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,
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. August 29 Exhibits Move-Out
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,
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.
Combinatorial geometry. Connections with sections 1, 2, 3, 6, 7,
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.)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