The Henri Poincaré Prize

For that reason I consider him as a great idealist. Certainly Faddeev had predecessors who shared the same point of view. He himself cites in that respect P.A.M. Dirac, H. Weyl and V.A. Fock. Our generation takes many things for granted. We know that Quantum Mechanics is a deformation of Classical Mechanics, we understand the importance of Hamiltonian Methods in that respect, we know that the functional integral is not just a fancy idea of Feynman but an important tool in Quantum Field Theory, that geometry plays a role in Quantum Field Theory along with the theory of Lie groups, that classical nonlinear equations admit non-trivial solutions which give rise to new particles after quantization, and hence it is not necessary for every particle to correspond to its own field. All that was taught to us, but for Faddeev this was a result of personal development. He had to understand all these matters himself and often in a hostile environment. That is why he understands them deeper than we do. Let me describe the main works of Faddeev. He started with the study of Quantum Mechanics in the framework of functional analysis as was usual at the time. The PhD thesis of Faddeev is devoted to the inverse problem for one-dimensional Schrödinger operator. Profound knowledge of the subject turned out, much later, to be of central importance in relation to the Korteweg-deVries equation. I think the main discovery of his early work was the recognition of the importance of functional determinants. I remember, at a later point, when I was surprised that the formulae for the form factors in integrable models are given by determinants, Faddeev told me: “Solution to any good problem is given by a determinant.” Certainly, this was a joke, but it is true that we find several remarkable determinants in Faddeev’s works. Then comes the three body problem with the famous Faddeev’s equations. This work combines an elegant original idea with very sophisticated techniques. Faddeev himself considered this work as a mathematical solution of a difficult physical problem. Actually its importance is much wider: all computer calculations needed for applications today are done using Faddeev’s equations. The sixties was a period of very diverse and successful scientific activity for Faddeev. It is difficult to establish an exact chronology because at that time he was working very actively in many different fields. For an ordinary person it would be impossible to deal with such different matters simultaneously. I have already said that I consider mathematical beauty as the main source of Faddeev’s inspiration in physics, but the opposite is also true: believing that physics is described by beautiful mathematics he naturally comes to the conclusion that a good problem in physics must provide new insight into pure mathematics. A realization of this idea is presented by his remarkable derivation of the Selberg trace formula by the methods of scattering theory. Determinants once again! The theory of automorphic functions is so far from his original area that once again he had to understand the subject by himself, and it is impressive how deep and clear this understanding is. Yet another work from the same period concerns the three-dimensional inverse problem. In all the years that followed very little was added to Faddeev’s work on the subject, and it remains a rare example of elegance in mathematical physics. Now comes the jewel of the scientific career of Faddeev: quantization of the Yang-Mills theory. I remember long ago C. Itzykson told me: “We were doing the usual perturbation theory, Faddeev taught us the functional integral.” I think this is the main discovery of Faddeev in Quantum Field Theory: the

functional integral and the measure of integration in it should be taken seriously. This discovery led him (with V.N. Popov) to the discovery of one more determinant which will surely stay in Quantum Field Theory forever. In the seventies Faddeev was one of the first to recognize the importance of the newly discovered solutions of non-linear PDE—solitons. He was not interested in applications to hydrodynamics, rather to Quantum Field Theory, being convinced that solitons would allow for the reduction of the number of fields in Lagrangians. The program which he developed with his colleagues and students was logically clear: to develop the Hamiltonian approach as the first step towards quantization, to find integrable relativistic models, to perform semi-classical quantization, to quantize exactly. All this was done during the seventies-eighties. As a result, unexpected connections have been found with works by H. Bethe, C.N. Yang, R. Baxter, and an entire new field of mathematics, the theory of Quantum Groups, appeared. In the early seventies Faddeev started to look for multidimensional solitons. He returned to this problem and to Yang-Mills theory in the nineties, and has been working in this direction up till the present time. I wish him many new achievements and I hope everybody will join me in echoing this sentiment. Congratulations Ludwig Dmitrievich!

David Ruelle

For his outstanding contributions to quantum field theory, both classical and quantum statistical mechanics, and dynamical systems theory.

Laudatio by Professor G. Gallavotti

Professor David Ruelle’s scientific carrier is remarkable for his various contributions and for the conceptual continuity of the development through them. He has been among the first to realize the relevance of a rigorous derivation of the properties of equilibrium Statistical Mechanics as an essential step towards understanding the theory of phase transitions. His work has been an important guide to the scientists who in the sixties were attempting accurate measurements of thermodynamic quantities, like critical exponents, in various statistical mechanics models using the newly available electronic computational tools in conjunction with the use of rigorous results for assessing the correctness and reliability of the computations. The treatise on Statistical Mechanics, 1969, has become a classic book and it is still the basis of the formation of the new generations of scientists interested in the basic aspects of the theory. He has written several other monographs which are widely known and used. His critical work on the structure of Equilibrium Statistical Mechanics led him to undertake in 1969 the analysis of the theory of turbulence. The first publication on the subject was the epoch making paper “On the nature of turbulence” in collaboration with Takens. The paper criticized the theory of Landau, based on the increasing complexity of quasi periodicity arising from successive bifurcations in the Navier Stokes equations. The main idea that only “generic” behavior should be relevant was a strong innovation at the time: this is amply proved by the hundreds of papers that followed on the subject, theoretical, numerical and experimental. The works making use of Ruelle’s ideas stem also, and perhaps mainly, from the innovative papers Ruelle wrote (and continues to write) after the mentioned one. There he developed and strongly stressed the role that dynamical systems ideas would be relevant and important in understanding chaotic phenomena. The impact on experimental works has been profound: one can say that after the first checks were performed, some by notoriously skeptical experimentalists, and produced the expected results we rapidly achieved, by the end of the seventies, a stage in which the “onset of turbulence” was so well understood that experiments dedicated to check

the so called “Ruelle-Takens” ideas on the onset of turbulence were no longer worth being performed as one would know what the result would be. The very fact that a study of the onset of turbulence was physically interesting was new at the time (the sixties). The ideas had been independently worked out by Lorenz, earlier (in a 1963 paper): this became clear almost immediately. However I think that Ruelle’s view, besides reviving the interest in Lorenz’ work, which had not been appreciated as it should have, were noticed by physicists and mathematicians alike, and perhaps had more impact, because they were more general and ambitious in scope and aimed at understanding from a fundamental viewpoint a fundamental problem. In 1973 he proposed that the probability distributions that describe turbulence be what is now called the “Sinai-Ruelle-Bowen” distribution. This was developed in a sequence of many technical papers and written explicitly only later in 1978. In my view this is the most original contribution of Ruelle: it has not been well understood for years although it has been quoted in impressively many works on chaos. It had impact mostly on numerical works, but it proposes a fundamental solution to one of the most outstanding theoretical questions: what is the analog of the Boltzmann-Gibbs distribution in non equilibrium statistical mechanics? His answer is a general one valid for chaotic systems, be them gases of atoms described by Newton’s laws or fluids described by Navier Stokes equations (or other fluid dynamics equations). Today the idea is still a continuous source of works both theoretical and experimental. Since the beginning of his work he has studied also problems concerning other fields like operator theory and operator algebras obtaining results remarkable for originality and depth: I mention here only his results on the LeeYang theorem on the location of the zeros of polynomials (a subject to which he continued to add new results and applications) and the Haag-Ruelle theory of scattering in relativistic quantum fields, very widely studied and applied, which is still today virtually the only foundation for relativistic scattering theory, employed in mathematical Physics, high energy phenomenology and theoretical Physics. In the last few years he has also provided important impulse to the development of non-equilibrium statistical mechanics: his work continues in this direction at the highest level. He has developed foundational papers for the theory of non-equilibrium Thermodynamics particularly with respect to the concept of entropy. The work of Ruelle is of mathematical nature: but it is an example of how important a conceptually rigorous and uncompromising approach can be fruitful and lead to progress in very applied fields like experimental fluid mechanics or numerical molecular dynamics simulations. His work is in the tradition of the 1800’s fundamental investigations in Physics. His work, books and papers, is always very careful, clear and polished: every word, however, is important and requires attention. The awarding of the prize recognizes the cultural influence that he has exercised in the last thirty years or so: and we are all here united in this recognition.

Edward Witten

For his work on string theory which laid down the foundation of this subject. His work has been most influential and inspiring also in mathematical subdisciplines like geometry and topology.

Laudatio by Professor A. Jaffe

Edward Witten is in the midst of an enormously productive career as a mathematical physicist. Born in 1951 in Baltimore, he began his undergraduate studies by majoring in history. Edward certainly had the opportunity for prior exposure to sophisticated physics as his father Louis is a noted expert on relativity and gravitation. After his undergraduate studies, Edward returned to physics, working with David Gross at Princeton, and receiving his doctorate in 1976.

Edward’s early work left an immediate impression on experts. He discovered a new class of instanton solutions to the classical Yang-Mills equations, very much a central subject at the time. He pioneered work on field theories with N-components and the associated “large-N limit” as N tends to infinity. Three years later as a Junior Fellow at Harvard he had already established a solid international reputation both in research and as a spell-binding lecturer. That year several major physics departments took the unusual step, at the time an extraordinary one, to attempt to recruit a young post-doctoral fellow to join their faculty as a full professor! At that point Edward returned to Princeton with Chiara Nappi, my post-doctoral fellow and Edward’s new wife. Edward has been in great demand ever since.

Edward already became well-known in his early work for having keen mathematical insights. He re-interpreted Morse theory in an original way and related the Atiyah-Singer index theorem to the concept of super-symmetry in physics. These ideas revolved about the classical formula expressing the Laplace-Beltrami operator in terms of the de Rham exterior derivative, = (d + d )2. This insight was interesting in its own right. But it inspired his applying the same ideas to study the index of infinite-dimensional Dirac operators D and the self-adjoint operator Q = D + D , known in physics as super-charges, related to the energy by the representation H = Q2 analogous to the formula for . This led to the name “Witten index” for the index of D, a terminology that many physicists still use.

In 1981 Witten also discovered an elegant approach to the positive energy theorem in classical relativity, proved in 1979 by Schoen and Yau. What developed as

Witten’s hallmark is the insight to relate a set of ideas in one field to an apparently unrelated set of ideas in a different field. In the case of the positive energy theorem, Witten again took inspiration from super-symmetry to relate the geometry of space-time to the theory of spin structures and to an identity due to Lichnerowicz. The paper by Witten framed the new proof in a conceptual structure that related it to old ideas and made the result immediately accessible to a wide variety of physicists and mathematicians. In 1986 Witten’s had a spectacular insight by giving a quantum-field theory interpretation to Vaughan Jones’ recently-discovered knot invariant. Witten showed that the Jones polynomial for a knot can be interpreted as the expectation of the parallel transport operator around the knot in a theory of quantum fields with a Chern-Simons action. This work set the stage for many other geometric invariants, including the Donaldson invariants, being regarded as partition functions or expectations in quantum field theory. In most of these cases, the mathematical foundations of the functional integral representations can still not be justified, but the insights and understanding of the picture will motivate work for many years in the future.

With the resurgence of “super-string theory” in 1984, Witten quickly became one of its leading exponents and one of its most original contributors. His 1987 monograph with Green and Schwarz became the standard reference in that subject. Later Witten unified the approach to string theory by showing that many alternative string theories could be regarded as different aspects of one grand theory.

Witten also pioneered the interpretation of symmetries related to the electromagnetic duality of Maxwell’s equations, and its generalization in field theory, gauge theory, and string theory. He pioneered the discovery of SL(2, Z) symmetry in physics, and brought concepts from number theory, as well as geometry, algebra, and representation theory centrally into physics.

In understanding Donaldson theory in 1995 Seiberg and Witten formulated the equations named after them which have provided so much insight into modern geometry. With the advent of this point of view and fueled by its rapid dissemination over the Internet, many geometers saw progress in their field proceed so rapidly that they could not hope to keep up.

Not only is Witten’s own work in the field of super-symmetry, string theory, M-theory, dualities and other symmetries of physics legend, but he has trained numerous students and postdoctoral coworkers who have come to play leading roles in string theory and other aspects of theoretical physics.

I could continue on and on about other insights and advances made or suggested by Edward Witten. But perhaps it is just as effective to mention that for all his mentioned and unmentioned work, Witten has already received many national and international honors and awards. These include the Alan Waterman award in 1986, the Fields Medal in 1990, the CMI Research Award in 2001, the U.S. National Medal of Science in 2002, and an honorary degree from Harvard University in 2005. Witten is a member of many honorary organizations, including the American Philosophical Society and the Royal Society. While Witten may not need any additional recognition, it is an especially great personal pleasure and honor, as one of the original founders of IAMP, to present Edward Witten to receive the Poincare´ prize in 2006.