Mathematics

Shing-Tung Yau

Dennis Sullivan 

Shing-Tung Yau (born in 1949, China) has linked partial differential equations, geometry, and mathematical physics in a fundamentally new way, decisively shaping the field of geometric analysis. He has developed new analytical tools to solve several difficult nonlinear partial differential equations, particularly those of the Monge-Ampere type, critical to progress in Riemannian, Kahler and algebraic geometry and in algebraic topology, that radically transformed these fields. The Calabi-Yau manifolds, as these are known, a particular class of Kahler manifolds, have become a cornerstone of string theory aimed at understanding how the action of physical forces in a high-dimensional space might ultimately lead to our four-dimensional world of space and time. Prof. Yau’s work on T-duality is an important ingredient for mirror symmetry, a fundamental problem at the interface of string theory and algebraic and symplectic geometry. While settling the positive mass and energy conjectures in general relativity, he also created powerful analytical tools, which have broad applications in the investigation of the global geometry of space-time.

Prof. Yau’s eigenvalue and heat kernel estimates on Riemannian manifolds, count among the most profound achievements of analysis on manifolds . He studied minimal surfaces,solving several classical problems, and then used his results, to create a novel approach to geometric topology. Prof. Yau has been exceptionally productive over several decades, with results radiating onto many areas of pure and applied mathematics and theoretical physics. In addition to his diverse and fundamental mathematical achievements, which have inspired generations of mathematicians, Prof. Yau has also had an enormous impact, worldwide, on mathematical research, through training an extraordinary number of graduate students and establishing several active mathematical research centers.

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Dennis Sullivan 

Shing-Tung Yau

Dennis Sullivan (born in 1941, USA) has made fundamental contributions in many areas, especially in algebraic topology and dynamical systems. His early work helped lay the foundations for the surgery theory approach to the classification of higher dimensional manifolds, most particularly, providing a complete classification of simply connected manifolds within a given homotopy type. He developed the notions of localization and completion in homotopy theory, and used this theory to prove the Adams Conjecture (also proved independently by Quillen). Profs. Sullivan and Quillen introduced the rational homotopy type of space.

Sullivan showed that it can be computed using a minimal model of an associated differential graded algebra. Sullivan’s ideas have had far-reaching influence and applications in algebraic topology. One of Sullivan’s most important contributions was to forge the new mathematical techniques needed to rigorously establish the predictions of Feigenbaum’s renormalization, as an explanation of the phenomenon of universality in dynamical systems. Sullivan’s “no wandering domains theorem” settled the classification of dynamics for iterated rational maps of the Riemann sphere, solving a 60 year-old conjecture by Fatou and Julia. His work generated a surge of activity, by introducing quasiconformal methods to the field and establishing an inspiring dictionary between rational maps and Kleinian groups of continuing interest. His rigidity theorem for Kleinian groups has important applications in Teichmuller theory and for Thurston´s geometrization program for 3-manifolds. His recent work on topological field theories and the formalism of string theory can be viewed as a byproduct of his quest for an ultimate understanding of the nature of space, and how it can be encoded in strange algebraic structures.

Sullivan’s work has been consistently innovative and inspirational. Beyond the solution of difficult outstanding problems, his work has generated important and active areas of research, pursued by many mathematicians.

Pierre R. Deligne

David B. Mumford

Phillip A. Griffiths

Central to modern algebraic geometry is the theory of moduli, i.e., variation of algebraic or analytic structure. This theory was traditionally mysterious and problematic. In critical special cases, i.e., curves, it made sense, i.e., the set of curves of genus greater than one had a natural algebraic structure. In dimensions greater than one, there was some sort of structure locally, but globally everything remained mysterious. The two main (and closely related) approaches to moduli were invariant theory on the one hand and periods of abelian integrals on the other. This key problem was tackled and greatly elucidated by Deligne, Griffiths, and Mumford.

Professor David B. Mumford revolutionized the algebraic approach through invariant theory, which he renamed “geometric invariant theory”. With this approach, he provided a complicated prescription for the construction of moduli in the algebraic case. As one application he proved that there were a set of equations defining the space of curves, with integer coefficients. Most important, Mumford showed that moduli spaces, though often very complicated, exist except for what, after his work, are well-understood exceptions. This framework is critical for the work by Griffiths and Deligne. Classically, the moduli space of curves was parameterized by using periods of the abelian integrals on them. Mathematicians, e.g., the Wolf Prize winner Andre Weil, have unsuccessfully tried to generalize the periods to higher dimensions.

Professor Phillip A. Griffiths had the fundamental insight that the Hodge filtration measured against the integer homology generalizes the classical periods of integrals. Moreover, he realized that the period mapping had a natural generalization as a map into a classifying space for variations of Hodge structure, with a new non-classical restriction imposed by the Kodaira-Spencer class action. This led to a great deal of work in complex differential geometry, e.g., his basic work with Deligne, John Morgan, and Dennis Sullivan on rational homotopy theory of compact Kaehler manifolds.

Building on Mumford´s and Griffiths´ work, Professor Pierre R. Deligne demonstrated how to extend the variation of Hodge theory to singular varieties. This advance, called mixed Hodge theory, allowed explicit calculation on the singular compactification of moduli spaces that came up in Mumford´s geometric invariant theory, which is called the Deligne-Mumford compactification. These ideas assisted Deligne in proving several other major results, e.g., the Riemann-Hilbert correspondence and the Weil conjectures.

Phillip A. Griffiths

David B. Mumford

Pierre R. Deligne

Central to modern algebraic geometry is the theory of moduli, i.e., variation of algebraic or analytic structure. This theory was traditionally mysterious and problematic. In critical special cases, i.e., curves, it made sense, i.e., the set of curves of genus greater than one had a natural algebraic structure. In dimensions greater than one, there was some sort of structure locally, but globally everything remained mysterious. The two main (and closely related) approaches to moduli were invariant theory on the one hand and periods of abelian integrals on the other. This key problem was tackled and greatly elucidated by Deligne, Griffiths, and Mumford.

Professor David B. Mumford revolutionized the algebraic approach through invariant theory, which he renamed “geometric invariant theory”. With this approach, he provided a complicated prescription for the construction of moduli in the algebraic case. As one application he proved that there were a set of equations defining the space of curves, with integer coefficients. Most important, Mumford showed that moduli spaces, though often very complicated, exist except for what, after his work, are well-understood exceptions. This framework is critical for the work by Griffiths and Deligne. Classically, the moduli space of curves was parameterized by using periods of the abelian integrals on them. Mathematicians, e.g., the Wolf Prize winner Andre Weil, have unsuccessfully tried to generalize the periods to higher dimensions.

Professor Phillip A. Griffiths had the fundamental insight that the Hodge filtration measured against the integer homology generalizes the classical periods of integrals. Moreover, he realized that the period mapping had a natural generalization as a map into a classifying space for variations of Hodge structure, with a new non-classical restriction imposed by the Kodaira-Spencer class action. This led to a great deal of work in complex differential geometry, e.g., his basic work with Deligne, John Morgan, and Dennis Sullivan on rational homotopy theory of compact Kaehler manifolds.

Building on Mumford’s and Griffiths’ work, Professor Pierre R. Deligne demonstrated how to extend the variation of Hodge theory to singular varieties. This advance, called mixed Hodge theory, allowed explicit calculation on the singular compactification of moduli spaces that came up in Mumford’s geometric invariant theory, which is called the Deligne-Mumford compactification. These ideas assisted Deligne in proving several other major results, e.g., the Riemann-Hilbert correspondence and the Weil conjectures.

David B. Mumford

Phillip A. Griffiths

Pierre R. Deligne

Central to modern algebraic geometry is the theory of moduli, i.e., variation of algebraic or analytic structure. This theory was traditionally mysterious and problematic. In critical special cases, i.e., curves, it made sense, i.e., the set of curves of genus greater than one had a natural algebraic structure. In dimensions greater than one, there was some sort of structure locally, but globally everything remained mysterious. The two main (and closely related) approaches to moduli were invariant theory on the one hand and periods of abelian integrals on the other. This key problem was tackled and greatly elucidated by Deligne, Griffiths, and Mumford.

Professor David B. Mumford revolutionized the algebraic approach through invariant theory, which he renamed ‘geometric invariant theory.’ With this approach, he provided a complicated prescription for the construction of moduli in the algebraic case. As one application he proved that there were a set of equations defining the space of curves, with integer coefficients. Most important, Mumford showed that moduli spaces, though often very complicated, exist except for what, after his work, are well-understood exceptions. This framework is critical for the work by Griffiths and Deligne. Classically, the moduli space of curves was parameterized by using periods of the abelian integrals on them. Mathematicians, e.g., the Wolf Prize winner Andre Weil, have unsuccessfully tried to generalize the periods to higher dimensions.

Professor Phillip A. Griffiths had the fundamental insight that the Hodge filtration measured against the integer homology generalizes the classical periods of integrals. Moreover, he realized that the period mapping had a natural generalization as a map into a classifying space for variations of Hodge structure, with a new non-classical restriction imposed by the Kodaira-Spencer class action. This led to a great deal of work in complex differential geometry, e.g., his basic work with Deligne, John Morgan, and Dennis Sullivan on rational homotopy theory of compact Kaehler manifolds.

Building on Mumford´s and Griffiths’ work, Professor Pierre R. Deligne demonstrated how to extend the variation of Hodge theory to singular varieties. This advance, called mixed Hodge theory, allowed explicit calculation on the singular compactification of moduli spaces that came up in Mumford´s geometric invariant theory, which is called the Deligne-Mumford compactification. These ideas assisted Deligne in proving several other major results, e.g., the Riemann-Hilbert correspondence and the Weil conjectures.

Harry Furstenberg 

Stephen Smale

Professor Harry Furstenberg is one of the great masters of probability theory, ergodic theory and topological dynamics. Among his contributions: the application of ergodic theoretic ideas to number theory and combinatorics and the application of probabilistic ideas to the theory of Lie groups and their discrete subgroups.

In probability theory he was a pioneer in studying products of random matrices and showing how their limiting behavior was intimately tied to deep structure theorems in Lie groups. This result has had a major influence on all subsequent work in this area–which has emerged as a major branch not only in probability, but also in statistical physics and other fields.

In topological dynamics, Furstenberg’s proof of the structure theorem for minimal distal flows, introduced radically new techniques and revolutionized the field. His theorem that the horocycle flow on surfaces of constant negative curvature is uniquely ergodic, has become a major part of the dynamical theory of Lie group actions. In his study of stochastic processes on homogenous spaces, he introduced stationary methods whose study led him to define what is now called the Furstenberg Boundary of a group. His analysis of the asymptotic behavior of random walks on groups, has had a lasting influence on subsequent work in this area, including the study of lattices in Lie groups and co-cycles of group actions.

In ergodic theory, Furstenberg developed the fundamental concept of dynamical embedding. This led him to spectacular applications in combinatorics, including a new proof of the Szemeredi Theorem on arithmetical progressions and far-reaching generalizations thereof.

Stephen Smale

Harry Furstenberg 

Professor Stephen Smale contributed greatly, in the late 50’s and early 60’s, to the development of differential topology, a field then in its infancy. His results of immersions of spheres in Euclidean spaces still intrigue mathematicians, as witnessed by recent films and pictures on his so-called “eversion” of the sphere. His proof of the Poincaré Conjecture for dimensions bigger or equal to 5 is one of the great mathematical achievements of the 20th Century. His h-cobordism theorem has become probably the most basic tool in differential geometry.

During the 60’s Smale reshaped the view of the world of dynamical systems. His theory of hyperbolic systems remains one of the main developments on the subject after Poincaré, and the mathematical foundations of the so-called “chaos-theory” are his work as well. In the early 60’s, Smale’s work contributed dramatically to change in the study of the topology and analysis of infinite-dimensional manifolds. This was achieved through his infinite-dimensional version of Morse’s critical point theory (known today as “Palais-Smale Theory”) and his infinite-dimensional version of Sard’s theorem.

In the 70’s Smale attention turned to mechanics and economics, to which he applied his ideas on topology and dynamics. For instance, his notion of “amended potential” in mechanics plays a key role in current developments in stability and bifurcation of relative equilibria. In economics, Smale applied an abstract theory of optimization for several functions, which he developed, to provide conditions for the existence of Pareto optima and to characterize this set of optima as a sub-manifold of diffeomorphic states to the set of Pareto equilibria. He also proved the existence of general equilibria under very weak assumptions and contributed to the development of algorithms for the computation of such equilibria.

It is this last activity that led Smale in the early 80’s to the longest segment of his career, his work on the theory of computation and computational mathematics. Against mainstream research on scientific computation, which focused on immediate solutions to concrete problems, Smale developed a theory of continuous computation and complexity (akin to that developed by computer scientists for discrete computations), and designed and analyzed algorithms for a number of specific problems. Some of these analyses constitute models for the use of deep mathematics in the study of numerical algorithms.

Gregory A. Margulis  

Sergei P. Novikov

At the center of Professor Gregory A. Margulis’s work lies his proof of the Selberg-Piatetskii-Shapiro Conjecture, affirming that lattices in higher rank Lie groups are arithmetic, a question whose origins date back to Poincaré. This was achieved by a remarkable tour de force, in which probabilistic ideas revolving around a non-commutative version of the ergodic theorem were combined with p-adic analysis and with algebraic geometric ideas showing that “rigidity” phenomena, earlier established by Margulis and others, could be formulated in such a way (“super-rigidity”) as to imply arithmeticity. This work displays stunning technical virtuosity and originality, with both algebraic and analytic methods. The work has subsequently reshaped the ergodic theory of general group actions on manifolds.

In a second tour de force, Margulis solved the 1929 Oppenheim Conjecture, stating that the set of values at integer points of an indefinite irrational non-degenerate quadratic form in ≥ 3 variables is dense in Rn. This had been reduced (by Rhagunathan) to a conjecture about unipotent flows on homogeneous spaces, proved by Margulis. This method transformed to this ergodic setting a family of questions till then investigated only in analytic number theory.

A third dramatic breakthrough came when Margulis showed that Kazhdan’s “Property T” (known to hold for rigid lattices) could be used in a single arithmetic lattice construction to solve two apparently unrelated problems. One was the solution to a problem posed by Rusiewicz, about finitely additive measures on spheres and Euclidean spaces. The other was the first explicit construction of infinite families of expander graphs of bounded degree, a problem of practical application in the design of efficient communication networks.

Margulis’ work is characterized by extraordinary depth, technical power, creative synthesis of ideas and methods from different areas of mathematics, and a grand architectural unity of its final form. Though his work addresses deep unsolved problems, his solutions are housed in new conceptual and methodological frameworks, of broad and enduring application. He is one of the mathematical giants of the last half a century.

Sergei P. Novikov

Gregory A. Margulis

Professor Sergei P. Novikov is awarded the Wolf Prize for his fundamental and pioneering contributions to topology and to mathematical physics. His early work in algebraic and differential topology includes such milestones as the calculation of cobordism rings and stable homotopy groups, proof of the topological invariance of rational Pontrjagin Classes, formulation of the “Novikov Conjecture” on higher signature invariants, and proof of the existence of closed leaves in two-dimensional foliations of the 3-sphere.

In the early 1970s, Novikov turned his attention to mathematical physics, initially contributing to general relativity and conductivity of metals. He constructed a global version of Morse Theory on manifolds and loop spaces that had novel applications to quantum field theory (multi-valued action functionals). His most significant achievements in mathematical physics flow from his introduction of algebraic-geometric methods to the study of completely integrable systems. These include a systematic study of finite-gap solutions of two-dimensional integrable systems, formulation of the equivalence of the classification of algebraic-geometric solutions of the KP equation with the conformal classification of Riemann surfaces, and work (with Krichever) on “almost commuting” operators, that appear in string theory and matrix models (“Krichever-Novikov algebras”, now widely used in physics).

Novikov made a fundamental and striking contribution to two separate fields in mathematics, while he is one of those rare mathematicians who brings deep, key mathematical ideas to bear on difficult pivotal problems of physics, in ways that are stunning and compelling for both mathematicians and physicists.

Mikio Sato

John T. Tate 

Professor Mikio Sato’s vision of “algebraic analysis” and mathematical physics initiated several fundamental branches of mathematics. Sato created the theory of hyperfunctions and invented microlocal analysis, that allowed for a description of the structure of singularities of (hyper) functions on cotangent bundles. Hyperfunctions, together with integral Fourier operators, have become a major tool in linear partial differential equations. Along with his students, Sato developed holonomic quantum field theory, providing a far-reaching extension of the mathematical formalism underlying the two-dimensional Ising model and introduced along the way the famous tau functions. Sato provided a unified geometric description of soliton equations in the context of tau functions and infinite dimensional Grassmann manifolds. This was extended by his followers to other classes of equations, including self dual Yang-Mills and Einstein equations.

Sato has generously shared his ideas with young mathematicians and has created a flourishing school of algebraic analysis in Japan.