Conduct research in fundamental mathematics or in application of mathematical techniques to science, management, and other fields. Solve or direct solutions to problems in various fields by mathematical methods.

Address the relationships of quantities, magnitudes, and forms through the use of numbers and symbols. Apply mathematical theories and techniques to the solution of practical problems in business, engineering, or the sciences. Assemble sets of assumptions and explore the consequences of each set. Conduct research to extend mathematical knowledge in traditional areas, such as algebra, geometry, probability, and logic. Develop new principles, and new relationships between existing mathematical principles, to advance mathematical science. Perform computations and apply methods of numerical analysis to data. Maintain knowledge in the field by reading professional journals, talking with other mathematicians, and attending professional conferences. Design, analyze, and decipher encryption systems designed to transmit military, political, financial, or law-enforcement-related information in code.

Mathematics -- Knowledge of arithmetic, algebra, geometry, calculus, statistics, and their applications.

Mathematics -- Using mathematics to solve problems. Learning Strategies -- Selecting and using training/instructional methods and procedures appropriate for the situation when learning or teaching new things. Active Learning -- Understanding the implications of new information for both current and future problem-solving and decision-making. Reading Comprehension -- Understanding written sentences and paragraphs in work related documents. Critical Thinking -- Using logic and reasoning to identify the strengths and weaknesses of alternative solutions, conclusions or approaches to problems. Writing -- Communicating effectively in writing as appropriate for the needs of the audience. Complex Problem Solving -- Identifying complex problems and reviewing related information to develop and evaluate options and implement solutions. Speaking -- Talking to others to convey information effectively. Active Listening -- Giving full attention to what other people are saying, taking time to understand the points being made, asking questions as appropriate, and not interrupting at inappropriate times.

Number Facility -- The ability to add, subtract, multiply, or divide quickly and correctly. Mathematical Reasoning -- The ability to choose the right mathematical methods or formulas to solve a problem. Deductive Reasoning -- The ability to apply general rules to specific problems to produce answers that make sense. Inductive Reasoning -- The ability to combine pieces of information to form general rules or conclusions (includes finding a relationship among seemingly unrelated events). Written Comprehension -- The ability to read and understand information and ideas presented in writing. Oral Comprehension -- The ability to listen to and understand information and ideas presented through spoken words and sentences. Information Ordering -- The ability to arrange things or actions in a certain order or pattern according to a specific rule or set of rules (e.g., patterns of numbers, letters, words, pictures, mathematical operations). Written Expression -- The ability to communicate information and ideas in writing so others will understand. Memorization -- The ability to remember information such as words, numbers, pictures, and procedures. Fluency of Ideas -- The ability to come up with a number of ideas about a topic (the number of ideas is important, not their quality, correctness, or creativity).

Processing Information -- Compiling, coding, categorizing, calculating, tabulating, auditing, or verifying information or data. Analyzing Data or Information -- Identifying the underlying principles, reasons, or facts of information by breaking down information or data into separate parts. Getting Information -- Observing, receiving, and otherwise obtaining information from all relevant sources. Interacting With Computers -- Using computers and computer systems (including hardware and software) to program, write software, set up functions, enter data, or process information. Identifying Objects, Actions, and Events -- Identifying information by categorizing, estimating, recognizing differences or similarities, and detecting changes in circumstances or events. Thinking Creatively -- Developing, designing, or creating new applications, ideas, relationships, systems, or products, including artistic contributions. Interpreting the Meaning of Information for Others -- Translating or explaining what information means and how it can be used. Updating and Using Relevant Knowledge -- Keeping up-to-date technically and applying new knowledge to your job. Making Decisions and Solving Problems -- Analyzing information and evaluating results to choose the best solution and solve problems.