Mathematicians are people who study areas in the broad field of mathematics. Mathematics is concerned with explaining change, quantitative relationships, and the structures of things within a framework of equations and numerical relationships. In school, mathematicians usually study a broad range of math topics as undergradates, then specialize into very specific subfields within mathematics. Unlike some other types of scientists, mathematicians support their statements with rigorous proofs that can be verified by their peers.
"Applied mathematics" is a broad term used to describe the field of mathematicians who address problems in the real world. Applied mathematicians work in all fields of science, engineering and industry. Applied mathematics focuses on studying and forming mathematical models that apply directly to practical problems. There are many branches of applied mathematical study. These traditionally include applied analysis, differential equations, numerical analysis, statistics and applied probability. Many other areas of math also can be used in the practice of applied math.
"Pure mathematics" is a broad term for the domain of mathematicians who study theoretical constructs in math. Pure mathematics is pursued largely to discover new insights into mathematics itself, not necessarily to address problems in the real world. A main concept of pure mathematics is generality. Pure mathematicians seek to generalize mathematical concepts to apply to a large variety of different branches of mathematics. Fields of study include such branches as functional analysis, number theory, abstract algebra, geometry and real analysis.
The field of "numerical analysis" deals with all kinds of numerical calculations, especially study and experimentation with computer algorithms to approximate solutions to problems in science and engineering. As the power of computers has increased exponentially, the field of numerical analysis has grown to utilize this power to great benefit. The primary goal for the numerical analyst is to give very accurate approximations to difficult mathematical problems. Some of the specialized fields of study within this broad branch include optimization problems, computing the values of difficult functions, solving systems of equations, and evaluating integrals.
Though "theoretical physics" is a more specialized term than applied or pure mathematics, the field remains quite broad and uses a variety of mathematical branches. These include electromagnetic theory, quantum mechanics and relativity, classical mechanics and statistical mechanics. Theoretical physicists are typically interested in physical systems and how they work. The main goal of the field is to understand the universe at its most basic level in a mathematical framework. These mathematicians attempt to discover patterns in the behavior of the universal makeup, and they use mathematics to create laws based on these patterns.
The field of statistics involves the use of mathematical techniques and computers to analyze data from the areas of business and science. Statisticians collect and organize data so that it can be accurately interpreted and utilized in the real world. Specialists such as mathematical statisticians are concerned with the theoretical study of the broader subject. A major focus of the study of statistics is to investigate causality, which allows statisticians to show the effects of changes in data on variables that depend on that data in some way. A few of the many specialized fields in statistics include actuarial science, econometrics, demography, data mining, image processing and biostatistics.