# Course 18 Option 3: Pure Option

Pure mathematics is the study of the basic concepts and structures that underlie mathematics. Its purpose is to search for a deeper understanding and an expanded knowledge of mathematics itself.

Traditionally, pure mathematics has been classified into three general fields: analysis, which deals with continuous aspects of mathematics; algebra, which deals with discrete aspects; and geometry. The undergraduate program is designed so that students become familiar with each of these areas. Students may also wish to explore other topics such as logic, number theory, complex analysis, and subjects within applied mathematics.

The subject 18.100 Real Analysis is basic to the program. Since this subject is strongly proof-oriented, some students find it useful to take an intermediate subject such as 18.06 Linear Algebra or 18.090 Introduction to Mathematical Reasoning or 18.700 Linear Algebra, before taking 18.100.

The subject 18.701 Algebra I is more advanced and should not be elected until the student has had some experience with proofs (as in 18.090 or 18.100 or 18.700).

## Required Subjects

• 18.03 or 18.032 (formerly 18.034) (Differential Equations)
[sufficiently advanced students may substitute 18.152 or 18.303]
• 18.100 (Real Analysis)
• 18.701 (Algebra I)
• 18.702 (Algebra II)
• 18.901 (Introduction to Topology)

## One of the following three Subjects

• 18.101 (Analysis and Manifolds)
• 18.102 (Introduction to Functional Analysis)
• 18.103 (Fourier Analysis — Theory and Applications)

## One of the following six Seminars

• 18.104 (Seminar in Analysis)
• 18.504 (Seminar in Logic)
• 18.704 (Seminar in Algebra)
• 18.784 (Seminar in Number Theory)
• 18.904 (Seminar in Topology)
• 18.994 (Seminar in Geometry)

## Two Restricted Electives

Two additional 12-unit Course 18 subjects of essentially different content with the first decimal digit one or higher.

A student may, with permission, substitute a first-year graduate subject in pure mathematics for the seminar. The graduate subject will not satisfy a CI-M requirement, however.