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Upper division undergraduate mathematics learning recommendations

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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
Probability Part of "probability and statistics" -- may also be covered in statistics courses Probability and statistics learning recommendations

Helpful general online resources

Question-and-answer websites

  • Math StackExchange is a good place to ask and answer questions as well as to browse an existing library of questions, provided your questions are related to highly specific aspects of mathematics.
  • Quora is a good place to look for questions and answers about the importance or about ways of studying particular branches of mathematics.
  • MathOverflow is a place for graduate-level and higher mathematics. MathOverflow emphatically does not encourage questions asking for undergraduate homework help, but if you have progressed to the stage where you are reading research papers or attempting your own original research, it might be a useful place to ask questions related to those.

Reference websites

Video resources

See also