In "Starting Graduate School: Mathematics Training, Part 1," I discussed the initial plan of attack  for the first year and probably one of the most important topics, selecting an adviser. Part 2 will mention a few more intangibles including grad school philosophy, social activities, and a look back at my first year.
First Year at the Great Show
You may remember that it was a big leap from high school to college. It was tough to be totally on your own. The pace was (or should have been) a lot faster than in high school, and competition was fierce. Mathematics students had to take history or biology classes with history and/or biology majors.
The leap from college to graduate school is even harder. In an attempt to catch up, it is not unusual to see graduate students enroll in senior-level undergraduate mathematics courses because those offered at their own undergraduate institutions were taught at a lower level.
Despite their best intentions, students can have a hard time adapting to the graduate school "philosophy." Class does not always end at the completion of textbook exercises but possibly at the library, mathematics lounge, or your faculty's blackboard. Doing the homework is essential, but it isn't enough. Faculty members often expect to see students try to push the envelope by attending relevant seminars and departmental colloquia. Students are supposed to teach themselves relevant research tools (computational, statistical, or technical word processing). Of course, there may be short courses or seminars that help, but it is assumed that students will find a way to grow in all possible academic directions on their own. Nobody will be watching you per se, but if a student does not show up for seminars or relevant academic activities, it will be noticed.
Grading in purely graduate-level courses can be misleading. You are expected to get mostly A's and B's, but some faculty members may routinely give A's and B's and limited homework. It is your responsibility to learn the material. You may have to make your own homework in preparation for passing key exams (qualifiers, comprehensives).
If you are supported by a research or teaching assistantship, then be prepared to work 15 to 20 hours per week. Be a professional and do the best possible work in this capacity, but do not forget that you are at this institution to get a Ph.D. Getting a Ph.D. is the priority.
It is customary to take at least three courses per semester (a load that will keep you off the streets). If you have a research or teaching assistantship, then taking two courses is often the recommended way to go. Getting lots of credits is not the key to success; mathematical growth is. Participation in activities (such as seminars) enhances your research skills while helping you find a research path that can be embraced with passion.
Obviously, it is important that you attempt to have a balanced life. Getting involved in social activities with your peers is highly desirable, but your social life does not end at the volleyball, basketball, or squash courts or at the local pub. Because you are starting an apprenticeship that will help you become a professional in the academic, research, or industrial arenas, you must spend time with all your peers. Be sure to occasionally have coffee or lunch with first-year and advanced graduate students, postdoctoral fellows, and faculty members. Try to participate in most (if not all) departmental social activities, and make it your job to meet new people. Do not just hang around with your friends.
Networking has tremendous advantages, and the earlier you start the better. As a result, you may get suggestions that will help you in your specific research or learn new connections between various types of mathematics. If being a mathematician is what you want to do for life, then meeting your future colleagues up close and personal should be exciting.
I attended a large program in mathematics at the University of Wisconsin, Madison (UW Madison). At UW Madison it was easy to remain anonymous--an excellent route toward isolation. However, my gregarious personality prompted me to visit the ninth floor mathematics lounge every morning. Here, I could not only see a great view of Lake Mendota, but I could also have a cup of not-so-good coffee and meet the faculty, visitors, and graduate students. From my very first trip to the ninth floor, I sat with a group that included faculty members. Soon, I became a "regular."
In the 3 years I spent at UW Madison, I was always welcomed and treated extremely well by the faculty. As a regular, I had the chance to get to know individuals such as Anatole Beck, Fred Brauer, Joshua Chover, Simon Hellerstein, and Mary Ellen and Walter Rudin. They are not only great mathematicians but also wonderful human beings.
The fortuitous decision to sit with faculty changed my life. I not only learned about mathematics and the profession but also about politics and life in general. Through this interaction, I experienced firsthand that a large mathematics department can be warm and caring.
I have maintained connections with some of my ninth-floor friends for the past 20 years, and we often have warm interactions at national meetings. I am sure that without this venue for social networking, I would have been lost. I may have never completed my degree. Being part of a community is fundamental.
Again, students should integrate themselves into their program or department during the first weeks of the first semester. In the process, students will get to know remarkable individuals, up close and personal. Students will find faculty members less perfect, more interesting, and more admirable. You will find many role models.
Carlos Castillo-Chavez is a Joaquin Bustoz Jr. Professor of Mathematical Biology at Arizona State University and can be reached at firstname.lastname@example.org .