I am a senior maths major (computer science minor) who is pretty worried about the next step in my academic career. First, let me state that I'm about as sure as I can be that I want to get a PhD in mathematics. Unfortunately, I didn't realize what the field entailed, or my passion for it, until I had made many really poor decisions - mostly in the form of bad attendance. For example, I basically just showed up for tests in calc 2, 3, linear algebra, and differential equations, and consequently, my field of potential letters of recommendation is quite small. To make things worse, I come from a party school - I need letters!
I've had one professor (abstract algebra) offer to write me a letter, and I've taken my advanced calc sequence under a professor who I think could write me a good recommendation (adv calc 2 was a graduate course; had [i think] the highest grade out of about 15 students). I'm also taking topology (graduate level) this semester, and am hoping to impress my way to a third letter.
My GPA is okay - cumulative about 3.61; math is all A's and one C in linear algebra. I've also been working through a few books (Spivak's "Calculus" and "Calculus on Manifolds," and am about to start Birkhoff and Maclane's "Survey of Modern Algebra." Although I love the material, and enjoy learning it, the independent studying probably stems from some feeling of inadequacy due to my past immaturity.
I got a 169/170 Q, 165/170 V on the general GRE. Also, I think I can crack 80% on the subject test, but am not overly confident about this. One glaring hole is that I have done zero research, and have done nothing extracurricular - I literally have nothing "extra" going for me.
My concern is that I've seen the resumes of many people accepted to top universities (PhD track), and I just don't stack up. But if my goal is to become a professor one day, it seems that where I go to school is extremely important. So should I just hope that I can get accepted into a top 30-50 school, or would it be beneficial to consider improving my resume in a solid Masters program so that better schools become available?
And if a masters is a viable option, what caliber of school would I need to excel at in order to be a competitive applicant for a top 10 PhD program?
I've always thought that one good indicator of whether or not you even have the motivation to complete an entire PHD is whether or not you do the practice problems in text books. If you enthusiastically do those practice problems, like you solve them in the shower, then I would say that, barring talent, you at least have the requisite level of enthusiasm for the subject. In other words if your not a fan of practice problems, your prolly not gunna like the 300 page writing part of the PHD, nor the fact that not all 100% of the work you do will make it into that writeup (there's alot of tangential calculation and verification). In this way, personal interest and commitment to mathematical activity is absolutely requisite.
You should wait until you finish that course in topology. Math takes on a different character when you get into analysis, manifolds, algebra and beyond. For me, smooth manifolds was as far as I needed to go in the analysis route to satisfy my curiosity. Then I became interested in other things. If I had had that shift of interest midPHD then I don't think I would have been able to finish.
You should also just sit down, learn LaTex if you haven't already, and write about something that interests you, exploring it to the absolute highest level of detail while always leaving an obvious path for generalization and application. Make it lucid and interesting. Convince the reader you have an idea and entertain them with it. Put it on the Internets, have a proff edit 1 or 2 pages, show it to a friend or classmate, stick it in a library book, whatever. This is one defining characteristic of a mathematician, communicating your thoughts to paper so that they may survive.
You should read this The Best Writing on Mathematics 2011, it will give you a good idea of what doing math as a profession is like. Regardless, you may have an excellent academic record, but what makes a good mathematician is a commitment to doing math and that should be your primary focus, grades second. Although there's nothing wrong with being competitive academically, if that's your thing then go for it.
Personally I didn't like Spivak's manifold calculus text. I went Munkres' Topology, to Lee's Topological Manifolds, and have yet to finish Lee's Smooth Manifolds. If your looking for a reliable publisher, just stick with the yellow covers.