ICM2002 Section Descriptions

1. Logic
Model Theory. Set theory. Recursion. Logics. Proof theory. Applications.
Connections with sections 2, 3, 14, 15.

2. Algebra
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.

5. Topology
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.

14. Combinatorics
Interaction of combinatorics with algebra, representation theory, topology, etc. Existence and counting of combinatorial structures. Graph theory. Finite geometries. Combinatorial algorithms. Combinatorial geometry.
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 mathematical journalism. Connections with section 19.

19. History of Mathematics