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I hope this question is not too opinion based, if it is, please tell me and I would make edits.

I am currently a junior student studying math in a state college. By now, I have took all the required courses along with several (4) lower level graduate level courses (i.e. courses for first year graduate students). I used to be really satisfied about my progress until I recently discovered that the math courses offered here is not comparable, in terms of level of difficulty, with math courses offered in those top math schools (which shouldn't be surprising, but I never thought the difference is this big). Namely, most of the upper level undergrad courses in my school are barely at the level of introductory courses at those top schools. Also students in those schools studies functional analysis and real analysis (measure theory) in their undergraduate years, which are offered as graduate level courses in my school. As a result, I start to question myself if I am really ready for graduate school, since apparently, getting A's in my school doesn't really mean anything.

So here are my questions:

1. 
   

   Is there a consensus on the math a (top) graduate school applicant must know?

   
   
   

   I talked to several professors and get two different answers: one view is that one should at least take all the lower level graduate courses, e.g real, complex and functional analysis, algebraic topology and abstract algebra. Another view is to study whatever one is interested in. He never used some certain courses in his research and one should just take some of the irreverent courses in graduate school and just know enough of the material to pass the qualifying exam.

   
   
   

   I assume the answer to this question depends heavily on the individual and schools he/she is applying to, let's assume for now that the individual seeks to do pure math and has no other specific preference beyond that. Therefore, we define a "top" school by it's overall ranking in math and not go down into any specific field. (This might be an awful assumption, but I believe this is the case for most students). Also, It seems that students in most Europe university also studies many geometry in their undergraduate years, while students in US don't. If you also believe background in geometry is necessary, please include it into your answer.

   
   


2. 
   

   How should an undergrad seeking to go to grad school study math?

   
   
   

   Of course, to study math, one needs to do many questions and review constantly. Answers to such questions often blame students for "wanting to do the bare minimum" and suggests that one should read math book from cover to cover and do all the questions. I strongly agree with this view, but for an undergraduate student, there are just too many fields (Well, five at least, lower level graduate courses mentioned in previous paragraph), and the time never seems to be enough (please correct me if I am wrong). If you believe time is enough for all the exercises, what are the book you used?
   If you also think that time is not that enough, what's your learning strategies back in undergraduate years? (I don't think homeworks are enough for learning. My real analysis professor loves to assign qualifying exam questions as homeworks. I managed to solved all of the question by myself, but I don't feel like I really understand measure theory well enough).

   
   


3. 
   

   How important is undergraduate research?

   
   
   

   This question might be a little personal. I was not a math major in my freshmen year, so this is just my second year in math and didn't have time to fit some research into my schedule. My professors claimed that undergraduate research is not really necessary, but just in case they are saying that out of sympathy, what is your experience of undergraduate research?

   
   

PS: I probably shouldn't include my background in the question in the first place, but I am asking this question not only for my sake not also for future students that aspires to go to graduate school in math. In other words, I am not looking for answers for my particular case. The goal of this question is to figure out what an undergraduate student should know and do when no related course is offered but is not offered at his/her institute at a reasonable level. This aim might be over ambitious, but let's see how close can we approach it.

Last but not the least, study plan is something that heavily depends on the individual, please don't make decision for other people.

Part of the answer here depends on the quality of the state college you are attending (for example, UCLA is a "state college" but nobody would imagine that the top math undergrad at UCLA should be rejected from all "top" math grad schools). From your question, your school is a University (since it offers grad courses, but I am assuming from your concerns that by "state college" you mean some school which is possibly deficient in the level of rigor or difficulty associated with a various math courses.

The first possibility your should be prepared for is that it is becoming too late for you to get into the "top school" math programs. For example, I doubt that Princeton accepts any math grad students from Southern Illinois University (to name a respectable but not prestigious institution) because one of their filters to get top-notch students is the past ability to get into a tougher school and to excel in the tougher classes.

That argues that although your professors are right in that undergraduate research is generally not a vital component of a math education, for your ambitions, it is more important. Forgetting about Princeton, a very good graduate institution such as University of Illinois might be more inclined to believe in a steller undergrad record from a mediocre school, if it is accompanied by a semi-impressive publication of original results in a respected journal. That is not at all easy for an undergraduate to do, but if you "have the stuff" to be a professional mathematician, maybe you can do it.

Examples of texts that you should compare your studies against are, for example:

• In analysis, Rudin (but not the "baby Rudin" text).




• In topology, Munkries.




• In group theory/algebra, Heurstein.




• In probability, I like Feller, but that just should set a level, because there are several others that are just as iconic.

As to doing all the problems in each book, once you have the material in a given section mastered to the point that further basic problems become just busy work, there is no need. But in most books, the later questions in each chapter are more challenging and you should savor those challenges by doing most of the tough problems. If you don't enjoy tackling those difficult problems, then you are probably not temperamentally suited to be a serious mathematician.

The fact is, you don't have enough time to treat every topic with the depth it deserves and still lead a normal undergrad life. But purely from the viewpoint of becoming a mathematician, you are competing with people who will sacrifice that normal life and make the time, and being at a lesser school you are already giving them a head start. That said, the decision to "become a tool" and spend 80 hours a week on course, supplemental problem, and research work is a serious matter, and it might not be right for you.

Plenty of people can have a profession outside of pure math, yet find enjoyment in mathematical activities.

That said, don't be too discouraged. The top student in any school will be attractive to various grad programs, just not the top-notch ones. It brings to mind a story I had heard:

A math department $D$ once wrote to a world-class mathematician $W$ who had worked with a person $P$ applying for a professorship. $W$ wrote back saying that $P$ is "a very good second-rate mathematician, and would be very well suited to your position." The school replied to him saying "but you refer to $P$ as a "second-rate" mathematician, shouldn't we reject him? To which $W$ replied "why would you be concerned, you have a respectable third-rate department!"

Everything in life is relative.

I can't give you a holistic answer, but I can give some of my direct experience as a first year master student

1) My undergrad lacked rigorous math, and in my masters I have come to find out that you need the epsilon-delta practice from real analysis, basic knowledge of group/ring theory, and basic knowledge of set theory, in order to tackle most other courses.

2) To save time, you can really only learn the material and do basic concepts. The more time you have, more questions you do

3) Not important1 It's an asset, but not a requirement

Good Luck :)