Basic mathematics & Advanced mathematics

Algebra

Algebra includes the study of algebraic structures, which are sets and operations defined on these sets satisfying certain axioms. The field of algebra is further divided according to which structure is studied; for instance, group theory concerns an algebraic structure called group.

Calculus and analysis

Calculus studies the computation of limits, derivatives, and integrals of functions of real numbers, and in particular studies instantaneous rates of change. Analysis studies the same subjects, but on a more rigorous level, and also topics that evolved from calculus.

Geometry and topology

Geometry is initially the study of spatial figures like circles and cubes, though it has been generalized considerably. Topology developed from geometry; it looks at those properties that do not change even when the figures are deformed by stretching and bending, like dimension.

Discrete mathematics

Discrete mathematics concerns itself with the study of finite sets. This is in contrast with analysis and geometry, which study continua: between every two distinct points, however close together, one can always find another point.

Logic

Logic is the foundation which underlies mathematical logic and the rest of mathematics. It tries to formalize valid reasoning. In particular, it attempts to define what constitutes a proof.

Number theory

Number theory studies the natural, or whole, numbers. One of the central concepts in number theory is that of the prime number, and there are many questions about primes that appear simple but whose resolution continues to elude mathematicians.

Applied mathematics
Computing

The fields of mathematics and computing intersect both in computer science, the study of algorithms and data structures, and in scientific computing, the study of algorithmic methods for solving problems in mathematics, science and engineering.

Dynamical systems

In a dynamical system, a fixed rule describes the time dependence of a point in a geometrical space. The mathematical models used to describe the swinging of a clock pendulum, the flow of water in a pipe, or the number of fish each spring in a lake are examples of dynamical systems.

Mathematical physics

Mathematical physics is concerned with "the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and for the formulation of physical theories

Probability and statistics

Probability theory is the formalization and study of the mathematics of uncertain events or knowledge. The related field of statistics concerns the mathematics and science of collecting and analyzing data.

Operations research

Operations research is the study and use of mathematical models, statistics and algorithms to aid in decision-making, typically with the goal of improving or optimizing performance of real-world systems.

The Mathematics Subject Classification (MSC) is a numerical classification scheme formulated by the American Mathematical Society.
It is used by many mathematics journals, which ask authors of research papers and expository articles to list subject codes from the Mathematics Subject Classification in their papers. The subject codes so listed are used by the two major reviewing databases, Mathematical Reviews and Zentralblatt MATH.