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”.