all mathematicians who have preliminarily preregistered for the
International Congress of Mathematicians 2002 in Beijing.
Subject: ICM2002-CL5: Section Descriptions
Date: January 2, 2001
Prof. Yuri I. Manin, the chairman of the ICM2002 International Program
Committee, announced a description of the sections planned for the
scientific program of ICM2002. Please find the description below.
If you have further comments or suggestions, please contact
Professor Yuri I. Manin
President of the
ICM2002 Organizing Committee
ICM2002 Section Descriptions
Model Theory. Set theory. Recursion. Logics. Proof theory. Applications.
Connections with sections 2, 3, 14, 15.
Finite and infinite groups. Rings and algebras. Representations of Finite dimensional algebras. Algebraic
K-theory. Category theory and homological algebra. Computational algebra. Geometric methods in group
theory.Operads and their applications.
Connections with sections 1, 3, 5, 6, 7, 13, 14, 15.
3. Number Theory
Algebraic and analytic number theory. Zeta and L-functions.
Modular functions (except general automorphic theory). Arithmetic on
algebraic varieties. Diophantine equations, Diophantine approximation.
Transcendental number theory, geometry of numbers. P-adic analysis.
Computational number theory. Arakelov theory. Galois representations.
Connections with sections 1, 2, 6, 7, 14, 15.
4. Differential geometry:
Geometry of smooth and partially smooth spaces, including degenerate limits of smooth geometric structures.
Linear and nonlinear PDE arising in geometry, e.g. Dirac, minimal surface, harmonic map, and Einstein
equations. Geometric structures e.g. Riemannian, K\"ahler, symplectic, Poisson and contact. Hamiltonian
systems. Metric geometry.
Connections with sections 4, 5, 7, 8, 9, 11, 12, 13.
Algebraic, differential, geometric and low dimensional topology.
4-manifolds and Seiberg-Witten theory. 3-manifolds including knot theory.
Connections with sections 2, 4, 6, 7, 13.
6. Algebraic and Complex Geometry
Algebraic varieties, their cycles, cohomologies and motives.
Singularities and classification. Includes moduli spaces. Low
dimensional varieties. Abelian varieties. Vector bundles. Real
algebraic and analytic sets.
Connections with sections 2, 3, 4, 5, 7, 14, 15.
7. Lie Groups and Representation Theory
Algebraic groups, Lie groups and Lie algebras, including infinite dimensional ones, e.g. Kac-Moody,
representation theory. Automorphic forms over number fields and function fields, including Langlands' program.
Quantum groups. Hopf algebras. Discrete groups. Shimura varieties, Vertex operator algebras. Enveloping
algebras. Super algebras.
Connections with sections 2, 3, 4, 5, 6, 9, 12, 13, 14.
8. Real and Complex Analysis
Classical and Fourier analysis. Complex analysis.
Connections with sections 4, 11, 12, 13.
9. Operator Algebras and Functional Analysis.
General theory, non-commutative geometry and topology, K-theoretic and
homological aspects, simple C*-algebras and classification, quantum
physical aspects, non-commutative dynamical systems, non-commutative
(free) probability and von Neumann algebras, operator spaces, similarity
theory, subfactor theory, Banach spaces and algebras.
Connections with sections 2, 4, 5, 7, 8, 10, 12, 13.
10. Probability and Statistics
Classical probability theory, limit theorems and large deviations. Combinatorial probability and stochastic
geometry. Stochastic analysis. Stochastic equations. Random fields and multicomponent systems. Statistical
inference, sequential methods and spatial statistics. Applications.
Connections with sections 8, 9, 11, 12, 13, 14, 15, 17.
11. Partial Differential Equations
Solvability, regularity and stability of equations and systems. Geometric properties (singularities,
symmetry). Variational methods. Spectral theory, scattering, inverse problems. Relations to continuous media
and control. Topological methods for non-linear PDEs
Connections with sections 4, 8, 9, 13, 17.
12. Ordinary Differential Equations and Dynamical Systems
Topological aspects of dynamics. Geometric and qualitative theory of ODE
and smooth dynamical systems, bifurcations, singularities (including
Lagrangian singularities), one-dimensional and holomorphic dynamics,
ergodic theory (including sensitive attractors).
Connections with sections 4, 7, 8, 9, 10, 13, 16.
13. Mathematical Physics
Quantum mechanics. Operator algebras. Quantum field theory. General relativity. Statistical mechanics and
random media. Integrable systems. Deformation quantization. Renormalization.
Connections with sections 2, 4, 6, 7, 8, 9, 11, 12.
Interaction of combinatorics with algebra, representation theory,
topology, etc. Existence and counting of combinatorial structures.
Graph theory. Finite geometries. Combinatorial algorithms.
Connections with sections 1, 2, 3, 6, 7, 10.
15. Mathematical Aspects of Computer Science
Complexity theory and efficient algorithms. Parallelism. Formal languages and mathematical machines.
Cryptography. Coding theory. Semantics and verification of programs. Computer aided conjectures testing and
theorem proving. Symbolic computation. Quantum computing. Graph and networks. Robotics.
Connections with sections 1, 2, 3, 6, 10.
16. Numerical Analysis and Scientific Computing
Difference methods, finite elements. Approximation theory.
Computational applications of analysis. Optimization theory.
Control, optimization and variational techniques. Linear, integer and
non-linear programming. Matrix calculation. Signal processing.
Simulations and applications.
Connections with sections 10.
17. Applications of Mathematics in the Sciences
Applications in the physical sciences including chemistry, combustion,
fluid dynamics, materials science, mechanics, and physics. Applications
in the biological sciences including genomics, neuroscience and physiology.
Applications in the social sciences including economics and finance.
Applications in the mathematical sciences including computational
geometry and networks. Control theory.
Connections with sections 10, 11, 12, 13, 15, 16.
18. Mathematics Education and Popularization of Mathematics
Education: Theories of mathematics teaching and learning, at all
levels. Teaching of particular mathematical topics (algebra,
geometry, etc.) and skills (proof, problem solving, etc.).
Curriculum and curriculum frameworks. Assessment. Teacher education
and professional development. Cultural aspects. International
comparisons. Mathematics competitions.
Popularization: Broadly accessible expositions of significant
mathematical concepts and developments. Narrative or dramatic
accounts of important mathematical events. High quality and creative
Connections with section 19.
19. History of Mathematics