Personal Goals for Math Learning

Michael Hartl • Aug 3, 2023

In Math Learning Project, I wrote in somewhat vague terms about my current math learning project. “Learning math” means different things to different people, so it’s unclear exactly what such a goal might mean in my case. Indeed, given that I have a Ph.D. in theoretical physics, by most people’s standards I already “know math”.

In this post, I offer some details about what my math learning goals are, including both realistic goals and stretch goals. In a couple of planned future posts, I’ll discuss strategies and resources I’m planning to use to accomplish these goals.

1 Academic math

My main focus in this project is on traditional academic (pure) mathematics at the intermediate and advanced undergraduate level, and possibly at the graduate level as well. As noted in my previous post, these topics include things like number theory, abstract algebra, topology, differential geometry, algebraic geometry, and complex analysis.

  • Realistic goal: Knowledge and skill level equivalent to a bachelor’s degree in math from a good university. My undergraduate and graduate coursework in physics included around eight semesters of pure and (mostly) applied mathematics, so I’ve got a good start on this one, but (as noted in Math Learning Project) my knowledge base in rigorous mathematics is shaky, so there’s still a lot of work to do.

  • Stretch goal: Be able to pass a math Ph.D. qualifying exam (a.k.a. “quals”) at a top university. This is a high bar to clear, but even if I decide not to take such an exam, it’s still a good direction to go.1

2 Contest math

This refers to the material covered in tests like the American Mathematics Competitions (AMC 10/12), the American Invitational Mathematics Examination (AIME), and the USA Mathematical Olympiad (USAMO) at the high-school level, and the Putnam Mathematical Competition at the college level.2 Being able to perform well on math competitions is neither necessary nor sufficient for a solid foundation in mathematics, but contest math is a great complement to the more abstract math covered in academic courses and helps ground such studies in concrete problems. It’s also nice for review since contest math starts with material that is nominally at the high-school level but is often exceptionally challenging nonetheless.

  • Realistic goal: Be able to get a decent score on the AIME. I qualified for the AIME in high school, but had absolutely no idea what to do with it. It would be a matter of great personal satisfaction if I could do better the second time around.

  • Stretch goal: Be able to get a decent score on the Putnam. I was friends with a couple of Putnam Fellows in college (meaning they were among the top five scorers in the nation3), so I know how high this bar is. Some years, the median score on the Putnam is zero, so even getting a nonzero score would be a significant accomplishment.

3 Computational math

Over the past several decades, computers have gotten exceptionally good at performing mathematical calculations, even symbolic ones like algebra and calculus. Accordingly, I consider fluency in at least one mathematics software system4 to be a requirement for a first-rate mathematics education in the 21st century. The question then becomes, which system to learn? In grad school, I was a heavy user of Mathematica, which is an elegant and powerful system, but Mathematica’s proprietary nature clashes a bit with the ethos of mathematics (and scholarship in general). Fortunately, a great open-source alternative now exists in the form of SageMath (also called simply “Sage” when the context is clear).

  • Realistic goal: Solid command of the Sage mathematics software system. Sage is based on the popular Python programming language and uses a lot of the same packages as Python tools for data science; since I recently wrote a monster 100-page chapter on the latter subject, I’ve got a nice head start on this one.

  • Stretch goal: Be able to write original packages in Sage. It seems like this could be a significant advantage when considering active areas of research.

4 Special topics

Although I plan to follow the standard curriculum to a significant degree, I’m also aware that “standard” doesn’t necessarily mean “best”, so I’m always on the lookout for worthy subjects that sometimes get short shrift in the current system. Such subjects may include Clifford (geometric) algebra, differential forms, computational algebraic geometry, and category theory. I’m especially interested in fields like geometric algebra that have significant applications to theoretical physics.

  • Realistic goal: Have a good grounding in a few important subjects currently underemphasized in the standard curriculum.

  • Stretch goal: Get to a research frontier in at least one such subject.

That’s it for now. Stay tuned for some of the strategies and resources I’m planning to use to reach (or at least move toward) the goals above.

1. I’m not suggesting taking quals as part of actually enrolling in a math Ph.D. program, which would be both difficult to achieve and generally superfluous given that I already have a Ph.D. in a related field. But, as I’ll discuss in the post on math learning resources, some math Ph.D. programs post their quals online.
2. Having already graduated from both high school and college, I don’t actually qualify to participate in any of the contests mentioned. Luckily, though, in many cases old exams (with solutions!) are available. I’ll discuss this more in the post on math learning resources.
3. Technically, the Putnam is open to students in Canada as well, but then again the National Basketball Association, National Hockey League, and Major League Baseball’s American League all include Canadian teams (and MLB’s National League used to), so we’re on solid ground using “nation” to refer collectively to the US and Canada in this case, too.
4. Also called a computer algebra system, though such systems typically do much more than just algebra.