Recommended Coursework for Graduate School
My advice in these pages can be derived from the principle
that you should figure out what you want to be doing in 5 years. And
then figure out what you need to do now to make that happen.
In this case, I will assume that you want to get a Ph.D. in pure
mathematics, but the same principles apply if you want to get a graduate
degree in Physics, Economics, or Computer Science. The summary: you
need letters, research experience, and coursework chosen to prepare
you for the graduate program in question.
Letters of Recommendation
To get into a mathematics graduate program, you will need letters of
recommendation. This means you need to take classes that are small
enough that your instructor becomes familiar with your work and has
some basis for writing the letter. (You can find my remarks about
letters here.) Of course it is an enormous
help if you excel in these classes, so your instructor finds it easy
to check the "top 2%" box that is inevitably on the form.
Research Experience
It is also desirable to have some familiarity with mathematical
research. One popular way to do this is the
REU (Research
Experience for Undergraduates). From the point of view of the
graduate school applicant, these programs have many advantages.
First, you find out whether you enjoy research generally, and whether
you're interested in the particular topic studied. Second, the
presence of the program on your application formally demonstrates your
interest. Third, the program's supervisors may be able to write you a
letter of recommendation. (This is often a concern at state
universities with very high student-to-faculty ratios!)
Coursework: Bachelor of Science
Finally, the coursework. Here, my recommendation is simple to state,
but will keep you busy for years: get the Bachelor of
Science and take the "mathematics track". (See
the undergraduate
handbook.)
For graduate work in pure math research, the
standard recommendation is one full year of study of each of the core
areas: algebra, analysis, and geometry/topology. At SUNY-Binghamton,
the specific courses I mean are 401-402, 478-479, and 461-462. Of
these courses, I would single out as most important Math 478. This is
because the definition of limit, which is the fundamental definition
in analysis, is more complex than the foundational definitions in, for
example, group theory. Thus, mastery of the material in introductory
analysis demonstrates correspondingly stronger proof-writing skills.
The previous paragraph need not be taken completely literally; there
are alternate programs which would fulfill the goal of "one year in
each of the core areas". For example, differential geometry or
partial differential equations, might be core areas, depending on your
interests. For more on the topic, consult the online
undergraduate handbook.
In any case, these recommendations should not be taken as a license to
ignore the fundamentals. For example, linear algebra is an incredibly
useful subject in virtually every part of mathematics, pure and
applied. And if you haven't taken Math 330 yet, there's
no need to worry to much about my advice here.
Coursework Modeled on Standard Graduate Program
The one-year-in-each-core-area recommendation is meant to provide
breadth and prepare for the standard first-year graduate
curriculum. (Which, coincidentally, is one year of study in each area,
followed by a comprehensive (or "qualifying", or "master's") exam.)
The ideal curriculum should also provide greater depth in at least one
area. This could be in one of the core areas, or some new direction,
like analytic number theory.
The principle which is applicable to future Ph.D. candidates in any
subject is to find out what the standard first-year graduate program
is, and model your coursework on that. This is the standard way to
have broad preparation. Depth is more a matter of individual preference.