Mathematics

Yakov Eliashberg

Simon Donaldson 

Yakov Eliashberg is one of the founders of symplectic and contact topology, a discipline originated as mathematical language for qualitative problems of classical mechanics, and having deep connections with modern physics. The emergence of symplectic and contact topology has been one of the most striking long-term advances in mathematical research over the past four decades. Eliashberg is among the main exponents of this development.

Professor Eliashberg was born in 1946 in Leningrad (now St. Petersburg), Russia. He received his doctoral degree in Leningrad University in 1972 under the direction of V.A. Rokhlin, and in the same year he joined Syktyvkar University in northern Soviet Union. Eliashberg’s route passed through the refusenik years in Leningrad (1980-1987) where he had to do software engineering in order to feed his family, and where he was virtually cut off from normal mathematical life. In 1988 he emigrated to the United States and in 1989 became a Professor at Stanford University. He is a Member of U.S. National Academy of Sciences. For his contributions, Eliashberg has received a number of prestigious awards, including the Guggenheim Fellowship in 1995, the Oswald Veblen Prize in 2001, the Heinz Hopf Prize in 2013 and the Crafoord Prize in 2016. Eliashberg is currently the Herald L. and Caroline L. Ritch Professor at Stanford University.

Eliashberg developed a highly ingenious and very visual combinatorial technique that led him to the first manifestation of symplectic rigidity: the group of symplectomorphisms is closed in the group of all diffeomorphisms in the uniform topology. This fundamental result, proved in a different way also by Gromov and called nowadays the Eliashberg-Gromov theorem, is considered as one of the wonders and cornerstones of symplectic topology. In a series of papers (1989-1992), Eliashberg introduced and explored a fundamental dichotomy “tight vs overtwisted” contact structure that shaped the face of modern contact topology. Using this dichotomy, he gave the complete classification of contact structures on

the 3-sphere (1992). In these papers Eliashberg laid foundations of modern contact topology and introduced mathematical language which is widely used by researchers in this rapidly developing field.

In a seminal 2000 paper Eliashberg (with Givental and Hofer) pioneered foundations of symplectic field theory, a powerful, rich and notoriously sophisticated algebraic structure behind Gromov’s pseudo-holomorphic curves. It had a huge impact and became one of the most central and exciting directions in symplectic and contact topology. It has led to a significant progress on numerous areas including topology of Lagrangian submanifolds and geometry and dynamics of contact transformations, and it exhibited surprising links with classical and quantum integrable systems.

In recent years (2013-2015), Eliashberg found a number of astonishing appearances of homotopy principle in symplectic and contact topology leading him to a solution of a number of outstanding open problems and leading to a “mentality shift” in the field. Before these developments the consensus among experts was that the symplectic world is governed by rigidity coming from Gromov’s theory of pseudo-holomorphic curves or, equivalently, by Morse theory on the loop spaces of symplectic manifolds. The current impression based on Eliashberg’s discoveries is that rigidity is just a drop in the ocean of flexible phenomena.

Professor Yakov Eliashberg is awarded the Wolf Prize for his foundational work on symplectic and contact topology changing the face of these fields, and for his ground-breaking contribution to homotopy principles for partial differential relations and to topological foundations of multi-dimensional complex analysis.

Simon Donaldson 

Yakov Eliashberg

Sir Simon Kirwan Donaldson (born 1957, Cambridge, U.K.) is an English mathematician known for his work on the topology of smooth (differentiable) four-dimensional manifolds and Donaldson–Thomas theory.

Donaldson’s passion of youth was sailing. Through this, he became interested in the design of boats, and in turn in mathematics. Donaldson gained a BA degree in mathematics from Pembroke College, Cambridge in 1979, and in 1980 began postgraduate work at Worcester College, Oxford.

As a graduate student, Donaldson made a spectacular discovery on the nature or 4-dimensional geometry and topology which is considered one of the great events of 20th century mathematics. He showed there are phenomena in 4-dlmenslons which have no counterpart in any other dimension. This was totally unexpected, running against the perceived wisdom of the time.

Not only did Donaldson make this discovery but he also produced new tools with which to study it, involving deep new ideas in global nonlinear analysis, topology, and algebraic geometry.

After gaining his DPhil degree from Oxford University in 1983, Donaldson was appointed a Junior Research Fellow at All Souls College, Oxford, he spent the academic year 1983–84 at the Institute for Advanced Study in Princeton, and returned to Oxford as Wallis Professor of Mathematics in 1985. After spending one year visiting Stanford University, he moved to Imperial College London in 1998. Donaldson is currently a permanent member of the Simons Center for Geometry and Physics at Stony Brook University and a Professor in Pure Mathematics at Imperial College London.

Donaldson’s work is remarkable in its reversal of the usual direction of ideas from mathematics being applied to solve problems in physics.

A trademark of Donaldson’s work is to use geometric ideas in infinite dimensions, and deep non-linear analysis, to give new ways to solve partial differential equations (PDE). In this way he used the Yang-Mills equations, which has its origin in quantum field theory, to solve problems in pure mathematics (Kähler manifolds) and changed our understanding of symplectic manifolds. These are the phase spaces of classical mechanics, and he has shown that large parts of the powerful theory of algebraic geometry can be extended to them.

Applying physics to problems or pure mathematics was a stunning reversal of the usual interaction between the subjects and has helped develop a new unification of the subjects over the last 20 years, resulting in great progress in both. His use of moduli (or parameter) spaces of solutions of physical equations – and the interpretation of this technique as a form of quantum field theory – is now pervasive throughout many branches of modem mathematics and physics as a way to produce “Donaldson-type Invariants” of geometries of all types. In the last 5 years he has been making great progress with special geometries crucial to string theory in dimensions six (“Donaldson-Thomas theory”), seven and eight.

Professor Simon Donaldson is awarded the Wolf Prize for his leadership in geometry in the last 35 years. His work has been a unique combination of novel ideas in global non-linear analysis, topology, algebraic geometry, and theoretical physics, following his fundamental work on 4-manifolds and gauge theory. Especially remarkable is his recent work on symplectic and Kähler geometry.

Jean Francois le Gall

Gregory Lawler

Jean-François Le Gall (born in 1959, France) is a French mathematician working in areas of probability theory such as Brownian motion, Lévy processes, superprocesses and their connections with partial differential equations, the Brownian snake, random trees, branching processes, stochastic coalescence and random planar maps. He received his Ph.D. in 1982 under the supervision of Marc Yor. He is currently professor at the University of Paris-Sud in Orsay and is a senior member of the Institut universitaire de France.

Le Gall’s generation has a great impact on Probability or demonstrated more power and beauty in their arguments. He has been at the forefront of Probability since 1983, when he established what still are the best results on pathwise uniqueness for one-dimensional stochastic differential equations. His current groundbreaking discoveries on the Brownian map ensure he remains at the cutting edge of the field today. He has received the 1986 Rollo Davidson Prize the 2005 Grand Prix Sophie Germain, French Academy of Science, the 2005 Fermat prize in mathematics and was elected to French academy of sciences, December 2013.

Jean-François Le Gall has made several deep and elegant contributions to the theory of stochastic processes. His work on the fine properties of Brownian motions solved many difficult problems, such as the characterization of sets visited multiple times and the behavior of the volume of its neighborhood – the Brownian sausage. Le Gall made groundbreaking advances in the theory of branching processes, which arise in many applications. In particular, his introduction of the Brownian snake and his studies of its properties revolutionized the theory of super-processes – generalizations of Markov processes to an evolving cloud of dying and splitting particles. He then used some of these tools for achieving a spectacular breakthrough in the mathematical understanding of 2D quantum gravity. Le Gall established the convergence of uniform planar maps to a canonical random metric object, the Brownian map, and showed that it almost surely has Hausdorff dimension 4 and is homeomorphic to the 2-spher.

Gregory Lawler

Jean Francois le Gall

Gregory Francis Lawler (born in 1955) is an American mathematician working in probability theory and best known for his work since 2000 on the Schramm–Loewner evolution. Gregory Lawler received his Ph.D. from Princeton University in 1979 under the supervision of Edward Nelson. He was on the faculty of Duke University from 1979 to 2001, of Cornell University from 2001 to 2006, and since 2006 is at the University of Chicago. Lawler received the 2006 SIAM George Pólya Prize, in 2012 he became a fellow of the American Mathematical Society and Member, National Academy of Sciences in 2013.

Lawler has made trailblazing contributions to the development of probability theory. He obtained outstanding results regarding a number of properties of Brownian motion, such as cover times, intersection exponents and dimensions of various subsets. Studying random curves, Lawler introduced a now classical model, the Loop-Erased Random Walk (LERW), and established many of its properties. While simple to define, it turned out to be of a fundamental nature, and was shown to be related to uniform spanning trees and dimer tilings. This work formed much of the foundation for a great number of spectacular breakthroughs, which followed Oded Schramm’s introduction of the SLE curves. Lawler, Schramm and Werner calculated Brownian intersection exponents, proved Mandelbrot’s conjecture that the Brownian frontier has Hausdorff dimension 4/3 and established that the LERW has a conformally invariant scaling limit. These results, in turn, paved the way for further exciting progress by Lawler and others.

Vladimir Drinfeld

Alexander Beilinson 

An “algebraic structure” is a set of objects, including the actions that can be performed on those objectas, that obey certain axioms. One of the roles of modern algebra is to research, in the most general and abstract way possible, the properties of various algebraic structures (including their objects), many of which are amazingly complicated.

Vladimir Drinfeld, born in Kharkov, Ukraine (1954), represented the USSR at the International Mathematical Olympiad at the age of 15 and won a gold medal. In the same year he also began his studies at the University of Moscow. Since the eighties he has been considered one of the world’s leading mathematicians. In 1990 he won the prestigious Fields Medal and in 2008 was elected to the National Academy of Sciences (USA). Drinfeld has contributed greatly to various branches of pure mathematics, mainly algebraic geometry, arithmetic geometry and the theory of representation – as well as mathematical physics. The mathematical objects named after him – the “Drinfeld Modules”, the “Drinfeld Chtoucas”, the “Drinfeld Upper Half Plane”, the “Drinfeld Associator”, and so many others that one of his endorsers jokingly said, “one could think that “Drinfeld” was an adjective, not the name of a person”.

In the seventies, Drinfeld began his work on the aforementioned “Langlands Program”, the ambitious program that aimed at unifying the fields of mathematics. This program was proposed by the American-Canadian mathematician Robert Langlands (winner of the Wolf Prize in 1996) and discovered for the first-time tight and direct links between different branches of mathematics. Numbers theory (the field based on arithmetic, “sums”), algebraic representation theory and another field called “automorphic forms”, (which is related to harmonic analysis and assists, for example, in the physical study of waves and frequencies). By means of a new geometrical object he developed, which is now called “Drinfeld Chtoucas”, Drinfeld succeeded in proving some of the connections that had been indicated by the Langlands Program. In the eighties he invented the concept of algebraic “Quantum Group”, which led to a profusion of developments and innovations not only in pure mathematics but also in mathematical physics (for example in statistical mechanics).

Drinfeld and Beilinson, together created a geometric model of algebraic theory that plays a key role in both field theory and physical string theory, thereby further strengthening the connections between abstract modern mathematics and physics. In 2004 they jointly published their work in a book that describes important algebraic structures used in quantum field theory, which is the theoretical basis for the particle physics of today. This publication has since become the basic reference book on this complex subject.

Alexander Beilinson 

Vladimir Drinfeld

An “algebraic structure” is a set of objects, including the actions that can be performed on those objectas, that obey certain axioms. One of the roles of modern algebra is to research, in the most general and abstract way possible, the properties of various algebraic structures (including their objects), many of which are amazingly complicated

Alexander Beilinson was born in Moscow (1957), won the Ostrowski Prize (together with a colleague) in 1999, for extraordinary achievements in mathematics, and in 2017 he was elected to the National Academy of Sciences of the United States. His outstanding achievements include proofs of the Kashdan-Lustig and Jantzen conjectures, which play a key role in the representation theory, the development of important conjectures (“Beilinson’s Conjectures”) in algebraic geometry, and a significant contribution to the interface between geometry and mathematical physics. The joint work of Beilinson and Vladimir Drinfeld on the Langlands Program – a woven fabric of theorems and conjectures designed to link key areas of mathematics – has led to impressive progress in implementing the program in important areas of physics, such as quantum field theory and string theory.

Drinfeld and Beilinson, together created a geometric model of algebraic theory that plays a key role in both field theory and physical string theory, thereby further strengthening the connections between abstract modern mathematics and physics. In 2004 they jointly published their work in a book that describes important algebraic structures used in quantum field theory, which is the theoretical basis for the particle physics of today. This publication has since become the basic reference book on this complex subject.

Richard Schoen

Charles Fefferman 

Richard Schoen (born in 1950) is one of the most recognized and outstanding mathematicians of our time. In the early years of his research career, he co-founded the then-new mathematical research field- geometric analysis. Since then, his vision, his insight, and his technical power led him to break countless barriers, open up numerous new research directions, and elevate our understanding of mathematical structure. Schoen is a leader in geometric analysis and its application to algebra, geometry, topology, differential equations, and mathematical physics.

Schoen’s fundamental contributions in several areas of geometric analysis fall into two main categories:

(a) the study of conformal geometry and scalar curvature, including applications to classical general relativity.

(b) the theory of minimal surfaces and harmonic maps, particularly the regularity theory of those.

The proof by Schoen (together with S.T. Yau) that energy is positive in General Relativity is a remarkable result, many people have tried to prove it without success, for half a century since Einstein invented his theory. What makes the problem so remarkable is that it is a fundamental statement, without which Einstein’s theory of Gravity would presumably not make much sense (at least from a quantum point of view), yet it is completely unclear why it is true. Neither Einstein nor any of his successors for half a century were able to clarify this. The theorem of Schoen and Yau is an unusual example in which a deep and rigorous mathematical theorem sheds light in an essential way on a central theory of modern physics.

Richard Schoen is awarded the Wolf Prize for being a pioneer and a driving force in geometric analysis. His work on the regularity of harmonic maps and minimal surfaces had a lasting impact on the field. His solution of the Yamabe problem is based on the discovery of a deep connection to general relativity. Through his work on geometric analysis, Schoen has contributed greatly to our understanding of the interrelation between partial differential equations and differential geometry. Many of the techniques he developed continue to influence the advance of non-linear analysis.

Charles Fefferman 

Richard Schoen

Charles Fefferman (born in 1949) evidenced an affinity for mathematics at an early age. As a 9-year-old in suburban Maryland, his interest in science fiction and rocketry led to a study of physics, and with it the desire to master the mathematics necessary to understand the physics textbooks he read. At age 12, he was being tutored by professors at the University of Maryland, where he matriculated at 14. He graduated in 1966 at the age of 17 with bachelor’s degrees in both mathematics and physics, earned with the highest distinction, having already published a paper on symbolic logic.

Fefferman earned his doctorate in mathematics in 1969 at Princeton University, supervised by Elias M. Stein, for a thesis entitled Inequalities for Strongly Regular Convolution Operators. After serving as a lecturer at Princeton for the 1969-1970 academic year, he took a position at the University of Chicago; one year later, in 1971, he was promoted to full professor there, making him the youngest full professor in the United States. In 1973 Fefferman returned to Princeton as a full professor, and was appointed the Herbert E. Jones ’43 Professor. He served as Chair of the Mathematics Department from 1999 to 2002.

Fefferman has been a dominant figure in analysis in the last 40 years, having made dramatic advances in a number of major new directions. It is no exaggeration to say that he is the most influential analyst of his generation. His major achievements include:

1. He revolutionized our understanding of fundamental parts or real analysis by proving the duality of BMO with Hardy space H¹, whose results nowadays have innumerable applications.
2. Fefferman made the first major breakthroughs in the areas related to Bochner-Riesz summability and restriction theorems, in finding a counter-example for the disc multiplier, and proving the first important positive results in this area.
3. Fefferman’s study of the Bergman kernel and mapping properties of strongly pseudo-convex domains is a landmark in the field of several complex variables and has also opened the way for important geometric consequences.
4. Fefferman obtained among the best results in the theory of pseudo-differential operators in the last 40 years:
   • Resolving with Beals a fundamental open problem of local stability.
   • Obtaining with Phong the optimal results for positive operators and sharp Garding inequalities.
   • Achieving the decisive results for second-order non-negative sub-elliptic operators.
5. In the last ten years he has obtained dramatic breakthroughs and important generalizations in the area of the Whitney extension problem. This fundamental work is contained in a series of works beginning with five papers in the Annals of Mathematics from 2005 to 2009, and including more recently collaborations with B. Klartag, A. Israel, and K. Luli. It deals with the general problem of given a set E in Rn and a function f0 on E, when (and how) does it extend to a function f on Rn that belongs to a given Banach space X of “smooth” functions (e.g. when X = Ck).

Charles Fefferman is awarded the Wolf Prize for making major contributions to several fields, including several complex variables, partial differential equations and subelliptic problems. He introduced new fundamental techniques into harmonic analysis and explored their application to a wide range of fields including fluid dynamics, spectral geometry and mathematical physics. This had a major impact on regularity questions for classical equations such as the Navier-Stokes equation and the Euler equation. He solved major problems related to the fine structure of solutions to partial differential equations.