Dear Physics Forum advisers, I am a college sophomore in US with a major in mathematics and an aspiring mathematician in the fields of computation theory and cryptography. I recently revised my course workload with my adviser, and he suggested some variations in the workload for me to choose. Ultimately, the choice is mine but I would like to discuss the matter with you. I was originally planned to take the Abstract Algebra I (text: Dummit/Foote), Linear Algebra with Proofs (text: Friedberg et al.), and Computational Multivariable Calculus (text: university course packet, around a level of Thomas' Calculus). Although the LA with Proofs is a prerequisite of AAI, I got a special enrollment permission from the instructor since I have been self-studying Artin's Algebra and Hoffman/Kunze's Linear Algebra, and my upcoming undergraduate research in the computation theory-computer security will involve a lot of abstract algebra. I discussed the matter with my adviser, and he told me to postpone taking either AAI or Multivariable Calculus to Spring 2016. I agree with him since I was hesitant about taking all of those courses at one semester. My research adviser favored taking AAI on Fall, while my academic adviser actually advised me to postpone taking the AAI to next Spring. Naturally, I am inclined to take AAI on the Fall since it will be heavily used in my upcoming research, but I fear that lack of knowledge in the vector calculus might hurt me the full understanding of abstract algebra and abstract linear algebra, and my preparation for Putnam competition (although it seems that the contest is more focused on algebra and number theory). What is your recommendation? Should I perhaps taking all of them at one semester? My adviser told me that my upcoming research will involve bits of measure theory and approximation theory, and he told me to study one of following real analysis books (he said they are introductory books): Apostol's Mathematical Analysis, Rudin's PMA, Pugh's Real Mathematical Analysis, or Folland/Royden's Real Analysis. What should I get? I only took computational 1-variable calculus course using Lang's A First Course in Calculus, but I do have good proof skills. Will my lack of knowledge in the multivariable calculus be a problem? Should I start with transition books of Spivak, Courant, or Apostol (Calculus)? Thank you very much for your time, and I look forward to hear back from you!