Fernando’s Notes for Majors in Mathematics and Mathematical Sciences

One of the messages we hear from alumni is that they would have liked to have some guidance as to what electives would have been most useful for them. To some extent, that is asking us to predict the future: we don’t really know what you will need. But it is certainly possible to list a few basic recommendations. I’ve never been shy about sharing my opinions with the world, so here goes.

These are my recommendations, not the department’s. But I suspect a lot of my colleagues would agree with much of this.

• Always remember the distinction between what is required and what is useful. No department wants to scare students by requiring too many courses for their major. That does not mean that they think it would be a good idea to take only the courses they require.
• Take Mathematical Reasoning early. No, don’t wait. Take it now. This is the course that defines the majors we offer, and also the course that will be most useful as you take other courses (and even more so if you go on to graduate school in any mathematical field). Students who are thinking of going to graduate school in Economics should also take this course.
• I once heard from a student that Linear Algebra was the most useful course she took, in the sense that ideas from that course kept coming up in all the other courses she took, both in Mathematics and in Computer Science. (She didn’t say Statistics and Data Science, but it would not be wrong to include them.)
• If you are seriously considering graduate school in mathematics, applied mathematics, or in mathematics-intensive areas such as economics, theoretical physics, or operations research, you really should be taking the heavy proof-oriented courses. Real Analysis, in particular, is crucial.
• Getting into a good graduate school is made much easier if you participate in a summer REU program. Not only does it give you a chance to see what doing mathematics research feels like, it also means you get to know a professor from some other college. It’s always good to have a recommendation from a professor in another college!
• The strongest possible mathematics major: take all the required courses for both the Mathematics and the Mathematical Sciences majors, as many 300-level and Topics courses as you can fit, and the complete Statistics minor. Do an REU in the summer. Do an honors thesis. Of course, if you can pull this off, you must have super-powers.
• Take a programming course. If you think you might go on to the “real world” and not graduate school, take a programming course. If you think you’re more likely to go to graduate school, take a programming course. Learn to use Excel, Mathematica, Python, Sage, R.
• If teaching is your goal, you should definitely take Abstract Algebra, Number Theory, History of Mathematics, Probability, and Mathematical Modeling. They will all be useful to you in the classroom.
• If you’re thinking you want to go to graduate school in mathematics or applied mathematics, take every course you can, but most especially Real Analysis, Abstract Algebra, Vector Calculus, Complex Analysis, Number Theory, and Differential Equations. And any geometry course you can find.
• If applied mathematics is your game, make sure you take Probability, Real Analysis, Differential Equations, and Mathematical Modeling. Oh yes, and a programming course.
• Talk to your advisor. Figure out what interests you, and ask your advisor what is relevant.
• Go to the library, check out books on mathematics, and read the first chapters. If you’re interested, read more. But just the first chapter will give you an idea of what that part of mathematics is about.
• Go to the colloquium talks. Ask questions. Sure, a goodly amount of what you hear will make little sense to you. That’s ok, learn to extract something worthwhile anyway.
• Be a member of your professional association. All mathematics majors are automatically members of the Mathematical Association of America. There are others you can belong to. In many cases the department can give you a free membership. When you graduate, continue your membership of at least one. This is an easy way to “pay it forward”, and you may also benefit directly.