Mathematics

Andrew J. Wiles 

Robert P. Langlands

Andrew J. Wiles has made spectacular contributions toward the resolution of long standing fundamental problems in number theory by introducing profound and novel methods. The problems that he has addressed on his own and jointly with others include; the Birch and Swinnerton-Dyer conjectures, the main conjecture of Iwasawa theory, and the Shimura-Taniyama-Weil conjecture. His work, to date, culminated in the proof of the celebrated Fermat’s last theorem. This problem is responsible for shaping much of number theory in the last two centuries.

Mikhael Gromov

Jacques Tits 

Professor Mikhael Gromov has introduced a large number of very original new concepts into the classical field of differential geometry which have led to the solution of a large number of important and apparently unsolvable problems. He has introduced new invariants linking the differential geometric structures with those of algebraic topology which have proved their value both in Gromov´s hands and in the hands of others. He succeeded, for example, in estimating the Betti numbers of Riemannian manifolds solely in terms of a lower bound on the sectional curvature. Much of his work makes use of concepts of the convergence of Riemannian manifolds. These daring ideas are also at the heart of his remarkable solution of the problem of groups of polynomial growth. This geometric approach together with an idea of E. Rips led him to the construction of the theory of hyperbolic groups which has revolutionized the theory of discrete groups. He has also made profound contributions to the global theory of partial differential equations and to the theory of symplectic manifolds, constructing invariants which, for the first time, give a description of the degree of symplectic rigidity. His works are characterized by their incomparable depth and startling originality and will continue to be an inspiration for geometers for many years to come.

Jacques Tits 

Mikhael Gromov

Professor Jacques Tits has made many fundamental contributions to the theory of groups and their interactions with geometry. He has developed the theory of buildings as a central organizing principle and powerful tool for an astonishingly wide range of problems in group theory and geometry. This is a geometric approach to group theory which applies to finite groups, to p-adic groups and to arithmetic groups and has been instrumental in many of the most important advances in the last twenty years. Conversely his work also makes use of group theory to make possible many profound contributions to geometry, both by Tits and others. Some of these were areas for which it was not originally designed, such as the theory of Riemannian spaces of rank exceeding 1. Tits ideas are now an essential ingredient in the arsenal of every geometer. Tits has also made many other important contributions to those parts of mathematics listed above, to the theory of Coxeter groups, of Lie groups and of Kac-Moody algebras. The work of Tits has enriched mathematics enormously and has opened the path for numerous further developments.

Lennart Carleson

John G. Thompson

Lennart Carleson (born in 1928, Sweden) earned his Ph.D. from Uppsala University in 1950. He conducted post-doctoral research at Harvard University, where he engaged in discussions about Fourier series and their convergence with Antoni Zygmund and Raphaël Salem, who were present during 1950 and 1951. He is a professor emeritus at Uppsala University, the Royal Institute of Technology in Stockholm, and the University of California, Los Angeles, Carleson has also held the position of director at the Mittag-Leffler Institute in Djursholm, outside Stockholm, from 1968 to 1984. Additionally, he served as the president of the International Mathematical Union between 1978 and 1982.

Professor Lennart Carleson’s fundamental contributions to Fourier analysis, complex analysis, quasiconformal mappings, and dynamical systems have clearly established his position as one of the greatest analysts of the twentieth century. His 1952 Acta paper on sets of uniqueness for various classes of functions was the breakthrough paper in that area. The 1958 and 1962 papers on interpolation and the Corona problem not only solved the Corona Conjecture but introduced a host of new methods and concepts (e.g. Carleson measures, the Corona construction, and the relations to interpolation). These concepts are now central to modern Fourier analysis as well as complex analysis in both one and several variables.

Carleson’s celebrated solution of the Lusin conjecture in 1965 gave a dazzling display of his technical mastery and proved the now famous result that the Fourier series of an L² function on [0,1] converges almost everywhere. In 1972 Carleson proved that in dimension two Bochner-Riesz means of any order are LP bounded, 4/3 ≤ p ≤ 4. The methods introduced are again of fundamental importance to this area of Fourier analysis.

In 1974 he proved that a quasiconformal selfmap of R³ can be extended to be quasiconformal in R4. The earlier known cases of R and R² can be solved by elementary arguments; the deep methods he introduced have now been modified so as to work in arbitrary dimension.

In 1984 Carleson and Benedicks introduced a new method to study chaotic behavior of the map 1 → ax², and in 1988 they extended this method in a tour de force to prove that the Henon map (x ,y) → (1+y –ax, bx) exhibits “strange attractors” for a nonempty (even positive measure) set of parameter values. This historic paper has opened an entire area in dynamical systems.

John G. Thompson

Lennart Carleson

John G. Thompson (born in 1932, USA) earned his Bachelor of Arts from Yale University in 1955 and completed his doctorate at the University of Chicago in 1959 under the guidance of Saunders Mac Lane. After a period on the mathematics faculty at the University of Chicago, he assumed the Rouse Ball Professorship in Mathematics at the University of Cambridge in 1970. Later, he transitioned to the Mathematics Department of the University of Florida, holding the position of Graduate Research Professor. Currently, Thompson is a professor emeritus of pure mathematics at the University of Cambridge and a professor of mathematics at the University of Florida.

Professor John G. Thompson’s work has changed the face of finite group theory. Already in his thesis he solved a long-standing conjecture reaching back to work of Frobenius at the turn of the century: if a finite group has a fixed-point-free automorphism of finite order, then the group is nilpotent. The solution was obtained by the introduction of novel and highly original ideas. He next turned his attention to the classification of the finite simple groups. The first astonishing achievement was his joint Mark with Walter Feit in which they prove that a finite non-abelian simple group must have even order. Thompson went on to classify the finite simple groups in which every soluble subgroup has a soluble normalizer. This work is a key element in the collective effort that led to the classification of finite simple groups.

In the late 1970’s he took up McKay’s remarkable observation, that there is a connection between the Fischer-Griess group and the modular j-function to formulate a series of conjectures relating modular functions and finite sporadic simple groups. These have now been verified and have led to deep and important questions which will occupy mathematicians for some time to come.

Also during this period he significantly contributed to coding theory and the theory of finite projective planes. The recent solution of the classical problem of the non-existence of a plane of order 10 owes much to his efforts.

During the past few years he has investigated the problem of constructing Galois groups over number fields, especially Q. The starting point here is Hilbert’s irreducibility Theorem and Thompson´s work may well be the most important advance since Hilbert’s time.

The penetrating power of Thompson’s genius is astonishing; his contributions to group theory and related subjects are of enduring significance.

Ilya Piatetski-Shapiro

Ennio De Giorgi 

Ilya Piatetski-Shapiro (born in 1929, Russia) developed a fascination for mathematics at the age of 10. His curiosity was ignited by the  unique beauty of negative numbers, a subject introduced to him by his father, a Ph.D. in chemical engineering. While still an undergraduate at Moscow University, Piatetski-Shapiro’s mathematical prowess earned him the Moscow Mathematical Society Prize for a Young Mathematician in 1952. The winning paper he presented contained a solution to the problem posed by the French analyst Raphaël Salem regarding sets of uniqueness of trigonometric series.

For almost 40 years Professor Ilya Piatetski-Shapiro has been making major contributions in mathematics by solving outstanding open problems and by introducing new ideas in the theory of automorphic functions and its connections with number theory, algebraic geometry and infinite dimensional representations of Lie groups. His work has been a major, often decisive factor in the enormous progress in this theory during the last three decades.

Among his main achievements are: the solution of Salem’s problem about the uniqueness of the expansion of a function into a trigonometric series; the example of a non symmetric homogeneous domain in dimension 4 answering Cartan´s question, and the complete classification (with E. Vinberg and G. Gindikin) of all bounded homogeneous domains; the solution of Torelli´s problem for K-3 surfaces (with I. Shafarevich); a solution of a special case of Selberg´s conjecture on unipotent elements, which paved the way for important advances in the theory of discrete groups, and many important results in the theory of automorphic functions, e.g., the extension of the theory to the general context of semi-simple Lie groups (with I. Gelfand), the general theory of arithmetic groups operating on bounded symmetric domains, the first “converse theorem” for GL(3), the construction of L-functions for automorphic representations for all the classical groups (with S. Rallis) and the proof of the existence of non arithmetic lattices in hyperbolic spaces of arbitrary large dimension (with M. Gromov).

Ennio De Giorgi 

Ilya Piatetski-Shapiro

The work of Professor Ennio De Giorgi is among the most important and creative accomplishments in the theory of partial differential equations and the calculus of variations. At the time he began his studies, mathematicians were not able to handle anything beyond second order nonlinear elliptic equations in two variables. In his first major break-through in 1957, De Giorgi proved that solutions of uniformly elliptic second order equations of divergence form with only measurable coefficients were Holder continuous. Probably his greatest contribution cue in 1960; this was a regularity theory for minimal hypersurfaces. Such surfaces arise as surfaces of smallest area spanning a given boundary. The proof required De Giorgi to develop his own version of what we now call geometric measure theory along with a related key compactness theorem. He was then able to conclude that a minimal hypersurface is analytic outside a closed subset of codimension at least two. Since then he and his school have settled litany of the outstanding problems in this area.

John W. Milnor 

Alberto P. Calderon

During his undergraduate years at Princeton University, John W. Milnor (born in 1931, USA)  was named a  Putnam Fellow in both 1949 and 1950. Notably, at the youthful age of 19, he successfully proved the Fáry–Milnor theorem. Upon completing his studies, Milnor earned an A.B. in mathematics in 1951, presenting a senior thesis entitled “Link groups,” which was supervised by Robert H. Fox. Remaining at Princeton for his graduate studies, Milnor continued his academic pursuit and obtained his Ph.D. in mathematics in 1954. His doctoral dissertation, titled “Isotopy of links,” was also conducted under the guidance of Fox.

The highly original discoveries of Professor John W. Milnor in geometry have exerted a major influence on the development of contemporary mathematics. The current state of the classification of topological, piecewise linear, and differentiable manifolds rests in large measure on his work in topology and algebra.

Milnor’s discovery of differentiable structures on S7 (the 7-dimensional sphere) which are exotic, i.e. different from the standard structure, came as a complete surprise and marked the beginning of differential topology. Later, in joint work with Kervaire, Milnor turned these structures (on any Sn) into a group which could then (in part) be computed; it turns out that there are over sixteen million distinct differentiable structures on S31 In his important work in algebraic geometry on singular points of complex hypersurfaces, exotic spheres are related to links around singularities. In the combinatorial direction, Milnor disproved the long-standing conjecture of algebraic topology known as the Hauptvermutung, by constructing spaces with two polyhedral structures that cannot have a common subdivision. This was based on an unexpected use of the previously known concept of torsion, which has since become, in its various algebraic and geometric versions, a basic tool.

These are just some highlights of Milnor’s impressive body of work. Beyond the research papers, a wealth of new results are contained in his books. These are famous for their clarity and elegance and remain a source of continuing inspiration.

Alberto P. Calderon

John W. Milnor 

Alberto P. Calderon (born in 1920, Argentina) commenced his academic journey at the University of Buenos Aires, focusing on engineering studies. Upon completing his degree in civil engineering in 1947, Calderon secured a position within the research laboratory of the geophysical division at YPF (Yacimientos Petrolíferos Fiscales), a state-owned oil company in Argentina. His work at the YPF Lab centered on exploring methods to determine the conductivity of a body by conducting electrical measurements at its boundary. Although he didn’t publish his findings until 1980 in a brief Brazilian paper, Calderon’s research marked an essential contribution to the field.Subsequently, Calderon transitioned to a role at the University of Buenos Aires. In 1948, Antoni Zygmund from the University of Chicago arrived at the university following an invitation by Alberto González Domínguez. Calderon was then designated as Zygmund’s assistant. Impressed by Calderon’s potential, Zygmund extended an invitation for Calderon to collaborate with him. In 1949, Calderon relocated to Chicago with the support of a Rockefeller Fellowship. Under the guidance of Marshall Stone’s encouragement to pursue a doctorate, Calderon utilized three recently published papers as his dissertation and earned his PhD in mathematics under Zygmund’s supervision in 1950.

The work of Professor Alberto P. Calderon has had a lasting impact on the shape of contemporary Fourier analysis and on its connections with real variables, complex analysis, and partial differential equations. In particular, his contributions to the theory of singular integral operators (SIO) have been decisive, both through the introduction of the sharpest technical tools into that theory and through its imaginative application to important problems in partial differential equations.

Together with his teacher, Antoni Zygmund, Calderon proved the famous theorem on the LP boundedness (1<P<∞) of SIO and introduced the important Calderon-Zygmund decomposition. Subsequently, he used SIO to establish the first general result on the uniqueness of the Cauchy problem. for higher order partial differential operators and systems. He then applied this machinery to obtain general results on local solvability for linear partial differential equations (independently of, and contemporaneously with, Hormander) and also gave the first general method of reducing elliptic boundary value problems for equations on the boundary. These papers provided the major stimulus for the development of pseudodifferential operators. At about the same time, he introduced the complex method of interpolation of operators in general banach spaces (contemporaneously with, J.L. Lions). with his celebrated theorem on the first commutator, Calderon initiated a new and fruitful attack on the theory of SIO, which reached a culmination in his paper on the Cauchy integral on Lipschitz curves, honored by the Bocher Prize. In this work, Calderon not only settled an important problem of long-standing in complex function theory but also apened up new vistas for research in classical analysis.