Fields Medallists' Lectures

The following sections are included:

• PREFACE

• RECIPIENTS OF FIELDS MEDALS

• CONTENTS

I was born on the eighteenth of April 1907 in Helsingfors, Finland. My father was a professor of mechanical engineering at the Polytechnical Institute. My mother died in childbirth when I was born.

At the time of my early childhood Finland was under Russian sovereignty, but with a certain degree of autonomy, sometimes observed and sometimes disregarded by the czar who was, by today's standards, a relatively benevolent despot. Civil servants, including professors, were able to enjoy a fairly high standard of living, a condition that was to change radically during World War I and the Russian revolution that followed…

In this work, I am no longer primarily interested in the problem of type although the connection with my earlier work is quite evident. My aim was to interpret the Nevanlinna theorems as geometric properties of covering surfaces. To set the stage, suppose first that 𝒲₀ is a compact surface, with or without boundary, and that 𝒲 is a ramified covering surface of 𝒲₀, which covers 𝒲₀ evenly N times, branch-points counted according to the local degree of the projection map. If the Euler characteristics of 𝒲_o and 𝒲 are ρ₀ and ρ, the Hurwitz relation reads ρ = Nρ₀ + 𝓌, where 𝓌 is the sum of the ramification orders. It was known that Nevanlinna's second main theorem was in some way connected with the Hurwitz relation, and I wanted to make this connection explicit…

The following sections are included:

• Historical remarks

• Beltrami coefficients

• Extremal length

• TeichmÜller's theorem

• TeichmÜller spaces

• Fuchsian and quasifuchsian groups

• The Bers representation

• Kleinian groups

• The zero area problem

• Several dimensions

• References

At a meeting of the organizing committee of the International Congress held in 1924 at Toronto the resolution was adopted that at each international mathematical congress two gold medals should be awarded, and in a memorandum the donor of the fund for the founding of the medals, the late Professor J. C. Fields, expressed the wish that the awards should be open to the whole world and added that, while the awards should be a recognition of work already done, it was at the same time intended to be an encouragement for further mathematical achievements. The funds for the Fields' medals were finally accepted by the International Congress in Zürich in 1932, and two Fields medals were for the first time awarded at the Congress in Oslo 1936 to Professor Ahlfors and Professor Douglas. And now, after a long period of fourteen years, the mathematicians meet again at an international congress, here in Harvard…

The following sections are included:

• Etudes et diplômes

• carrière

• Prix

Cet article ne contient presque pas de résultats nouveaux, mais donne, à partir de rien, et sans démonstrations, de nombreux résultats personnels du calcul infinitésimal stochastique. Je suis très heureux de le dédier à Jacques-Louis Lions.

On the three previous occasions on which Fields Medals have been presented, the addresses on the achievements of the recipients have been given either by the Chairman or by a member of the awarding Committee. On this occasion, Professor Siegel was to have spoken about the work of Dr Roth, but as he is unfortunately unable to be present the duty has devolved on me. It is a pleasant duty, in that it requires me to pay tribute to the work of a colleague and friend…

The following sections are included:

• Education

• Academic career

• Visiting appointments

• Medals

• Memberships

• Address

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René Thom was born in Montbéliard in 1923. He studied at the Ecole Normale Supérieure in Paris from 1943 to 1946 and then went to the University of Strasbourg where he is now “Professeur sans Chaire”. In Strasbourg, he prepared his thesis “Espaces fibrés en sphères et carrés de Steenrod”. He presented it in Paris in 1951 in order to get the degree of Docteur ès Sciences. In 1954 his paper “Quelques propriétés globales des variétés différentiables” appeared in Comm. Math. Helvet. 28. It was prepared from his thesis. There, one finds the foundations of the theory of cobordism. Today, four years later, one can say that, for a long time, only few events have so strongly influenced Topology and, through topology, other branches of mathematics as the advent of this work…

The following sections are included:

• Childhood: 1923–1940

• 1940 – The War

• ENS, 1943–1946

• Strasbourg. Entering the CNRS

• After 1958

Lars Hörmander was born 1931, studied at Lund University and is now professor of mathematics at Stockholm University. He has been given a Fields Medal for his outstanding work in the theory of partial differential equations. Before trying to describe some of his results I shall have to lead you through a somewhat lengthy introduction…

I was born on January 24, 1931, in a small fishing village on the southern coast of Sweden where my father was a teacher. After elementary school there and “re alskola” in a nearby town which could be reached daily by train I went to Lund to attend “gymnasium”, as my older brothers and sisters had done before me. I was more fortunate than they, for the principal was just starting an experiment which meant that three years were decreased to two with only three hours daily in school. This meant that I could mainly work on my own, with much greater freedom than the universities in Sweden offer today, and that suited me very well. I was also lucky to get an excellent and enthusiastic mathematics teacher who was a docent at the University of Lund. He encouraged me to start reading mathematics at the university level, and it was natural to follow his advice and go on to study mathematics at the University of Lund when I finished “gymnasium” in 1948…

The following sections are included:

• Introduction

• Pseudodifferential Operators and The Wave Front Set

• Fourier Intergral Operators

• Hypoellipticity

• Solvability and Propagation of Singularities

• Holmgren's Uniqueness Theorem

• Analytic Hypoellipticity and Propagation of Analytic Singularities

• Carleman Estimates

• Spectral Asymptotics

• References

The following sections are included:

• La K-théorie

• Le théorème de l'indice

• Formules de points fixes

• Réféences

Michael F. Atiyah was born in 1929. His father was a distinguished Lebanese and his mother came from a Scottish background. He was educated at Victoria College, Cairo, and Manchester Grammar School. After National Service, he went to Trinity College, Cambridge, where he obtained his BA and PhD degrees and continued with further research, finally as a University lecturer and Fellow of Pembroke College. In 1961 he moved to Oxford, initially appointed to a Readership, and later to the Savilian Professorship of Geometry. From 1969 he was Professor of Mathematics at the Institute for Advanced Study in Princeton, USA (where he had held a Commonwealth Fund Fellowship in 1955) until 1972 when he returned to Oxford as a Royal Society Research Professor and Fellow of St Catherine's College. He held this post until 1990 when he became Master of Trinity College, Cambridge, and Director of the new Isaac Newton Institute for Mathematical Sciences…

The following sections are included:

• Introduction

• The Classical Period
  • Algebraic curves
  • Algebraic surfaces
  • Higher dimensions

• Sheaf Theory

• K-Theory

• The Index Theorem

• References

• Postscript (1996)

Le premier travail scientifique de S. Smale est sa thèse de Ph.D. soutenue en 1956 à l'Université de Michigan (Ann Arbor). Faite sous la direction de Raoul Bott, elle témoigne déjà d'une éclatante maîtrise. Le résultat essentiel, maintenant bien connu, est le théorème du relèvement des homotopies des immersions d'une variété modulo une sous-variété. Etabli par des constructions géométriques raffinées, ce résultat témoignait chez son auteur de capacités d'intuition de tout premier ordre. Grâce à lui, on pouvait établir une conjecture — vieille alors d'une dizaine d'années — de C. Ehresmann sur la classification des immersions d'une variété dans une autre; il en résultait qu'il était possible, par une déformation régulière (c'est-à-dire sans sortir des immersions) de transformer le plongement canonique de la ≪ 2-sphère ≫ dans l'espace euclidien R³ à trois dimensions en un plongement antipodique; ce réultat n'alla pas sans soulever la curiosité des topologues, dont beaucoup s'ingénièrent à préciser cette déformation. Mais, la thèse de Smale donnait plus que cette curiosité, elle ouvrait une voie d'attaque dans tout un domaine de questions jusqu'alors inabordables, et tout un chapitre de Topologie Différentielle, l'étude des immersions et plongements d'une variété différentiable dans une autre, marqué par les travaux de M. Hirsh, Haefliger, etc., en est plus ou moins directement sorti…

The following sections are included:

• Honors

• Bibliography

Please refer to the full text

The theory of transcendental numbers, initiated by Liouville in 1844, has been enriched greatly in recent years. Among the relevant profound contributions are those of A. Baker, W. M. Schmidt and V. A. Sprindzuk. Their work moves in important directions which contrast with the traditional concentration on the deep problem of finding significant classes of functions assuming transcendental values for all non-zero algebraic values of the independent variable. Among these, Baker's have had the heaviest impact on other problems of mathematics. Perhaps the most significant of these impacts has been the application to diophantine equations. This theory, carrying a history of more than thousand years, was, until the early years of this century, little more than a collection of isolated problems subjected to ingenious ad hoc methods…

Professor Alan Baker obtained his first degree from University College, University of London and M.A., Ph.D. from Trinity College, Cambridge University. He has been Professor of Pure Mathematics at Cambridge University since 1974.

He was awarded a Fields Medal in 1970 at the International Congress of Mathematicians. In addition, he has a distinguished career with many awarded honours which include Fellow of Trinity College, Cambridge (1964), Adams Prize, Cambridge (1972), Fellow of the Royal Society (1973), First Turán Lecturer, Hungary (1978), Fellow, UCL (1979), Foreign Honorary Fellow of the Indian National Science Academy (1980)…

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The following sections are included:

• Introduction

• On the Logarithms of Algebraic Numbers

• Lower Bounds for Linear Forms

• On Imaginary Quadratic Fields with Class Number 1

• On the Representation of Integers by Binary Forms

• The Elliptic and Hyperelliptic Equations

• On the Weierstrass Elliptic Functions

• Concluding Remarks

• References

The following sections are included:

• Introduction

• Lower Bounds for Linear Forms

• Diophantine Equations

• Class Numbers

• Elliptic Functions

• Further Results and Problems

• References

Although the theory described in the preceding papers has progressed greatly in the intervening 25 years or so since they were published and modern surveys of the field would look very different, it became apparent on reading through them that I could not possibly reflect again the novelty of the results and the excitement of their discovery that is evident in these works. They have therefore been left as they stand and I give now a short note to indicate some of the main developments…

It gives me great pleasure to report on the work of Serge Novikov. For many years he has been generally acknowledged as one of the most outstanding workers in the fields of Geometric and Algebraic Topology. In this rapidly developing area, which has attracted many brilliant young mathematicians, Novikov is perhaps unique in demonstrating great originality and very powerful technique both in its geometric and algebraic aspects…

The following sections are included:

• Papers before 1971

• Papers after 1971

• Awards and Honors

• Students of Sergei Novikov

The history of mathematics and theoretical physics shows that the starting ideas of the best mathematical methods were discovered in the process of solving integrable models. Mathematical discoveries of the last twenty years will be especially discussed as by products of the famous integrable systems of the soliton and quantum theories.

It is a great pleasure for me to report on Mumford's work. However I feel there are many people more qualified than I to do this. I have consulted with some of them and would like to thank them all for their help, especially Oscar Zariski.

Mumford's major work has been a tremendously successful multi-pronged attack on problems of the existence and structure of varieties of moduli, that is, varieties whose points parametrize isomorphism classes of some type of geometric object. Besides this he has made several important contributions to the theory of algebraic surfaces. I shall begin by mentioning briefly some of the latter and then will devote most of this talk to a discussion of his work on moduli…

I was born in 1937 in an old English farm house. My father was British, a visionary with an international perspective, who started an experimental school in Tanzania based on the idea of appropriate technology and worked during my childhood for the newly created United Nations. He imbued me with old testament ideas of your obligation to fully use your skills and I learned science with greed. My mother came from a well-to-do New York family. I grew up on Long Island Sound, and went to Exeter and Harvard…

The following sections are included:

• Introduction

• Mathematical Formulation of the Problem

• Four Universal Transformations of Perceptual Signals

• Pattern Analysis Requires Pattern Synthesis

• Examples
  • Pyramids and wavelets
  • Segmentation as a free-boundary value problem
  • Random diffeomorphisms and template matching
  • The Stereo correspondence problem via minimum description length
  • Pattern Theory and Cognitive Information Processing

• References

The following sections are included:

• Discrete subgroups of Lie groups. The origin

• The noncocompact case. Selberg's conjecture

• The cocompact case. Rigidity

• Other results
  • S-arithmetic groups
  • “Abstract” isomorphisms
  • Normal subgroups
  • Action on trees

• Conclusion

• References

The following sections are included:

• Date of birth:

• Place of birth:

• Education:

• Scientific degrees:

• Academic career:

• Visiting positions held:

• Honors and awards:

The following sections are included:

• Analytic Number Theory Methods
  • Chowla's result and the distribution of values of positive definite quadratic forms at integer points
  • Diagonal forms in n≥5 variables
  • A modification of the theorem of Davenport and Heilbronn
  • Results for general quadratic forms

• Oppenheim Conjecture And Subgroup Actions On Homogeneous Spaces
  • Paper [CaS] of Cassels and Swinnerton-Dyer
  • Paper [CaS] (continuation)
  • Closures of orbits of orthogonal groups and integer solutions of quadratic inequalities
  • Quantitative results
  • Markov spectrum

• Unipotent Flows On Homogeneous Spaces
  • Growth properties of polynomials
  • Unipotent groups of linear transformations
  • Recurrence to compact sets
  • Finiteness of ergodic measures
  • General properties of minimal invariant sets
  • Topological limits of double cosets in algebraic groups
  • Proof of Theorem 2.3.4 (a)
  • Conjectures of Dani and Raghunathan. Uniform distribution
  • Ratner's results on unipotent flows

• References

The following sections are included:

• Classification of Type III Factors

• Structure Analysis of Type III Factors

• Classification of Automorphisms of the Hyperfinite Factor

• Classification of AFD Factors

• Other Works

• References

The following sections are included:

• Généralités

• Un Calcul Infinitésimal

• La Formule De L'indice Locale Et La Classe Fundamentale, Transverse

• La Notion De Variété Et Les Axiomes De La Géométrie

• La Géométrie Spectrale De L'espace Temps

• Références

Thurston has fantastic geometric insight and vision; his ideas have completely revolutionized the study of topology in 2 and 3 dimensions, and brought about a new and fruitful interplay between analysis, topology and geometry…

In 1982, when he was a second-year graduate student, Simon Donaldson proved a result [1] that stunned the mathematical world. Together with the important work of Michael Freedman (described by John Milnor), Donaldson's result implied that there are “exotic” 4-spaces, i.e., 4-dimensional differentiable manifolds which are topologically but not differentiably equivalent to the standard Euclidean 4-space R⁴. What makes this result so surprising is that n = 4 is the only value for which such exotic n-spaces exist. These exotic 4-spaces have the remarkable property that (unlike R⁴) they contain compact sets which cannot be contained inside any differentiably embedded 3-sphere!…

Please refer to the full text

The following sections are included:

• Reminiscences, 1980–83

• Gauge Theory and 4-Manifolds: Survey of Surveys

• Elliptic Functions and Modular Forms

• Stability and Kahler-Einstein Geometry

• References

Michael Freedman has not only proved the Poincaré hypothesis for 4-dimensional topological manifolds, thus characterizing the sphere S⁴, but has also given us classification theorems, easy to state and to use but difficult to prove, for much more general 4-manifolds. The simple nature of his results in the topological case must be contrasted with the extreme complications which are now known to occur in the study of differentiable and piecewise linear 4-manifolds…

The following sections are included:

• Personal Data

• Education

• Membership

• Honors and Awards

• Professional Experience

We prove an extension of the following result of Lubotzky and Magid on the rational cohomology of a nilpotent group G: b₁ < ∞ and G ⊗ ℚ ≠ 0, ℚ, ℚ² then b₂ > b²₁. Here the b_i are the rational Betti numbers of G ⊗ ℚ denotes the Malcev completion of G. In the extension, the bound is improved when we know that all relations of G all have at least a certain commutator length. As an application of the refined inequality, we show that each closed oriented 3-manifold falls into exactly one of the following classes: it is a rational homology 3-sphere, or it is a rational homology S¹ × S², or it has the rational homology of one of the oriented circle bundles over the torus (which are indexed by an Euler number n ε ℤ, e.g. n = 0 corresponds to the 3-torus) or it is of general type by which we mean that the rational lower central series of the fundamental group does not stabilize. In particular, any 3-manifold group which allows a maximal torsion-free nilpotent quotient admits a rational homology isomorphism to a torsion-free nilpotent group.

The following sections are included:

• The Index Theorem

• Knots and Links

• Statistical Mechanics

• Quantum Groups and Representations of Lie Algebras

• Dynkin Diagrams

• Concluding Remarks

• References

The following sections are included:

• Date and Place of Birth:

• Schooling

• Employment:

• Special Awards:

A theorem of J. Alexander [1] asserts that any tame oriented link in 3-space may be represented by a pair (b, n), where b is an element of the n-string braid group B_n. The link L is obtained by closing b, i.e., tying the top end of each string to the same position on the bottom of the braid as shown in Figure 1. The closed braid will be denoted b^{^}…

The following sections are included:

• Introduction

• Generalities
  • The global index
  • The local index
  • Examples

• The Basic Construction
  • Extending finite von Neumann algebras by subalgebras
  • Inclusions of complex semisimple algebras
  • Finite dimensional C^*-algebras
  • Two extensions; Goldman's theorem

• Possible Values of the Index
  • Certain algebras generated by projections
  • Restrictions on τ
  • Values of the index

• The Bratteli Diagram; the Relative Commutant
  • The Bratteli Diagram when τ≦¼
  • The Bratteli Diagrams
  • The relative commutant when τ≦¼

• References

The most profound and exciting development in algebraic geometry during the last decade or so was the Minimal Model Program or Mori's Program in connection with the classification problems of algebraic varieties of dimension three. Shigefumi Mori initiated the program with a decisively new and powerful technique, guided the general research direction with some good collaborators along the way, and finally finished up the program by himself overcoming the last difficulty. The program was constructive and the end result was more than an existence theorem of minimal models. Even just the existence theorem by itself was the most fundamental result toward the classification of general algebraic varieties in dimension 3 up to birational transformations. The constructive nature of the program, moreover, provided a way of factoring a general birational transformation of threefolds into elementary transformations (divisorial contractions, flips and flops) that could be explicitly describable in principle. Mori's theorems on algebraic threefolds were stunning and beautiful by the totally new features unimaginable by those algebraic geometers who had been working, probably very hard too, only in the traditional world of algebraic or complex-analytic surfaces. Three in dimension was in fact a quantum jump from two in algebraic geometry…

The following sections are included:

• Personal Data:

• Education:

• Professional Experience:

• Visiting Positions:

• Academic Honors:

The following sections are included:

• Introduction

• Birational Classification

• Surface Case

• The Extremal Ray Theory (The Minimal Model Theory)

• Applications of the Minimal Model Program (MMP) to 3-Folds

• Comments on the Proofs for 3-Folds

• Related Results

• References

The following sections are included:

• Morse Theory

• Index Theorem

• Rigidity Theorems

• Knots

The following sections are included:

• General

• Influential Papers

• The Positive Mass Conjecture

• Rigidity Theorems

• Topological Quantum Field Theories

• Conclusion

• References

The following sections are included:

• Born:

• Education:

• Employment:

• Honors and Awards:

• Publications:

The following sections are included:

• Introduction

• Physical Hilbert Spaces and Transition Amplitudes

• Axioms of Quantum Field Theory

• Construction of Quantum Field Theories

• Conclusion

• References

The following sections are included:

• Introduction

• The solution of the Λ_p, problem

• The converse of Santalo's inequality

• Ergodic theory

• Oscillatory integrals

• The circle maximal function

• Nonlinear partial differential equations

The following sections are included:

• Education:

• Appointments:

• Honors:

• Invited speaker:

• Editorial Board:

The following sections are included:

• Introduction

• Nonlinear Schrödinger Equations and Invariant Gibbs Measures

• Symplectic Capabilities, Squeezing and Growth of Higher Derivatives

• Persistency of Periodic and Quasi-periodic Solutions under Perturbation

• References

Pierre-Louis Lions has made unique contributions over the last fifteen years to mathematics. His contributions cover a variety of areas, from probability theory to partial differential equations (PDEs). Within the PDE area he has done several beautiful things in nonlinear equations. The choice of his problems has always been motivated by applications. Many of the problems in physics, engineering and economics when formulated in mathematical terms lead to nonlinear PDEs; these problems are often very hard. The nonlinearity makes each equation different. The work of Lions is important because he has developed techniques that, with variations, can be applied to classes of such problems. To say that something is nonlinear does not mean much; in fact it could even be linear. The entire class of nonlinear PDEs is therefore very extensive and one does not expect an all-inclusive theory. On the other hand, one does not want to treat each example differently and have a collection of unrelated techniques. It is thus extremely important to identify large classes that admit a unified treatment…

The following sections are included:

• ETUDES

• ACTIVITIES PROFESSIONNELLES

• DISTINCTIONS

• RESPONSABILITIES COLLECTIVES
  • Responsabilités administratives
  • Comité de rédaction de “séries”
  • Membre des comités de rédaction de:
  • Organisateur de colloques
  • Comités scientifiques
  • Comités de prix

• ENCADREMENT

The following sections are included:

• Introduction

• Boltzmann Equation
  • Existence and compactness results
  • Velocity averaging
  • Gain terms and Radon transforms

• Compressible Euler and Navier-Stokes Equations
  • 1D isentropic gas dynamics
  • Compensated compactness and Hardy spaces
  • Isentropic Navier–Stokes equations

• Perspectives, Trends, Problems and Methods

• References

The following sections are included:

• Curriculum

• La Théorie des Systèmes Dynamiques

• Conjugaison C^∞ à la Rotation

• Conditions Diophantiennes

• Les Cas R-analytique

• Réciproque du Théorème de Bruno

• Jeu de Cadres

• MLC: la Taille des Membres

• MLC: les Puzzles de Yoccoz

• Conjugaison C^∞

• Autres Travaux avec Palis

• Travaux avec Le Calvez et Raphael Douady

• Cocorico

The following sections are included:

• Introduction

• Quasiperiodic Dynamics

• Hyperbolic Dynamics

• Parameter Space

• References

The following sections are included:

• Introduction

• Jordan Algebras

• Burnside Problems

• Results

• Some Consequences

• Lie Algebras

• The Case of Exponent 2^k

• General References

The following sections are included:

• Date of Birth:

• Place of Birth:

• Education:

• Academic Positions:

• Other Scientific Activities:

• Awards:

• Invited Addresses:

In 1902 W. Burnside formulated his famous problems on periodic groups.

A group G is said to be periodic if for an arbitrary element g ε G there exists a natural number n = n(g) depending on g such that g = 1. A group G is said to be periodic of bounded exponent if there exists n ≥ 1 such that for an arbitrary element g ε G there holds gⁿ = 1. The minimal n with this property is called the exponent of G…