Lists of mathematics topics

As a rough guide, this list is divided into pure and applied sections although in reality, these branches are overlapping and intertwined.

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.

Number theory is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions.

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.

Historically, information theory was developed to find fundamental limits on compressing and reliably communicating data.

Processing of such signals includes filtering, storage and reconstruction, separation of information from noise, compression, and feature extraction.

Statistics, the science concerned with collecting and analyzing data, is an autonomous discipline (and not a subdiscipline of applied mathematics).

It has applications in a variety of fields, including economics, anthropology, political science, social psychology and military strategy.

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 the performance of real-world systems.

Such statements include axioms and the theorems that may be proved from them, conjectures that may be unproven or even unprovable, and also algorithms for computing the answers to questions that can be expressed mathematically.