adapt these methods to the proof of universal optimality for the only remaining candidate: a lattice of equilateral triangles on a two-dimensional plane. For mathematicians, the fact that no one could give proof in such simple conditions is considered “a terrible disgrace for the whole community,” said Edward Saff , a mathematician at Vanderbilt University in Nashville.

In contrast to E ₈ and the Lich lattice, a two-dimensional triangular lattice appears in various places in nature, from honeycomb 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 for why a triangular lattice should be universally optimal — something that, hopefully, will give a 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 indicates that there must be a magic function 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 distance between points in E ₈ and the Lich lattice behave particularly well, which does not occur in two dimensions. So far, this dimension “appears to be beyond human capabilities,” said Kohn.

So far, mathematicians are celebrating their new insight related to 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 for my life."