Proceedings of the International Congress of Mathematicians, 1995
August 3–11, 1994 Zürich, Switzerland

Since the first ICM was held in Zürich in 1897, it has become the pinnacle of mathematical gatherings. It aims at giving an overview of the current state of different branches of mathematics and its applications as well as an insight into the treatment of special problems of exceptional importance. The proceedings of the ICMs have provided a rich chronology of mathematical development in all its branches and a unique documentation of contemporary research. They form an indispensable part of every mathematical library. The Proceedings of the International Congress of Mathematicians 1994, held in Zürich from August 3rd to 11th, 1994, are published in two volumes. Volume I contains an account of the organization of the Congress, the list of ordinary members, the reports on the work of the Fields Medalists and the Nevanlinna Prize Winner, the plenary one-hour addresses, and the invited addresses presented at Section Meetings 1 - 6. Volume II contains the invited address for Section Meetings 7 - 19. A complete author index is included in both volumes. '...the content of these impressive two volumes sheds a certain light on the present state of mathematical sciences and anybody doing research in mathematics should look carefully at these Proceedings. For young people beginning research, this is even more important, so these are a must for any serious mathematics library. The graphical presentation is, as always with Birkhäuser, excellent....' (Revue Roumaine de Mathematiques pures et Appliquées)

Invited Forty-Five Minute Addresses at the Section Meetings.- Section 1. Logic.- New trends in partition calculus (Manuscript not submitted).- On the size of quotients by definable equivalence relations.- Model theory, differential algebra, and number theory.- Logic and complexity: Independence results and the complexity of propositional calculus.- Section 2. Algebra.- The path model for representations of symmetrizable Kac-Moody algebras.- Subgroup growth.- Constructions polynomiales et théorie de Galois.- Study of quadratic forms — Some connections with geometry.- Invariant differential operators.- Algebraic K-theory and motivic cohomology.- Modules of covariants.- Section 3. Number theory.- Linearized algebra (Manuscript not submitted).- Bounds for L-functions.- Polylogarithms in arithmetic and geometry.- Unexpected irregularities in the distribution of prime numbers.- Fonctions L p-adiques.- The role of smooth numbers in number theoretic algorithms.- Non-archimedean period domains.- Representations of Galois groups associated to modular forms.- Section 4. Geometry.- Einstein metrics and metrics with bounds on Ricci curvature.- Minimal foliations and laminations.- Harmonic maps, rigidity and Hodge theory.- Homological geometry and mirror symmetry.- Constant mean curvature surfaces in Euclidean spaces.- Intersection pairings on quotients and moduli spaces, and Witten’s nonabelian localisation.- Anti-self-dual metrics and Kähler geometry.- The geometry of moduli spaces of vector bundles over algebraic surfaces.- Embedded surfaces and the structure of Donaldson’s polynomial invariants (Manuscript not submitted).- Spaces with curvature bounded below.- Lagrangian intersections, 3-manifolds with boundary, and the Atiyah-Floer conjecture.- Generating functions, symplectic geometry and applications.- Section 5. Topology.- Smooth 4-manifolds and symplectic topology.- Topological modular forms, the Witten genus, and the theorem of the cube.- Applications dont la source est un classifiant.- Spaces of algebraic cycles: Levels of holomorphic approximation.- Dehn surgery on knots in the 3-sphere.- Cyclic splittings of finitely presented groups and the canonical JSJ-decomposition.- An algorithm to recognize the 3-sphere.- Characteristic classes, lattice points, and Euler-MacLaurin formulae.- Positive scalar curvature metrics — Existence and classification questions.- Nonlocally linear manifolds and orbifolds.- Section 6. Algebraic geometry.- Mumford-Stabilität in der algebraischen Geometrie.- Fundamental groups of curves in characteristic p.- Mixed motives, motivic cohomology, and Ext-groups.- Rational curves on algebraic varieties.- Some Eisenstein cohomology classes for the integral unimodular group.- Rigid and exceptional vector bundles and sheaves on a Fano variety.- Variations of Hodge structure and algebraic cycles.- Section 7. Lie groups and representations.- Genuine representations of the metaplectic group and epsilon factors.- The irreducible characters for semi-simple algebraic groups and for quantum groups.- Automorphic forms on Os+2,2(?)+ and generalized Kac-Moody algebras.- Spherical varieties.- Rigidity properties of group actions on CAT(0)-spaces.- Smooth representations of p-adic groups: The role of compact open subgroups.- Flows on homogeneous spaces and diophantine approximation.- Singular automorphic forms.- Gradings on representation categories.- Comparaison d’integrales orbitales pour des groupes p-adiques.- Section 8. Real and complex analysis.- L2-methods and effective results in algebraic geometry.- Meromorphic functions: Progress and problems.- Teichmüller space, dynamics, probability.- Fibering compact Kähler manifolds over projective algebraic varieties of general type.- Regularity of Fourier integral operators.- Finding structure in sets with little smoothness.- Topological, geometric, and complex analytic properties of Julia sets.- Smoothing estimates for the wave equation and applications.- A survey of Möbius groups.- Geometric and dynamical aspects of real submanifolds of complex space.- Section 9. Operator algebras and functional analysis.- The classification problem for amenable C*-algebras.- Recent results in the theory of infinite-dimensional Banach spaces.- Exact C*-algebras, tensor products, and the classification of purely infinite algebras.- Distortion and stabilized structure in Banach spaces; New geometric phenomena for Banach and Hilbert spaces.- Operator algebras and conformal field theory.- The algebra of functional analysis (Manuscript not submitted).- Section 10. Probability and statistics.- Brownian motion, heat kernels, and harmonic functions.- Interaction and hierarchy in measure-valued processes.- Abstract statistical estimation and modern harmonic analysis.- Probabilistic methods in finance (Manuscript not submitted).- Parabolic systems in random media and aspects of intermittency (Manuscript not submitted).- Quasi-regular Dirichlet forms and applications.- A surface view of first-passage percolation.- Quantum stochastic calculus.- Measure-valued branching diffusions and interactions.- Diffusion processes in random environments.- Section 11. Partial differential equations.- Analytical and numerical aspects of fluid interfaces.- Morse theory in differential equations.- Analyse microlocale et mécanique des fluides en dimension deux.- Some mathematical problems of fluid mechanics.- Hyperbolic systems of conservation laws.- Eigenfunctions and harmonic functions in convex and concave domains.- Singularities of solutions of nonlinear parabolic and elliptic equations (Manuscript not submitted).- Kinetic equations and hyperbolic systems of conservation laws.- The Cauchy problem for harmonic maps on Minkowski space.- Interface dynamics in phase transitions.- Fully nonlinear elliptic equations and applications to geometry.- Lower-semicontinuity of variational integrals and compensated compactness.- Section 12. Ordinary differential equations and dynamical systems.- The Riemann-Hilbert problem and Fuchsian differential equations on the Riemann sphere.- Invariant sets of hamiltonian systems and variational methods.- Rotation vectors for surface diffeomorphisms.- A priori estimates and regularity of nonlinear waves.- Applications of dynamics to compact manifolds of negative curvature.- On the borderline of real and complex dynamics.- Ergodic variational methods: new techniques and new problems.- Homoclinic bifurcations and persistence of nonuniformly hyperbolic attractors (Talk not delivered at the congress).- Ergodic theory of attractors.- Section 13. Mathematical physics.- Noncommutative geometry and the quantum Hall effect.- Conformal field theory and integrable systems associated to elliptic curves.- Free field realizations in representation theory and conformal field theory.- Homogenized models of strongly inhomogeneous media.- Fluxes and dimers in the Hubbard model.- Von Neumann algebras and quantum field theory.- Two-dimensional Yang-Mills theory and topological field theory.- Mirror symmetry and moduli spaces of superconformal field theories.- The critical behaviour of random systems.- A rigorous (renormalization group) analysis of superconducting systems.- Section 14. Combinatorics.- Eigenvalues of graphs.- Extremal hypergraphs and combinatorial geometry.- Asymptotics of hypergraph matching, covering and coloring problems.- Combinatorics and convexity.- Probabilistic methods in combinatorics.- Asymptotic combinatorics and algebraic analysis.- Section 15. Mathematical aspects of computer science.- Probabilistic proof systems.- The computational complexity of counting.- Methods for message routing in parallel machines.- Randomness and non determinism.- Pseudorandom generators for derandomization of algorithms (Manuscript not submitted).- Section 16. Numerical analysis and scientific computing.- Mathematical modeling and numerical analysis of linearly elastic shells.- Multiscale techniques — Some concepts and perspectives.- Matrix computation and the theory of moments.- Subscale capturing in numerical analysis.- Analysis-based fast numerical algorithms of applied mathematics.- Solving numerically Hamiltonian systems.- Section 17. Applications of mathematics in the sciences.- Methods of control theory in nonholonomic geometry.- Homogenization, renormalization and the mathematics of inhomogeneous media (Manuscript not submitted).- High-resolution computations of incompressible fluid flow (Manuscript not submitted).- Whiskered tori and chaotic behavior in nonlinear waves.- Formulas for finding coefficients from nodes/nodal lines.- Mechanochemical energy conversion in biological systems (Talk not delivered at the congress and manuscript not submitted).- Backward stochastic differential equations, and applications.- Max-plus algebra and applications to system theory and optimal control.- Statistical mechanics and hydrodynamical turbulence.- Spaces of observables in nonlinear control.- Section 18. Teaching and popularization of mathematics.- Changes in the teaching of undergraduate mathematics: The role of technology.- Issues for the popularization of mathematics.- Number theory as a core mathematical discipline.- Section 19. History of mathematics.- Mathematics in medieval Islamic Spain.- Mathematics in national contexts (1875–1900): An international overview.- Hermann Weyl’s “Purely infinitesimal geometry”.