Mathematics / Interview

Vol. 5, NO. 2 / May 2020

In late 2019, Sergiu Klainerman was interviewed in Paris by Jean-Michel Kantor. The following is a transcript of their conversation.

Kantor: Can you tell us about your education in Romania and how it influenced your career in the United States?

Klainerman: At the time I studied mathematics at the university in Bucharest, Romania was isolated not only from the West, like all other communist countries, but also from Russian mathematics. This result of Nicolae Ceauşescu’s phony claim of independence made it very difficult for Romanian mathematicians to have access to and navigate the vast research literature of the time. Isolation led to poorly chosen subjects, heavy formalism, and a terrible lack of intuition. The typical attitude among my fellow students was to postpone real understanding while learning. “Climb the mountain first and the true meaning of the subject will be revealed at the top” was what we told each other as encouragement.

In spite of these enormous difficulties, there was among my colleagues at the university an enthusiasm for mathematics I have rarely seen anywhere else. This, more than anything else, explains why Romanian-trained mathematicians did so well once they left the country and were exposed to mathematics at the main universities in the US and western Europe. Isolation led me and a few other friends to study partial differential equations (PDEs) from Lars Hörmander’s books and articles, even as we had very little exposure to classical material. By the time I left Romania, at the age of twenty-five, I was terribly depressed by the fact that, while I could follow the arguments step by step, I had no sense of the motivation behind Hörmander’s theorems. It was in a first-year graduate course with Fritz John, at the Courant Institute in New York, that I had this incredible revelation that you can understand the motivation behind a result while you study it and not some time later when you reach the top. This experience, which was replicated in many other courses at Courant, more than anything else made me develop my taste for a practice of PDEs.

Your vision of PDEs gives a central role to physics. How do you connect with other domains of PDEs?

I feel that physics provides us with a remarkable source of great problems that are not only compelling but where there is no way to play with their formulation to make them easier to solve. But geometry and topology are also, of course, sources of great problems. One of the most impressive mathematical results in the last twenty years is the solution to the Poincaré conjecture using Hamilton’s Ricci flow, a huge achievement of geometric PDEs and a dramatic demonstration of how ideas originating in physics percolate in areas which seem to have nothing to do with them. How on earth does a heat flow, originating in Joseph Fourier’s study of heat conduction, have anything to do with the Poincaré conjecture?

Good PDE problems may also arise in biology or economics, but I am, to my shame, quite ignorant in these matters.

What do you consider the most important recent developments in PDEs?

There are three areas that I am most familiar with: geometric PDEs, general relativity, and fluid dynamics.

The last twenty years in geometric PDEs have been dominated by the work of Grigori Perelman. Though mathematicians, alas, have the habit of attaching only the name of the last contributor to a major development, his achievement is the culmination of the immense progress in geometric PDEs of the elliptic and parabolic types made in the last century. The introduction of the Ricci flow itself and first important results based on it are, of course, due to Richard Hamilton. The geometrization conjecture that put the Poincaré conjecture in a full classification setting of 3-compact manifolds was introduced by William Thurston. The techniques of dealing with nonlinear parabolic and elliptic equations is due to great mathematicians such as Sergei Natanovich Bernstein, Ennio de Giorgi, David Hilbert, Eberhard Hopf, John Nash, Louis Nirenberg, Aleksei Pogorelov, Henri Poincaré, Bernhard Riemann, Juliusz Schauder, Sergei Sobolev, Hermann Weyl, and many others throughout last century. The more recent blending of Riemannian geometry with PDEs was pioneered by people such as Thierry Aubin, Richard Schoen, Karen Uhlenbeck, and Shing-Tung Yau.

Though general relativity is, together with quantum mechanics, one of the pillars of modern physics and, arguably, the most mathematically sophisticated of all physical theories, it has been, with few notable exceptions (Yvonne Choquet-Bruhat, Jean Leray, André Lichnerowicz, Roger Penrose, etc.), mostly neglected by mathematicians until quite recently. Things have drastically changed since the proof of the positive mass theorem by Schoen and Yau in the early eighties and the proof of the stability of the Minkowski space by Demetrios Christodoulou and me somewhat later. Today there are two distinct directions of research in mathematical general relativity. The first concerns itself with Riemannian geometric issues related to the constraint equations, influenced by the work of Schoen and Yau. The second is centered around the problem of evolution, stimulated by my work with Christodoulou. Today mathematical general relativity is a rapidly developing field with an impressive list of results.

Fluid mechanics, or more broadly, continuum mechanics, unlike general relativity, has played a central role within the mathematics community for the entire last century. I used to think that progress on the main mathematical problems in fluids is extremely slow, but in fact I see, with the advent of an impressive list of young people, a renewed energy in the field. One of the important recent developments is the pioneering work in connection with the Onsager conjecture started with Camillo De Lellis and László Székelyhidi and continued by Tristan Buckmaster, Philip Isett, and Vlad Vicol. The conjecture, made by the mathematical physicist Lars Onsager, is intimately connected to the phenomenon of anomalous dissipation of energy, and used as an important hypothesis by Andrey Kolmogorov in his well-known theory of turbulence. More precisely the conjecture asserts that, below a certain level of regularity, solutions to the incompressible Euler equations may dissipate energy. This is in contrast to the fact that, above this level, energy is conserved and seems to have something to do with Kolmogorov’s hypothesis. Progress on the conjecture, which was finally proved by Isett, was based on convex integration, a method introduced by Nash in a completely different setting in relation to the isometric embedding problem in geometry.

Another recent and important development, under the name of inviscid damping, originates in the work of Clément Mouhot and Cédric Villani on Landau damping in plasma physics, work for which the latter got the Fields Medal in 2014.

Can you elaborate on general relativity, which seems to be your main research interest?

I will talk only about the problem of evolution where most progress was made in the last twenty years.

First I need to recall that the Einstein field equations are a system of nonlinear partial differential equations that connect the Lorentz metric of space with various matter fields acting in, and on, the space. In the simplest case, when no matter fields are present, the field equations take a simple geometric form: the Ricci curvature of the metric vanishes identically. It is called the Einstein vacuum equations (EVEs). Albert Einstein observed in 1916 that in special coordinates of the space, and ignoring all but linear terms, one can derive a simple wave equation verified by the components of the metric. This led Einstein to predict the existence of gravitational waves, even though he later expressed doubts about their existence. In any case, his derivation was the first manifestation of the hyperbolic character of the equations, a fact that was often misunderstood until the work of Choquet-Bruhat. 

The mathematical treatment of the problem of evolution started with that work and a companion result by Choquet-Bruhat and Robert Geroch which associates, to any initial data set, a unique maximal development. The study of mathematical general relativity can then be said to be that of understanding the global features of this maximal development.

The most remarkable feature of EVEs is the presence of an explicit two-parameter family (mass m and angular momentum a) of stationary solutions of the equation. They exhibit well-defined regions of space-time from which not only no physical object, or observer, can escape, but there is also no way by which the observer can communicate with the exterior. These are the famous black holes, the most dramatic prediction of general relativity.

The simplest solution of the family (a = m = 0) is the space-time of special relativity: the flat Minkowski space. Minkowski space admits no black hole region and is the space relative to which all other space-times are compared. 

The first nontrivial family (a = 0, m > 0) exhibiting black hole regions was discovered by Karl Schwarzschild in 1916, within months of the formulation of general relativity, though it took at least another half century to understand the solution and appreciate the full magnitude of the discovery. The manifest singularity of the solution was deeply puzzling and gave a lot of headaches to people, in particular Einstein himself.

The next step was made only much later, in 1963, by Roy Kerr who discovered the full range (0 ≤ a ≤ m) of rotating black holes solutions.

The discovery of black holes has not only revolutionized our understanding of the universe. It also gave mathematicians a monumental task: to test the physical reality of these solutions. This may seem nonsensical since physics tests the reality of its objects by experiments and observations and, as such, it needs mathematics to formulate the theory and make quantitative predictions, not to test it. The problem, in this case, is that black holes are by definition non-observable and thus no direct experiments are possible. Astrophysicists ascertain the presence of such objects through indirect observations and numerical experiments, but more caution is clearly required. One can rigorously check that the Kerr solutions have vanishing Ricci curvature, that is, their mathematical reality is undeniable. But to be real in a physical sense, they have to satisfy certain properties that can be neatly formulated in unambiguous mathematical language. Chief among them is the problem of stability, that is, to show that if the precise initial data corresponding to Kerr are perturbed a bit, the basic features of the corresponding solutions do not change much. I usually include two other problems among the mathematical tests of the physical reality of black holes. The first is the rigidity problem—that is, to show that there are no other stationary solutions of the EVEs besides the Kerr family. The second is the problem of collapse—that is, to understand the mathematical mechanism by which black holes can form in the first place. There has been recent progress on both problems. A beautiful, old result by Brandon Carter and David Robinson establishes rigidity under the additional assumption of axial symmetry. A later result, due to Stephen Hawking, replaces axial symmetry with an unrealistic analyticity assumption. In work with Alexandru Ionescu and Spyros Alexakis, we have been able to prove a more realistic perturbative rigidity result in the class of smooth solutions. We show that any stationary solution sufficiently close to a Kerr solution must itself be a Kerr solution. The full scope of the problem remains wide open. Regarding collapse, we owe to Christodoulou the first result on the formation of black holes for EVEs. It shows that a large, uniformly distributed pulse of gravitational waves can produce a trapped surface. This was somewhat counterintuitive, as people thought that only large concentrations of matter, outside the scope of the EVEs, could produce black holes. Together with Jonathan Luk and Igor Rodnianski, we have significantly strengthened Christodoulou’s result by showing, even more counterintuitively, that even a large pulse concentrated in only one direction can produce the same effect.

The first nontrivial result concerning stability was the proof of the nonlinear stability of the Minkowski space, obtained together with Christodoulou in 1993.

It took twenty-five more years to derive the first nonlinear stability result of the Schwarzschild family. With Jérémie Szeftel in 2018, we were able to prove that, under a certain class of restricted perturbations, the final state of the evolution of the perturbed data is another Schwarzschild solution. An important feature of this type of result, expected to be true in general for Kerr perturbations, is that one does not prove the stability of one member of the family but rather of the family itself. Note that the Minkowski space is exceptional in this respect; the final state of its perturbation has to be a Minkowski space. This means, to our peace of mind, that small disturbances cannot lead to black holes.

My result with Szeftel takes advantage of the remarkable progress made in the last twenty years by mathematicians, most prominently my Princeton colleagues Mihalis Dafermos and Rodnianski and their collaborators, on the stability of the linearized EVEs. This required the development of powerful tools to understand the decay properties of gravitational waves. The main new difficulty in the nonlinear setting was to identify the mass m of the final state as well as the coordinate, or gauge, condition relative to which convergence to this final state takes place.

To understand this latter difficulty, it helps to point out that the presence of a small amount of gravitational radiation, due to the perturbation, affects the black hole dynamically by changing its center of mass. Convergence to the final state requires decay with respect to a well-adapted center of mass frame which, like the final mass, has to be itself dynamically identified. This, the so-called recoil, which does not occur in linear theory, is intertwined with all the other difficulties of the problem and required us to make use of the full covariance properties of the equations. The proof, alas, takes over 900 pages. But, as it often happens if the work is deemed important, people will find shortcuts in the future.

Do you have any predictions for the next twenty years?

For one thing, I am now sure that a full stability result for the Kerr family will be proved in the next few years. Beyond this, I find it difficult to predict anything. Twenty years ago, who could have predicted the solution to the Poincaré and geometrization conjecture, the solution to the Onsager conjecture, or the progress made on the mathematical theory of black holes?

I would love to see progress with respect to one of my favorite problems: the issue of optimal well-posedness for EVEs and other strongly nonlinear equations. Though the problem was and still is out of reach, I have conjectured that well-posedness holds, in the context of EVEs, for initial data with bounded L2 curvature. The conjecture is now a theorem proved in collaboration with Rodnianski and Szeftel. But this is by no means the end of the road; in fact just the beginning. Finding an optimal scale invariant quantity which insures global well-posedness is, I think, one of the most pressing and difficult problems in general relativity intimately tied to the famous cosmic censorship conjectures proposed by Penrose. Similar problems can be formulated for other important quasilinear equations appearing in mathematical physics.


Sergiu Klainerman is Higgins Professor of Mathematics at Princeton University.

Jean-Michel Kantor is a mathematician at the Institut de Mathématiques de Jussieu in Paris.

More from this Contributor

More on Mathematics


Copyright © Inference 2024

ISSN #2576–4403