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This page provides an overview of the typical course structure that people pursuing undergraduate studies in mathematics have, along with important points of difference between different colleges in how they structure their programs.
Beginning non-major-specific material
- Single-variable calculus
- Multivariable calculus
- Linear algebra
- Differential equations (for some majors)
These topics are also studied by many students who do not intend to major in mathematics. The following are some variations:
- Some colleges allow students to place out of single-variable and multivariable calculus courses based on credit from exams taken in high school (such as the Advanced Placement or International Baccalaureate) or from placement examinations conducted by the college. Generally, students who intend to major in mathematics are more likely to avail of the opportunity to place out so that they can start their major-specific studies more quickly (they are also more likely to be qualified to place out).
- Some colleges have slightly different versions of these courses targeted at students who intend to major in mathematics:
- There may be an "honors" version of single-variable calculus (and in some cases also of multivariable calculus and linear algebra) that is more heavily focused on proofs.
- In some colleges, the multivariable calculus and linear algebra classes are not targeted at mathematics majors. Instead, mathematics majors are exposed to these topics as part of their analysis sequence. This means that the students can get started more quickly on their major-specific courses.
Major-specific material typically includes:
- A year of real analysis: This includes some topological, measure-theoretic, and metric concepts associated with the real line, and an understanding of the real numbers and spaces like in that context. Some multivariable calculus and linear algebra may be covered here.
- A year of abstract algebra: This includes group theory, ring theory, and field theory. Some linear algebra may be covered here.
- A number of other topics such as topology (point set topology + a glimpse of algebraic topology), complex analysis, functional analysis, commutative algebra, differential geometry, algebraic number theory, analytic number theory, etc. The range of courses offered, and the degree of flexibility students have, depend on the college.
In some colleges, the convention is to do the year of real analysis first, and then proceed to the year of abstract algebra. In other colleges, the year of abstract algebra precedes the year of real analysis. In yet other places, the courses may be mixed with each other. Generally, whichever sequence is taught first will also have the responsibility of giving students some general mathematical knowledge (set theory, logic, proof-writing). Places that have an honors calculus option may use that to provide general mathematical knowledge.