TITLE: How to tell if I'm good enough for graduate school?
QUESTION [59 upvotes]: I'm concerned about my level of preparation for graduate school. I have decent grades (3.64 GPA, 4.0 in math), great recommendations lined up, and some OK research experience (no publications), but I really don't feel like I'm ready for the next step. 
I'm interested in applied math, and was a physics/computer science double major for most of undergrad, switching into math just last semester. I'm taking Analysis 1 now as a first semester senior, and it is the only pure math class I've ever taken (though I've taken a bunch of applied courses and semi-pure courses, like advanced udergrad ODEs). I'm doing very well in the course, and I'm really enjoying it, but I feel that, because I'm so late in joining mathematics, I'll be really behind everyone when I get to graduate school, if I do get in somewhere decent. My scores on practice tests for the math GRE are in the mid 600's, and that makes me feel like a phony as well (although I haven't really studied that hard yet, maybe they'll go up? Also, its a good thing many applied programs don't care about the subject test...). Sure, I have some skills from my background that other students don't have, such as non-trivial software development and an ability to solve difficult physics problems, but when I look at the profiles of the 'good' people on this website, or talk to my friends that have been doing math olympiads since they were 3, I feel like there is just no way I can be good enough for a career in mathematics given my 'late' start.
Is it possible to catch up with others at this stage? Will I be doomed to a life of constantly being behind those who are smarter than me? Has anyone ever built a strong career from my kind of situation before?
I just wish I had some more time...

REPLY [5 votes]: I think you are over-worried, because most graduate schools require students to go over fundamental materials in the first year. It is true that some material will almost never be covered in such classes and assumed to be known by everyone, like implicit function theorem. But usually it does not take a lot of time and effort to catch up with other people on a specific area, unless that subject is extremely advanced or needs a lot of preparation by the student. However, most of the time students do not know what is their future area of study until second or third year of graduate school, and a lot of them even change thesis direction during the research process. I do not think there is a perfect map for success in graduate school, in the sense that if one entered into graduate school without knowing a certain subject he/she is expected to be in deep trouble; and students who know all these subjects are adequately prepared. 
However, it require a huge amount of personal work to push oneself beyond the amateur stage. I think you should have the confidence that you can compete with other people more advanced than you currently are not based on track record, but based on your interest in mathematics. If you believe you can contribute something unique belonging to yourself that has its importance to be justified in future, then you should focus your time and energy to build your career to expand these ideas into reality. On the other hand, if you found what you can do in mathematics is no different from other people, then maybe it is a wise choice to abandon it.  
I think only because we believe we can contribute something so creative that deserves to be immortal then shall we consider pursuing a career in mathematics. In this way, when Prof. Clark or someone boldly suggested you that the competition in job market is extremely keen and the chance to get tenure is even slimmer, you would be able to refute him or her by the fact you still managed to contribute something. Perelman considered himself an unsuccessful mathematican because he was having trouble to get a tenured professorship right after post doc. But to us his deep contribution is at a level much higher than that of most other experts in his research field. A very basic fact of a mathematican's career is that a proof is a proof, such that when it is finished and checked to be correct, one does not need other external incentive to judge its importance. 
Background: I am a second year graduate student in Binghamton University, ranked around 80-90 (when I applied here I did not even check this, as it is not important to me). Prof Clark specifically asked people to list it at here. So without disclosing further personal information this is my background. I just passed my quals and started research. I used to receive the same piece of advice as his from other people, and my old advisor (also Harvard graduated, has a respectable advisor, etc) suggested otherwise. There is obviously no right or wrong at here, but to me I really appreciate my advisor's support during a difficult time. 
(edit):
While having little time to respond to Prof. Clark's further detailed comments(it is 3am now), I wish to make the following (practical) suggestion to OP or anyone in OP's situation. I strongly recommend him or her to consider having an appointment with his/her advisor, and see what he/she says. In my own case, I was informed by my advisor that my mathematical caliber is enough for any top ranked graduated school, but my credentials(GPA, REU, etc) did not support that. And I should be prepared to enter a much lower ranked state university. While this was extremely disappointing I learned to prepare for my career early this way, and I am confident that I am the "best" graduate student in our department this year, even though I did not do very well as an undergraduate. 
I also want to suggest OP to consider staying for an extra year at his home university taking a few graduate level courses or considering applying for a master program catering his taste. Personally I have known quite a few fellow students followed this plan and done quite well(she was admitted by University of Chicago, Cornell, U of Michigan, Columbia, etc and chose Chicago in the end). I strongly recommend OP to try Cambridge's Part III program, which opens for students without a pure math background and do not require any GRE subject tests, but does care about GPA and reference letters. From what I heard of from friends, it is a great program. In case OP wants to apply PhD later, he/she would have improved his/her resume (and chance of grad school admittance) by showing he/she did well in Cambridge. My other friend who could only got into Arizona state university in the first year was accepted by NYU in the second year. 
I hope these suggestions might help, even though I have never been enrolled in any top graduate program(defined by rank) or served in its admission committee to make any suggestion with authority.