# Learning Goals in the Mathematics Major

Mathematics majors are expected to master a significant body of material, including differential and integral calculus of one and several variables, discrete mathematics, and linear algebra. Upper-level courses in abstract algebra, real and complex analysis, and probability provide the theoretical underpinnings for much of modern mathematics, both pure and applied, including techniques and concepts encountered in earlier courses. Students also take a variety of electives, chosen to reflect their own interests, to represent the breadth of the discipline, and to introduce connections to other subjects. These electives may include differential equations, graph theory, mathematical methods for the physical sciences, theory of computation, topology, mathematical statistics, and a variety of other topics. Students may select a specialized course of study that leads to a concentration in statistics. Students are also exposed to a variety of special topics through colloquia and seminar talks sponsored by the department. All students are expected, at some point during their junior or senior year, to give a talk at the departmental seminar on a topic they have independently researched under the guidance of a faculty member. Many majors further develop their mathematical and expository skills by working as student tutors in the Math Help Center.

Mathematics majors acquire a substantial body of mathematical knowledge, become proficient with a wide array of problem-solving techniques, and develop an awareness and appreciation for the vast scope of the discipline. Successful majors are able to employ the techniques they have learned, aided by technology when appropriate, to solve problems in mathematics itself, in statistics, and in a number of other fields, including computer science, the natural and social sciences, engineering, and finance. The techniques and arguments they employ may be geometric, algebraic, analytic, graphical, probabilistic, or statistical, and may include constructing mathematical models. Students also develop the ability to communicate their solutions cogently, both orally and in writing. Most importantly, successful majors learn to construct valid mathematical proofs; that is, to make rigorous arguments to prove or disprove mathematical conjectures. All of these skills help prepare students for a wide variety of potential careers (such as secondary education, financial services, and information technology), as well as graduate study in a number of disciplines (including mathematics, applied mathematics, and statistics).

In summary, students will be able to:

- Acquire a comprehensive knowledge of the fundamental concepts underlying the discipline of mathematics, as well as material from specific courses of their own selection.
- Use mathematical methods and skills to solve a wide variety of problems, both within mathematics and in other disciplines.
- Analyze and prove mathematical statements, effectively communicating their ideas both orally and in writing.
- Become fluent with increasing levels of mathematical abstraction.
- Master sophisticated techniques from advanced courses.
- Attend and participate in talks from both local and visiting mathematics faculty on advanced topics.
- Research new topics independently, analyze them, and present them in a cogent way to their peers and professors.