Identification of Schools:
I read that it is a good idea to have about 10 to 12 schools identified as possible choices. I reviewed U.S. News & World Report's issue on America's Best Graduate Schools, which also ranks colleges and universities by discipline. Graduate programs vary from institution to institution in their reputation and focus, so you have to be careful. Remember, our professors are a great resource in helping to identify the good schools.
I knew that I had to be realistic in my approach, so I chose and grouped my selections by target schools, reach schools, and safety schools. Included in my research of the schools were the research interests of the faculty (I wanted to make sure that most of the schools to which I applied offered strong support in analysis, topology, and dynamical systems.), the size of the school, the testing requirements (was the GRE subject test required or just recommended), financial packages (teaching vs. research assistantships), etc.
Being a warm weather person, I first focused on schools in temperate climates. So, some of my "southern" possibilities were Duke, University of North Carolina at Chapel Hill, North Carolina State, University of Virginia, Emory, Rice, Virginia Tech, Vanderbilt University, and the University of South Carolina. (A SMCM grad was a PhD candidate at USC, where he was doing very well and having a really good experience.) I also "threw in" University of Maryland and Johns Hopkins, really good schools near my "home base." My professors advised me to also include in the mix the University of Wisconsin at Madison, University of California at Davis, and University of Michigan at Ann Arbor.
I ended up with 4 schools in the "reach" range, 6 in the "target" range, and 2 "safeties." This made the process expensive, since the application fees charged by the schools ranged from $40 to around $60, but was worth it in the long run. You'll also find that some schools waive or reduce the fee if you apply on-line rather than by mail. The participants of a graduate school panel presented as part of a summer math program I attended advised not to visit campuses prior to acceptance. They were right, since the schools to which I received letters of admission paid for a good part of my travel expenses to visit and attend open houses.
Application Process:
It is important to carefully research application instructions through both the graduate admissions and math department websites. You might have to send transcripts, test scores, statement of purpose, letters of recommendation, and the application to one or two different offices.
I realized quickly that procrastination would not enhance the process, so I was prepared to apply early - deadlines vary from school to school and can come upon you quickly. Some schools to which I applied recommended that all information be in by December 1. This included test scores, statement of purpose, and letters of recommendation.
It helped me to be really organized. I created an Excel spreadsheet that listed the various schools, their application deadlines, where the various supporting documentation had to be sent, etc.
It is also important to start working on the statement of purpose as soon as possible in the school year. For most schools, you can use the same, core essay and then modify it slightly according to the requirements of the respective schools.
You also have to check with each school to make sure that your application is complete and all materials have been received. This is the applicant's responsibility. If you apply on-line (the best way), you can periodically check the Web site; you can also phone or email the graduate coordinator or adviser.
Offer of Admission:
The offer of acceptance typically comes with a financial package, which can include a waiver of tuition plus a teaching or research assistantship worth between $13,500 to around $16,000 and health insurance. Sometimes there is a first-year "signing bonus" and/or the promise of summer employment. The department also sponsors graduate weekends or open houses. I went to as many as I could, although the times were often inconvenient relative to my coursework at St. Mary's (mid-terms and my senior thesis), and tried to attend graduate classes if offered. The open houses provided a great opportunity to meet the faculty and present grad students and learn what the department was all about. Some questions that need to be asked: "What are the placement rates and types of jobs obtained by recent graduates?"; "Is the financial support reliable year to year?"; "What are the primary responsibilities of the teaching assistant?"; "What are some of the research topics of present doctoral candidates?" Visiting the campus also provides an opportunity to explore the town/city and research apartment complexes and availability.