Upper division undergraduate mathematics learning recommendations

This page contains some learning recommendations for people undertaking undergraduate studies in mathematics. This includes:

• People who are considering going on to graduate studies in mathematics (with the possible intention of continuing into academia)
• People who intend to do graduate work and research in a natural or social science subject that uses mathematics heavily, and may be doing a double major in mathematics and that subject, or a minor in mathematics. For instance, many leading economists did their undergraduate studies in mathematics.
• People who intend to go into a quantitative field like computer programming, finance, or actuarial sciences, for which a mathematics degree is a logical starting point.

The focus of our recommendations differs in the various cases.

Learning recommendations by subject area

Subject area	Brief description	Top recommendations	Page with more detailed recommendations
Undergraduate analysis (focused on real analysis)	Generally designed to serve the dual purpose of introducing students to basic ideas of analysis and to expose them to general mathematical ideas, including both proof-writing skills and a few definitions from algebra, logic, and set theory	Books by Maxwell Rosenlicht, Charles Pugh, and Walter Rudin	Undergraduate analysis learning recommendations
Point set topology	A rigorous foundation that helps provide sufficiently general definitions of continuity, connectedness, compactness, open and closed sets, and related ideas. Typically studied after undergraduate analysis, but smart and motivated people familiar with abstraction may choose to study it before.	Books by James Munkres and Gamelin and Greene	Topology learning recommendations
Abstract algebra	Groups, rings, and fields	Books by Charles Pinter and Dummit and Foote	Abstract algebra learning recommendations