Well, as it so happens, I already had some responses prepared for another person on this forum that should be quite helpful. Tamar Arnon is the assistant director of student affairs, for reference below.
I considered taking the comprehensive exams and signed up twice, but as I prepared I realized I had many gaps from my undergraduate coursework (I went to a small liberal arts school and even though I was one of the top math students there, I learned at NYU that there was much I did not know, which was already expected for the exams). Another piece of advice is that some professors at NYU teach so much more and more in-depth than others, which makes a difference when considering the exams (according to my friends). For instance, Professor Hameiri is excellent, and he incorporates questions similar to those on the written exams to his homework assignments, midterms, and finals. Another example is Professor Vanden Eijnden, who can usually lecture well when he is there, but he often has someone substitute because he is a top researcher and unexpectedly misses course and office hours, among other things. Beware of visiting professors and postdocs because they tend to repeat what's in the textbook, word for word, but with a few extra errors. There are exceptions, so find out who you think is a good lecturer. Some TA's are great for teaching you tips and tricks (even the ones who are supposedly just graders-try to befriend these because they clearly can elucidate the problems). For the written exams, look into:
MATH-GA 1410.001, 1420.001 INTRODUCTION TO MATHEMATICAL ANALYSIS I, II
MATH-GA 2430.001 REAL VARIABLES (one-term format)
MATH-GA 2110.001, 2120.001 LINEAR ALGEBRA I, II (OR)
MATH-GA 2111.001 LINEAR ALGEBRA (one-term format)
MATH-GA 2450.001,2460.001 COMPLEX VARIABLES I, II (OR)
MATH-GA 2451.001 COMPLEX VARIABLES (one-term format)
The one-term courses may cover more advanced material, but they go so much faster, so you may not learn the material so well. Also, my friends in the Master's program who took these courses had a harder time getting A's. No matter what, you will need to teach yourself most of the material for the written exams. There are written exam workshops, which do brief reviews. I got an email saying that now there is a section for Master's students (at a slower pace). It is perceived that these students have a weaker background, which may (my case) or may not be true (see below). Also, Tamar has free copies of the written exams, which are already bound for your convenience. Also, if you are disciplined and motivated enough, look up the wikis with partial solutions now. They should be quite helpful. The common belief is that you should work through as many problems as possible and redo all the old tests. Sure, some of the questions will be basic, but many are quite hard or were incorrectly formulated or stated in a weird way. Not knowing what the correct is also does not help. Here is where the wikis help.
Taking five courses per term is highly discouraged. Tamar got furious whenever students signed up for an extra course and then wanted to drop it right at the cutoff date for add/drop. Again, it may depend on the professors if it is manageable. Personally, if you are going to do it anyway, don't bring attention to yourself and make sure that you pick up the material quickly. The courses that require weekly assignments are labor intensive to say the least. Some professors like Cappell give optional homework, so try to mix and match. My suggestion is to sit-in for all the classes you are interested until the week of the add/drop deadline. Then register for your top picks. The classes usually don't fill up (not in my experience). That way you don't lose money in class fees, which are not refundable, by dropping a course last minute. Tamar taught me that, but again she hates hearing about 5 classes per term. My friends started joking that it might be best not to tell anyone to avoid being yelled at (two of my Korean friends, the same ones below, endured this).
The TA's for some analysis courses are usually really nice. Evan and Jose (don't know if they are still there), for instance, were quite helpful. Also, the professors that are helpful during office hours help everyone and find time for your questions and concerns, e.g. Hameiri, Hoppensteadt, and Cappell. I graded for Spencer (super nice and helpful), Novikoff, and Tsichancka (may have misspelled his name). Faculty heavily involved in their research may not have the time for you unless you are one of their research students. It's understandable to a degree. That being said, I remember the reaction of new PhD students (my second year) in my classes when I told them I was Master's student. Their attitude seemed disinterested, and they were not interested in forming study groups. There are exceptions; I noted that those who became TAs or workshop instructors were more courteous. It seems from the mass emails that some of the newer Masters students may have felt alienated and wanted to feel more integrated. One sent out a survey asking:
Have you ever wished things were different in the MS programs?
Have you yearned for a work space to relax, collaborate with colleagues, and call your own?
Have you searched for more direction and wished there was someone to help you?
Do you have suggestions or complaints but don't know who to tell?
Eventually, you will make good friends with your fellow Masters classmates, but at the beginning, it was a bit lonely. It's like a commuter campus, unless you live and work close by. PhD students definitely must have a different experience because many have housing in Stuyvesant Town. During office hours, ask questions no matter how silly they seem. Don't worry about what others think, but of course, read the book and review your notes beforehand (maybe several times), so you know it's not something you simply overlooked. I got a few "you must be naive" as well as the usual "this is clearly true or obvious" responses. Your experience may vary.
Hmm, I don't remember, but for the department's Master's students orientation session about 30 people came. So, maybe double or triple that for a total in any given year. For the two term courses, you can have 30 to about 100 students, maybe more. Introduction to Analysis I, Complex Variables I, Linear Algebra I, Basic Probability, and Topology I were packed. The second term you would see at most 40 students, usually less. These courses had many Master's students, but also many undergraduates, and even high school students (who are quite advanced, mainly from Stuyvesant). PhD students normally take the abbreviated one-term courses. Also, analysis and probability courses get many Financial Mathematics students and people from the Stern School of Business. Algebra (abstract version) courses are not as densely populated. Both Real Variables and Algebra I had more PhD students, naturally.
Technically, all you need is 3 A's (one in each of the written exams) to go from the Master's to the PhD program. One of my classmates from Taiwan studied from May to August, got some of the highest scores (or 3 perfect scores) and was automatically accepted into the PhD program. He had only completed 1 year of the Master's program. Unfortunately, he did not get funding until year 3 or 4, I forgot to find out if he did get it in the end. So, it is not impossible. Be sure you are into: analysis and applied mathematics, including partial differential equations, differential geometry, dynamical systems, probability and stochastic processes, scientific computation, mathematical physics, or fluid dynamics. I was interested in another area, but I found a good adviser. You can apply to all the math PhD programs in the US you can afford (see application, transcript, and exam score fees) even while completing your Master's degree, but make sure you do well on the Math GRE, have some true research experience, and excellent recommendations that testify to your potential. Two of my Korean friends, who both completed their undergraduate work at Berkeley, got into PhD programs elsewhere. One got into UCLA in biomathematics (I think he applied twice to PhD programs after he started at NYU and got in a year after he finished) and the other one went to a school in Germany after supposedly talking to a professor (strange I know...I guess she was interested in the research and gave a good impression at a conference she attended).
If you go for a thesis, just try to get it done before two years since you started the program have lapsed. It does not really matter if you work on it during the fall, spring, or summer. Ask a faculty member in a field you are interested in to be your adviser, and work out a timeline that suits you. I entered the program in Fall 2008. I started my Master's thesis in Spring 2010, but I finished it over the summer. All of my friends that did a thesis, except one, also did that. My adviser was helpful, but he was away during the summer, so I had to figure a lot of stuff on my own. I should have asked in advance since I knew from my undergraduate experience that was not a good setup for me. By Fall 2010, my adviser sent me an email saying "How are things? It is time to complete the thesis." Due to my shaky basis, it was especially hard for me. I thought I knew algebra well, but graduate school taught me differently. In the end, I finished it, but it was unreasonably long (my fault, I guess). If you start sooner, you finish sooner. Keep working on it all the time because classes and jobs (I worked as a private tutor; many of my friends and I also worked as graders for undergraduate courses; ask Jillian Kerlin, who is in charge of that, if you are interested) won't let you focus. It may also be discouraging when you are stuck, but sit in a quiet place and reread and think for countless hours until you have an AHA! moment since this tends to be a solitary activity. In hindsight, I should have decided to do the thesis during the spring of my first year, worked on it during the summer, and finished it during the fall of my second year to finish my degree in 3 terms.
I wanted to mention a couple of things that may or may not be useful to you.
*Make sure you spend this summer studying like crazy even if you are not planning to take the written exams. It will make your life easier once you start school at NYU if you so choose. Review all your calculus, analysis, topology, abstract algebra, probability, and linear algebra notes or try to teach yourself from textbooks now. This summer is the time to go over those sections your undergraduate professors skipped or glossed over because they were too advanced, or otherwise, there was not enough time. I learned it's better if you adopt this habit before classes start because it's more stressful when you realize you need to catch up to advanced high school and/or PhD students. If you are good at teaching yourself using terse textbooks, then you are pretty much set. I wished I had done this and learned to start sooner every term. Definitely, study during all your breaks.
*Don't ever get sick. This may sound unrealistic, but try to stay as healthy as possible. I contracted swine flu, so it made staying on top of things much more challenging, especially when I got bronchitis a week later. Few things make me more miserable than working on complicated math problems that require all of my concentration when I can barely breathe.
*Unless you make her mad with a 5th. course or some other administrative no no, Tamar is a fountain of useful information. She may not know advanced theorems, but she can help you with many other things, including explaining what you need to do to complete your degree, help you sign up for written exams, and help with with full-time or half-time equivalency for loans or outside funding and/or visas. Also, I did not bother to do this my first year (as it turned out, it would have been better), but it's good if you talk to the Department Chair (Tschinkel) and Bogomolov, or whoever is in charge of the program (PhD and Masters) when you start. Really spend time with them talking about your goals and expectations, and what resources are available. Avoid being a hermit at all costs (as tempting and effortless as it seems when you commute from far away). Also, ask all of your questions that are not addressed online. Repeat this procedure with current Master's and PhD students. If you wait for these conversations to take place organically, it may not be to your advantage time-wise. Do this when your coursework is not overwhelming and get it out of the way.
*If you start at NYU and are fortunate enough to take several classes (2 or more) with the same instructor and you get along, always go to office hours and develop a professional relationship that may yield a more recent recommendation letter for PhD programs. Trust me, it comes up in PhD applications for those who complete Master's degrees. Try to pick a thesis adviser that can write a good letter and is helpful with your research.
*Go to the math talks. They may all be over your head, but I believe that faculty may take these students more seriously as future PhD candidates. I don't have anything more than a hunch, but I believe that networking is just as important in academia as it is elsewhere. So force yourself to go if needed. I had gone to many during a summer program, but I went only to one at NYU since I knew I rarely understood much of the presentation past the first few minutes. Nonetheless, sometimes what is presented is later discussed in higher-level courses, e.g. p-adic analysis.
*Moreover, teach yourself LATEX. If you type your assignments, you will get better at using it. If you write a thesis, this is necessary. The following book will help: http://www.google.com/url?sa=t&rct=j&q= ... LaTeX(2007
*NYU is a secure campus, so get your ID card and NetID. The security guards can be annoying if you forget it. Signing in to go to class is a drag. Thus, never leave it behind by mistake.
*Learn how to program in C++, Matlab, etc. before you start your classes. This is especially useful for scientific computing and numerical analysis and modeling courses and research with the applied faculty. Again, the better prepared you are before you begin, the more enjoyable (and less stressful) your experience will be.
*Get acquainted with the LINUX environment as the free computer labs in Courant use it. Bobst has both Windows and MAC computers. There is a photocopier in the 10th. or 12th.? floor. Any random number sequence seems to work, but you may need permission now. So, print any documents you need in the lab (problem sets, code, etc.), and make copies using the copier. Your ink is otherwise rationed to 200 or 300 pages per term.
*Also, I learned that I would have made friends much sooner if I spent more time in the department library. My classmates that lived close to campus hanged out there all the time studying, but since I lived farther away, I was drained by the evening classes and only stayed late or came early to campus to finish up assignments. I do not know if I miss the hectic subway switches between 2 (sometimes 3) trains. Study groups, even if it is just you and one other friend, may help a lot for the written exams and coursework in general. Then again, try to meet after you are mostly done and are only missing a problem or two. The rule is 1) mostly teach yourself all you can and think about it hard and long enough to finish most of the assignment, 2) then google anything that you could not figure out (become adept at using search engines, if you are patient, you will probably find what you are looking for or the book you need for a better explanation), and 3) if all that fails, you were going to office hours anyway. After step 1, you can meet with friends to discuss your findings, but scheduling a time that works for everyone may be an issue. Sometimes, all you need is a little insight or a hint.
As for incurring debt, a member of the faculty from Cornell told me it was never a good idea to pay for graduate school. I should have listened. NYU was amazing and completely different from my undergraduate experience. I also fell in love with Brooklyn and Washington Square Park and St. Mark's Place. But none of this is free (except for my friend from Mexico; her government gave her a scholarship).
Let me know if this was at all useful and if you have any other specific questions.