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08.09.2013

David Hilbert, probably the most prominent mathematician around the turn of the previous century, gave a talk at the 1900 International Mathematicians Congress in Paris in which he presented a list of outstanding math challenges for the new era. His full list included 23 problems. He explained to those in attendance: “For the close of a great epoch not only invites us to look back into the past but also directs our thoughts to the unknown future.” Hilbert expected that his 23 problems would be solved by the end of the 20^{th} century. Indeed, most have been solved, but a few have resisted all efforts to find solutions.

Under number 16 on Hilbert’s list appears the “problem of the topology of algebraic curves and surfaces.” This is really a two-part problem: The first involves closed ovals defined by algebraic equations and the second planar problem concerns closed trajectories of differential equations.

The solutions to planar differential equations are expressed as curved lines in a plane that do not intersect themselves. When such a line comes back around to its starting point, thus “closing,” it is called a cycle. Often the other solutions close to such a cycle spiral gradually inward toward it; the cycle is then called a “limit cycle,” and it becomes the limit of all the solutions around it. Such limit cycles are considered important not just for what they reveal about their own solutions, but because they can be used to reveal all the solutions adjoining them.

The second part of the problem had essentially been put forward earlier by Henri Poincaré, yet even today it is far from being solved. Despite the difficulty in resolving it, one can state the problem in somewhat simple terms: How many limit cycles can there be in a planar polynomial differential equation of a given order?”

A number of attempts to solve the Hilbert 16^{th} problem had been proposed over the years, but these were eventually found to contain fatal flaws. The most significant progress was made by the mathematicians Yulij Ilyashenko and Jean Ecalle, who had each independently proved in the early 1990s that for every planar differential equation there is a finite number of limit cycles. Though it was a tremendous breakthrough, this finding was still far from a complete solution of the problem. At some point, the mathematical community realized that the problem, in its entirety, was resisting their best efforts to solve it; and they turned to intermediate, “weakened” problems in hopes of gaining insight through dealing with these simpler cases.

One such relaxed problem, put forward by Ilyashenko, was to focus on equations that are small perturbations of a particular case known as Hamiltonian equations. Hamiltonian equations, among other things, are often used to describe the physics of mechanical systems. They have been used to show, for instance, that when such a system is in perfect energetic balance, it contains no limit cycles.

But following a small change in the system, that perfect energy balance can be broken; when this happens, limit cycles arise in the system as if appearing out of nowhere. Ilyashenko proposed to investigate the question of how many cycles can be created this way from a Hamiltonian system and thus to determine the upper limit to this number. Though this problem has been known under a number of different names, today it is most commonly referred to as the “infinitesimal Hilbert 16^{th} problem.”

Prof. Sergei Yakovenko, today head of the Weizmann Institute’s Mathematics Department, focused on a particular case of the infinitesimal Hilbert 16^{th} problem in his Masters’ thesis work under Ilyashenko at Moscow University. “Since then,” he says, “I have returned to the question again and again. It is, for me, a sort of beacon pointing me in the most intriguing direction.” Years later, together with his then student Dmitry Novikov – now, himself, a professor in the Institute’s Mathematics Department – Yakovenko succeeded in obtaining several intermediate results. But a comprehensive solution remained elusive.

Fast forward eight years: A visitor poking his head into Yakovenko’s office would have encountered the silence of deep thought – and three pairs of feet propped up on the low table in the middle of the room. These feet belonged to Yakovenko, Novikov and research student Gal Binyamini. Or, as Yakovenko describes this scene: “We were banging our heads against a solid brick wall, and suddenly we heard the sounds of cracks appearing. It was an exciting period in my career, and one of the most turbulent.”

Binyamini says: “One day, as I was taking my usual walk around campus, I thought of the work of another of Prof. Yakovenko’s students, Dr. Alexei Grigoriev. Incidentally, it occurred to me that it might be possible to ‘rescale’ or ‘stretch’ the arguments he put forward in his thesis in various and different ways.”

Rounds of rescaling enabled the team to uncover new properties of a very classical object – linear differential equations with polynomial coefficients – which had previously gone unnoticed. As they built on the information they obtained, the three Institute mathematicians were able to determine an upper limit that represents a full solution to the infinitesimal Hilbert 16^{th} problem.

“That,” says Yakovenko, “is still a far cry from the complete solution to Hilbert’s 16^{th} problem. But then, even the infinitesimal Hilbert 16^{th} problem had stood open for 50 years, until we managed to find an explicit upper bound for the number of newborn limit cycles and thus solve the infinitesimal version. It appears to be one of the most significant advances in this field in the past few decades.”

*Prof. Sergei Yakovenko is the incumbent of the Gershon Kekst Professorial Chair.*

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