National Science Foundation,
Here are the subdivisions of mathematics classified according to the system used at the (U.S.) National Science Foundation. This scheme is used for the assignment of proposals for sponsored projections. You can click here to go directly back to the index arranged according to the AMS/Zbl Mathematics Subject Classification system used to organize materials at this site; no links are yet provided below to match the NSF program areas with the corresponding MSC areas.
There are also separate divisions in the MPS Directorate for Astronomical Sciences, Chemistry, Physics, Materials Research, and Multidisciplinary Activities.
Besides the MPS, the NSF includes separate directorates for
The remainder of this page is taken nearly verbatim from the NSF's DMS Programs page.
Algebra, including algebraic structures, general algebra, and linear algebra; number theory, including algebraic and analytic number theory, quadratic forms, and automorphic forms; and combinatorics and graph theory and algebraic geometry.
Properties and behavior of solutions of differential equations; variational methods; approximations and special functions; analysis of several complex variables and singular integrals; harmonic analysis and wavelet theory; Kleinian groups and functions of one complex variable; real analysis; Banach spaces; Banach algebras and function algebras; Lie groups and their representations; harmonic analysis; ergodic theory and dynamical systems; some aspects of mathematical physics (e.g., Schrödinger operators, quantum field theory); and operators and algebras of operators on Hilbert space.
Supports research in any area of mathematics except probability or statistics; research is expected to be motivated by or have effect on problems arising in science or engineering, though intrinsic mathematical merit is the major decision factor. Areas of interest include, but are not limited to, partial differential equations modeling natural phenomena or arising from problems in science or engineering; continuum mechanics; reaction-diffusion and wave propagation; dynamical systems; numerical analysis; control theory; asymptotic methods; variational methods; optimization theory; inverse problems; mathematics of biological or geological sciences; and mathematical physics.
Computation is increasingly important to all sciences. Mathematics plays a unique role in providing the development of basic algorithms and techniques necessary to carry out computations. Proposals from interdisciplinary teams of mathematical, computer, and general scientists are encouraged in an effort to develop critical computational techniques from algorithm development through implementation. Proposals for innovative, computational methods within the mathematical sciences are also encouraged.
Differential geometry and its relation to partial differential equations and variational principles; aspects of global analysis including the differential geometry of complex manifolds and geometric Lie group theory; geometric methods in modern mathematical physics and dynamical systems; and geometry of convex sets, integral geometry, and related geometric topics.
Statistical theory and methods are used to plan scientific experiments, and to understand and analyze data. Major subfields include parametric and nonparametric inference, sequential analysis, multivariate analysis, Bayesian analysis, experimental design, time series analysis, resampling methods, and robust statistics. Almost all these subfields have become computationally intensive in recent times.
Probability theory is the study of mathematical structures that provide tractable models to statistics as well as many diverse areas such as physics, chemistry, biology, and engineering. Major subfields include stochastic processes, limit theory, infinite particle systems, stochastic analysis in Banach spaces, martingales, and Markov processes.
Algebraic topology, including homotopy theory, ordinary and extraordinary homology and cohomology, cobordism theory, and K-theory; topological manifolds and cell-complexes, fiberings, knots, and links; differential topology and actions of groups of transformations; general topology and continua theory; and mathematical logic, including proof theory, recursion theory and model theory, foundations of set theory, and infinite combinatorics.