An overview of the Barnumbia mathematics program.

The mathematics major at Barnard/Columbia is one that, despite how daunting it may sound, offers a lot of flexibility and support from a tight-knit community. This article aims to spotlight the generic “mathematics” major. However, there are also a variety of other majors under the math major umbrella such as applied mathematics, mathematical sciences, and mathematical computer science with similar requirements.

Requirements for the major:

The major requires 14 courses and a minimum of 35 credits.

• Calc I-IV: You can test out of I-II with AP, IB, etc. credits, but plenty of math majors start at Calc I and work their way up! Regardless, you will either need to take Calc III-IV or some substitution like Accelerated Multivariable or Honors Math A-B.
• Linear Algebra: This is the most useful foundational topic in theoretical math, as well as a lot of compsci and physics. You’ll learn about vectors and vector spaces, matrices, and solving systems of linear equations. You can also replace this with Honors Math A-B.
• Honors Math A & B: An optional, intensive year-long course replacing the above two requirements. You learn linear algebra, multivariable calculus, and depending on the instructor, some rudimentary things from analysis or manifolds, all from a rigorous proof-based perspective. The course is definitely challenging, but it excellently prepares you for more advanced proof-based math courses, and usually involves fun things like writing projects about a mathematical topic of your choice. Plus, because it’s a year long, the class and instructor develop a strong sense of camaraderie.
• Modern Algebra I-II: No, not the same algebra you did in high school (although you see how they’re related towards the end). This involves sets, groups, rings, and fields; basically, how do we combine objects in a set and what structure does that give us? This sequence is super important, especially if you are interested in number theory or anything with “algebra” in its name. It’s also a prerequisite for basically everything (especially Modern Algebra I), so better to take it sooner rather than later.
• Modern Analysis I-II: This class involves actually proving the theorems/techniques you learn in calculus, and then moves on to more complex topics like doing calculus on spaces of functions, etc. It’s generally considered to be one of the more challenging courses in the major, but like modern algebra it opens tons of doors once you do take it.
• Undergraduate Seminars: Generally taken in your junior or senior year, you sign up for the general course. Then, in the beginning of the semester, the grad students/faculty running seminars announce their topics, and you can try to join one that sounds interesting to you. Each student gives at least one talk for a grade. You get as much out of these as you put in—the classes are best when the students are really interested in learning the topic and put effort into asking questions and giving clear talks.

After that, the courses you take are really up to you! One of the best things about the math major is the flexibility it allows you in what you study. Some of the authors’ favorite courses are Topology, Algebraic Number Theory, Probability Theory, Differentiable Manifolds, and ODEs. It’s also easy to get into PhD level courses after your sophomore year if you are interested in going to graduate school.

Notes on the classes:

Many people find the Algebra and/or Analysis sequences particularly challenging because it’s their first proof-based course. If you haven’t seen any formal proofs before taking one of these classes, I highly recommend either taking Introduction to Higher Mathematics beforehand, signing up for the Intro to Proof Workshops that Undergraduate Math Society runs at the start of each semester, or reviewing some books with proofs with them on your own.

Notes on research/guided readings:

Directed Reading Program: Each semester, students can apply to be paired with a volunteer graduate student to participate in the Directed Reading Program (DRP). Students pick a topic to study along with a book to study it from (the grad students usually have good suggestions), and then students read the book, meeting once a week to talk about their progress with the graduate student. At the end of the semester, students give a ten-minute presentation about what they learned.

I’m currently in the DRP, studying p-adic analysis. So far, the experience has been really enriching. Also, the math department will buy the book for you!

Research: There are plenty of summer REUs (research experiences for undergraduates) around the country you can apply for if you’re interested in doing research, as well as one right here at Columbia, specifically for Columbia students. Another good alternative is reaching out to a professor about doing a guided reading with you over the summer; this option is particularly good for underclassmen who may not have the requirements to apply for an REU yet.

Department communications:

There’s a super secret math major mailing list. You can also join the Columbia Undergraduate Math Society (see Gina’s article) mailing list, which will not only tell you about events for that club but also general useful resources within the math department.

Declaring the major:

This is pretty straightforward and doesn’t require any approval. For Barnard students, I recommend reaching out to potential advisors (currently, this is either Dusa McDuff or Daniela de Silva) to ask if they would be okay with you requesting them as an advisor before doing so.

Last minute tips:

Form study groups and get to know other people in the major! Despite the anti-social reputation of mathematicians, the math department has a well-maintained social scene and most people, from undergrads to grad students to faculty, are very excited to help others learn math.

Math lawns via Bwarchives