Mathematicians typically do the following:
Some mathematicians apply theories and techniques, such as mathematical modeling, to solve practical problems. These mathematicians, sometimes known as applied mathematicians, typically work with individuals in other occupations to solve these problems. For example, they may work with chemists, materials scientists, and chemical engineers to analyze the effectiveness of new drugs. Other applied mathematicians may work with industrial designers to study the aerodynamic characteristics of new automobiles.
Other mathematicians may study theoretical or abstract concepts in mathematics. Sometimes called theoretical mathematicians, they identify, research, and resolve unexplained issues in mathematics. They are concerned primarily with exploring new areas and relationships of mathematical theories to increase knowledge and understanding about the field.
Despite the differences between applied and theoretical mathematics, these areas frequently overlap. Many mathematicians, particularly those in government or private industry, will use both applied and theoretical knowledge in their job duties.
However, most people with a degree in mathematics or who develop mathematical theories and models are not formally known as mathematicians. Instead, they work in related fields and professions. In the computer systems design and related services industries, for example, they may be known as computer programmers or systems analysts. In finance, they may be known as quantitative analysts or statisticians. Other industries may refer to them as data scientists.
Computer and information research scientists, physicists and astronomers, economists, actuaries, operations research analysts, engineers, and many other occupations also use mathematics extensively.
Some people with a mathematics background become middle school or high school math teachers.
Many people with a Ph.D. in mathematics, particularly theoretical mathematics, work as postsecondary teachers in education institutions. They usually have a mix of teaching and research responsibilities. Some may conduct individual research or collaborate with other professors or mathematicians. Collaborators may work together at the same institution or from different locations.