Stage 5

By completing stage 4, you should be ready for graduated schools, a.e. My list ends here. You've completed an undergraduate mathematics program (who cares about certificate?), and possibly more than that. Believe it or not, I've seen mathematics graduates haven't taken any analysis, abstract algebra or topology courses. This is (partly) because the entry requirement of mathematics degree is usually not so high (but don't expect it's easy) and folks fall into such program accidentally and they don't have to pick those abstract courses. Now you probably know what's going on and you should have an idea what you can read. Several books in the list above give suggestions for further reading, you may refer to them according to your interest. Moreover, choices decrease as stages increase.

For more advanced FREE! materials, check out:

• American Mathematical Society Books Online
• MIT's graduate courses list
• The Electronic Library of Mathematics
• FreeScience
• Scanned text (mostly Chinese, e.g. Baby Rudin and Big Rudin, Riesz and Sz.-Nagy, Yosida, Harthsorne, Halmos, etc)
• Stein W.'s Modular Abelian Varieties page (I've listed a few of them in the above list)

List of books:

• American Mathematical Society: Graduate Studies in Mathematics
• Springer: Graduate Text in Mathematics GTM
• Springer: Grundlehren der mathematischen Wissenschaften
• Springer: Universitext
• Cambridge University Press: Cambridge Studies in Advanced Mathematics
• Cambridge University Press: London Mathematical Society Student Texts
• Cambridge University Press: London Mathematical Society Lecture Notes Series
• Wiley-Interscience: Wiley Series in Probability and Statistics
• CRC: CRC Press Statistics series
• Dover: books in mathematics

Want challenging questions? See:

• Mathematical Tripos Examination Papers
• Harvard Mathematics Qualifying Exam

After several extra stages, if you feel like you are ready to read journals, try:

• Annals of Mathematics
• arXiv.org
• Project Euclid

After all, if it turned out that you're not going to further your study in pure mathematics, what can you do with your mathematical knowledge? There are a few options (ideas by James Franklin, given in the 'Professional Issues and Ethics in Mathematics' course):

• Modelling of environment (related areas are global warming, weather prediction, etc)
• Optimization for resources planning
• Statistical research on effectiveness of drugs and medical procedures, better diagnosis
• Teaching, to inspire (or corrupt?) the next generation