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Proceedings of the International Congress of Mathematicians, Madrid 2006
Edited by: Marta Sanz-Solé and Javier Soria, University of Barcelona, Spain, Juan Luis Varona, University of La Rioja, Logroño, Spain, and Joan Verdera, University Autònoma, Barcelona, Spain
A publication of the European Mathematical Society.
2007; 4500 pp; hardcover
ISBN-10: 3-03719-022-1
ISBN-13: 978-3-03719-022-7
List Price: US$428
Member Price: US$342.40
Order Code: PICM/2006
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The International Congress of Mathematicians (ICM) is held every four years. It is a major scientific event, bringing together mathematicians from all over the world and demonstrating the vital role that mathematics play in our society. In particular, the Fields Medals are awarded to recognize outstanding mathematical achievement. At the same time, the International Mathematical Union awards the Nevanlinna Prize for work in the field of theoretical computer science.

The proceedings of ICM 2006, published as a three-volume set, present an overview of current research in all areas of mathematics and provide a permanent record the congress. The first volume features the works of Fields Medallists and the Nevanlinna Prize winner, the plenary lectures, and the speeches and pictures of the opening and closing ceremonies and award sessions. The other two volumes present the invited lectures, arranged according to their mathematical subject.

A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society.


Undergraduates, graduate students, and research mathematicians.

Table of Contents

Volume I
The work of the Fields Medalists, the Rolf Nevanlinna Prize Winner and the Gauss Prize Winner
  • G. Felder -- The work of Andrei Okounkov
  • J. Lott -- The work of Grigory Perelman
  • C. Fefferman -- The work of Terence Tao
  • C. M. Newman -- The work of Wendelin Werner
  • J. Hopcroft -- The work of Jon Kleinberg
  • H. Föllmer -- On Kiyosi Itô's work and its impact
Plenary lectures
  • P. Deift -- Universality for mathematical and physical systems
  • J.-P. Demailly -- Kähler manifolds and transcendental techniques in algebraic geometry
  • R. A. Devore -- Optimal computation
  • Y. Eliashberg -- Symplectic field theory and its applications
  • E. Ghys -- Knots and dynamics
  • H. Iwaniec -- Prime numbers and \(L\)-functions
  • I. M. Johnstone -- High dimensional statistical inference and random matrices
  • K. Kato -- Iwasawa theory and generalizations
  • R. V. Kohn -- Energy-driven pattern formation
  • I. Madsen -- Moduli spaces from a topological viewpoint
  • A. Nemirovski -- Advances in convex optimization: conic programming
  • S. Popa -- Deformation and rigidity for group actions and von Neumann algebras
  • A. Quarteroni -- Cardiovascular mathematics
  • O. Schramm -- Conformally invariant scaling limits: an overview and a collection of problems
  • R. P. Stanley -- Increasing and decreasing subsequences and their variants
  • T. Tao -- The dichotomy between structure and randomness, arithmetic progressions, and the primes
  • J. L. Vázquez -- Perspectives in nonlinear diffusion: between analysis, physics and geometry
  • M. Vergne -- Applications of equivariant cohomology
  • A. Wigderson -- \(\mathcal{P, NP}\) and mathematics--a computational complexity perspective
Special activities
  • J. W. Morgan -- The Poincaré Conjecture
  • Panel discussion organised by the European Mathematical Society
  • ICM 2006 closing round table
Cultural activities
  • J. M. Sánchez-Ron -- From the private to the public: The road from Zurich (1897) to Madrid (2006)
  • List of participants
  • Participants by country
  • Author index
Volume II
Logic and foundations
  • R. Downey -- Algorithmic randomness and computability
  • I. Neeman -- Determinacy and large cardinals
  • M. Rathjen -- The art of ordinal analysis
  • T. Scanlon -- Analytic difference rings
  • S. Thomas -- Borel superrigidity and the classification problem for the torsion-free abelian groups of finite rank
  • W. Crawley-Boevey -- Quiver algebras, weighted projective lines, and the Deligne-Simpson problem
  • M. du Sautoy and F. Grunewald -- Zeta functions of groups and rings
  • B. Keller -- On differential graded categories
  • R. Rouquier -- Derived equivalences and finite dimensional algebras
  • M. Sapir -- Algorithmic and asymptotic properties of groups
  • Seress -- A. unified approach to computations with permutation and matrix groups
  • A. Smoktunowicz -- Some results in noncommutative ring theory
Number theory
  • M. Bhargava -- Higher composition laws and applications
  • C.-L. Chai -- Hecke orbits as Shimura varieties in positive characteristic
  • H. Darmon -- Heegner points, Stark-Heegner points and values of \(L\)-series
  • K. Fujiwara -- Galois deformations and arithmetic geometry of Shimura varieties
  • B. Green -- Generalising the Hardy-Littlewood method for primes
  • G. Laumon -- Aspects géométriques du Lemme Fondamental de Langlands-Shelstad
  • P. Michel and A. Venkatesh -- Equidistribution, \(L\)-functions and ergodic theory: on some problems of Yu. Linnik
  • W. Nizioł -- \(p\)-adic motivic cohomology in arithmetic
  • C. Skinner and E. Urban -- Vanishing of \(L\)-functions and ranks of Selmer groups
  • V. Vatsal -- Special values of \(L\)-functions modulo \(p\)
Algebraic and complex geometry
  • V. Alexeev -- Higher-dimensional analogues of stable curves
  • J.-B. Bost -- Evaluation maps, slopes, and algebraicity criteria
  • T. Bridgeland -- Derived categories of coherent sheaves
  • L. Ein and M. Mustaţǎ -- Invariants of singularities of pairs
  • T. Graber -- Rational curves and rational points
  • J.-M. Hwang -- Rigidity of rational homogeneous spaces
  • T. Terasoma -- Geometry of multiple zeta values
  • Y. Tschinkel -- Geometry over nonclosed fields
  • J. Włodarczyk -- Algebraic Morse theory and the weak factorization theorem
  • C. Böhm and B. Wilking -- Manifolds with positive curvature operators are space forms
  • S. Brendle -- Elliptic and parabolic problems in conformal geometry
  • K. Honda -- The topology and geometry of contact structures in dimension three
  • M. Kapovich -- Generalized triangle inequalities and their applications
  • B. Kleiner -- The asymptotic geometry of negatively curved spaces: uniformization, geometrization and rigidity
  • F. Lalonde -- Lagrangian submanifolds: from the local model to the cluster complex
  • X. Liu -- Gromov-Witten invariants and moduli spaces of curves
  • T. Mabuchi -- Extremal metrics and stabilities on polarized manifolds
  • G. Mikhalkin -- Tropical geometry and its applications
  • W. P. Minicozzi II -- Embedded minimal surfaces
  • Y.-G. Oh and K. Fukaya -- Floer homology in symplectic geometry and in mirror symmetry
  • A. Ros -- Properly embedded minimal surfaces with finite topology
  • C.-L. Terng -- Applications of loop group factorization to geometric soliton equations
  • I. Agol -- Finiteness of arithmetic Kleinian reflection groups
  • M. R. Bridson -- Non-positive curvature and complexity for finitely presented groups
  • M. Khovanov -- Link homology and categorification
  • Y. N. Minsky -- Curve complexes, surfaces and 3-manifolds
  • F. Morel -- \(\mathbb{A}^1\)-algebraic topology
  • K. Ono -- Development in symplectic Floer theory
  • P. Ozváth and Z. Szabó -- Heegaard diagrams and Floer homology
  • K. Vogtmann -- The cohomology of automorphism groups of free groups
Lie groups and Lie algebras
  • R. Bezrukavnikov -- Noncommutative counterparts of the Springer resolution
  • A. Braverman -- Spaces of quasi-maps into the flag varieties and their applications
  • G. Henniart -- On the local Langlands and Jacquet-Langlands correspondences
  • N. Monod -- An invitation to bounded cohomology
  • B.-C. Ngô -- Fibration de Hitchin et structure endoscopique de la formule des traces
  • E. M. Opdam -- Hecke algebras and harmonic analysis
  • P. Schneider -- Continuous representation theory of \(p\)-adic Lie groups
  • Y. Shalom -- The algebraization of Kazhdan's property (T)
  • D. Soudry -- Rankin-Selberg integrals, the descent method, and Langlands functorality
  • B. Speh -- Representation theory and the cohomology of arithmetic groups
  • T. A. Springer -- Some results on compactifications of semisimple groups
  • M. Bonk -- Quasiconformal geometry of fractals
  • S. Hofmann -- Local \(Tb\) theorems and applications in PDE
  • S. V. Konyagin -- Almost everywhere convergence and divergence of Fourier series
  • L. P. Rothschild -- Iterated Segre mappings of real submanifolds in complex space and applications
  • S. Smirnov -- Towards conformal invariance of 2D lattice models
  • E. J. Straube -- Aspects of the \(L^2\)-Sobolev theory of the \(\overline\partial\)-Neumann problem
  • V. N. Temlyakov -- Greedy approximations with regard to bases
  • X. Tolsa -- Analytic capacity, rectifiability, and the Cauchy integral
Operator algebras and functional analysis
  • F. Barthe -- The Brunn-Minkowski theorem and related geometric and functional inequalities
  • B. Klartag -- Isomorphic and almost-isometric problems in high-dimensional convex geometry
  • N. Ozawa -- Amenable actions and applications
  • M. Rørdam -- Structure and classification of \(C^\ast\)-algebras
  • S. J. Szarek -- Convexity, complexity, and high dimensions
  • G. Yu -- Higher index theory of elliptic operators and geometry of groups
Ordinary differential equations and dynamical systems
  • O. N. Ageev -- On spectral invariants in modern ergodic theory
  • V. Bergelson -- Ergodic Ramsey theory: a dynamical approach to static theorems
  • N. Chernov and D. Dolgopyat -- Hyperbolic billiards and statistical physics
  • R. de la Llave -- Some recent progress in geometric methods in the instability problem in Hamiltonian mechanics
  • M. Einsiedler and E. Lindenstrauss -- Diagonalizable flows on locally homogeneous spaces and number theory
  • Author index
Volume III
Ordinary differential equations and dynamical systems (continued)
  • R. Ghrist -- Braids and differential equations
  • A. Gorodetski, B. Hunt, and V. Kaloshin -- Newton interpolation polynomials, discretization method, and certain prevalent properties in dynamical systems
  • B. Kra -- From combinatorics to ergodic theory and back again
  • P. Le Calvez -- From Brouwer theory to the study of homeomorphisms of surfaces
  • M. Shub -- All, most, some differentiable dynamical systems
  • A. Zorich -- Geodesics on flat surfaces
Partial differential equations
  • S. Bianchini -- Asymptotic behavior of smooth solutions for partially dissipative hyperbolic systems and relaxation approximation
  • P. Gérard -- Nonlinear Schrödinger equations in inhomogeneous media: wellposedness and illposedness of the Cauchy problem
  • F. Golse -- The periodic Lorentz gas in the Boltzmann-Grad limit
  • M. J. Gursky -- Conformal invariants and nonlinear elliptic equations
  • H. Ishii -- Asymptotic solutions for large time of Hamilton-Jacobi equations
  • M. Pulvirenti -- The weak-coupling limit of large classical and quantum systems
  • O. Savin -- Symmetry of entire solutions for a class of semilinear elliptic equations
  • S. Serfaty -- Vortices in the Ginzburg-Landau model of superconductivity
  • N. S. Trudinger -- Recent developments in elliptic partial differential equations of Monge-Ampère type
  • L. Vega -- The initial value problem for nonlinear Schrödinger equations
  • J. J. L. Velázquez -- Singular solutions of partial differential equations modelling chemotactic aggregation
Mathematical physics
  • A. S. Cattaneo -- From topological field theory to deformation quantization and reduction
  • B. Derrida -- Matrix ansatz and large deviations of the density in exclusion processes
  • J.-M. Maillet -- Correlation functions of the \(XXZ\) Heisenberg spin chain: Bethe anastz approach
  • M. Mariño -- Gromov-Witten invariants and topological strings: a progress report
  • I. Rodnianski -- The Cauchy problem in general relativity
  • C. Schweigert, J. Fuchs, and I. Runkel -- Categorification and correlation functions in conformal field theory
  • A. Soffer -- Soliton dynamics and scattering
  • C. Villani -- Hypocoercive diffusion operators
Probability and statistics
  • A. Bovier -- Metastability: a potential theoretic approach
  • R. Cerf -- On Ising droplets
  • A. Dembo -- Simple random covering, disconnection, late and favorite points
  • P. Donnelly -- Modelling genes: mathematical and statistical challenges in genomics
  • K. D. Elworthy and X.-M. Li -- Geometric stochastic analysis on path spaces
  • J. Fan and R. Li -- Statistical challenges with high dimensionality: feature selection in knowledge discovery
  • A. Guionnet -- Random matrices and enumeration of maps
  • S. P. Lalley -- The weak/strong survival transition on trees and nonamenable graphs
  • Y. Le Jan -- New developments in stochastic dynamics
  • P. McCullagh and J. Yang -- Stochastic classification models
  • A. Okounkov -- Random partitions and instanton counting
  • D. Picard and G. Kerkyacharian -- Estimation in inverse problems and second-generation wavelets
  • W. Werner -- Conformal restriction properties
  • A. Barvinok -- The complexity of generating functions for integer points in polyhedra and beyond
  • M. Bousquet-Mélou -- Rational and algebraic series in combinatorial enumeration
  • J. Geelen, B. Gerards, and G. Whittle -- Towards a structure theory for matrices and matroids
  • M. Haiman -- Cherednik algebras, Macdonald polynomials and combinatorics
  • J. H. Kim -- Poisson cloning model for random graphs
  • T. Łuczak -- Randomness and regularity
  • I. Z. Ruzsa -- Additive combinatorics and geometry of numbers
  • F. Santos -- Geometric bistellar flips: the setting, the context and a construction
  • R. Thomas -- A survey of Pfaffian orientations of graphs
Mathematical aspects of computer science
  • M. Agrawal -- Determinant versus permanent
  • A. S. Holevo -- The additivity problem in quantum information theory
  • J. Kleinberg -- Complex networks and decentralized search algorithms
  • O. Reingold -- On expander graphs and connectivity in small space
  • T. Roughgarden -- Potential functions and the inefficiency of equilibria
  • R. Rubinfeld -- Sublinear time algorithms
  • L. Trevisan -- Pseudorandomness and combinatorial constructions
Numerical analysis and scientific computing
  • P. Bochev and M. Gunzburger -- Least-squares finite element methods
  • Z. Chen -- A posteriori error analysis and adaptive methods for partial differential equations
  • R. G. Durán -- Error estimates for anisotropic finite elements and applications
  • N. Dyn -- Linear subdivision schemes for the refinement of geometric objects
  • R. J. LeVeque -- Wave propagation software, computational science, and reproducible research
  • Y. Maday -- Reduced basis method for the rapid and reliable solution of partial differential equations
  • E. Süli -- Finite element algorithms for transport-diffusion problems: stability, adaptivity, tractability
Control theory and optimization
  • V. S. Borkar -- Ergodic control of diffusion processes
  • S. Boyd -- Convex optimization of graph Laplacian eigenvalues
  • O. Y. Emanouilov (Imanuvilov) -- Controllability of evolution equations of fluid dynamics
  • A. van der Schaft -- Port-Hamiltonian systems: an introductory survey
  • O. J. Staffans -- Passive linear discrete time-invariant systems
  • E. Zuazua -- Control and numerical approximation of the wave and heat equations
Application of mathematics in the sciences
  • R. E. Caflisch -- Multiscale modeling for epitaxial growth
  • E. J. Candès -- Compressive sampling
  • V. Caselles -- Total variation based image denoising and restoration
  • M. Griebel and J. Hamaekers -- A wavelet based sparse grid method for the electronic Schrödinger equation
  • C. Le Bris -- Mathematical and numerical analysis for molecular simulation: accomplishments and challenges
  • M. A. Nowak -- Evolutionary dynamics of cooperation
  • D. Nualart -- Fractional Brownian motion: stochastic calculus and applications
  • A. Szepessy -- Atomistic and continuum models for phase change dynamics
Mathematics education and popularization of mathematics
  • P. S. Kenderov -- Competitions and mathematics education
  • A. Siegel -- Understanding and misunderstanding the Third International Mathematics and Science Study: what is at stake and why K-12 education studies matter
  • I. Stewart -- Mathematics, the media, and the public
  • M. Artigue, E. de Shalit, and A. Ralston -- Panel A: Controversial issues in K-12 mathematical education
  • L. P. Yee, J. de Lange, and W. Schmidt -- Panel B: What are PISA and TIMSS? What do they tell us?
  • Fr. Nebres, S.-Y. Cheng, K. Osterwalder, and H.-H. Wu -- Panel C: The role of mathematicians in K-12 mathematics education
History of mathematics
  • L. Corry -- On the origins of Hilbert's sixth problem: physics and the empiricist approach to axiomatization
  • N. Guicciardini -- Method versus calculus in Newton's criticisms of Descartes and Leibniz
Special activity
  • S. X. Descamps, H. Bass, G. B. Evia, R. Seiler, and M. Seppälä -- e-learning mathematics
  • Author index
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