[mrjangles]

Yep, I came here to say this.

I think it is important that anyone who wants to study math understand that real math is not at all like what you learn in a physics or engineering department. In these departments you will always hear people say things like

>"proofs are not useful, all you have to do is memorize the 'trick' they use. Once you know which trick to use, it is easy"

or you will hear them say.

>"Math isn't about understanding, it is just about learning rules and symbols on paper".

This is not mathematics. These things do happen.... in a physics and engineering department. It is, in fact, a descriptions of a physics education, not a description of a mathematics education.

For this reason I would be careful taking mathematics advice from physicists too seriously as they may, unintentionally, lead you very far astray.

[susanrigetti]

For what it’s worth, the curriculum in this guide is modeled after the math major maps of many universities, including the one I attended (Penn). I would be curious to know what part of an undergraduate math curriculum will lead people very far astray…

[mrjangles]

It's just that it is very much a "mathematics for engineers" style course. I think very few of the subjects outlined there give you a flavor for what "real" mathematics is really all about at all (except for algebra, which you do mention).

Apart from the applied stuff you mention, the real core of a mathematics education involves, I think, 4 main areas with significant overlap

Group A: number theory, graph theory, combinatorics

which shares concepts with

Group B: Algebra, Topology, complex analysis, differential geometry, metric spaces...etc

which shares concepts with

Group C: Functional analysis, measure theory

which shares concepts with

Group D: probability and statistics.

As for the applied math that you mention, you should really need to add vector calculus and I'd highly encourage anyone to take a course on fluid mechanics (from a mathematics department instead of an engineering department) to get a real feel for vector calculus in action.

[susanrigetti]

I suggest taking another look at the list and comparing it to the required courses of the undergraduate math majors at the top 20 universities in the USA.

Real analysis, complex analysis, topology, and number theory are there (topology and number theory are both listed as electives since most math programs categorize them as such). Graph theory, functional analysis, differential geometry, probability, and statistics are almost always either electives or graduate courses.

It’s funny, because most of the things you mention as “real math” are things that many math undergraduates don’t learn (not until graduate school at least) but that physics students learn as undergraduates (differential geometry, measure theory, functional analysis, etc.).

[mrjangles]

I have never met a physics student that even knows what functional analysis even is, despite it being at the core of quantum mechanics. I can't even imagine why someone in a physics department would learn about measure theory. True that any student learning General Relativity will get an introduction to differential geometry.

[adastra22]

Except for the course ordering, it lines up almost identically with the math major at my university. I'm not sure what the other posters are going on about. Most of their preferred topics that they feel you missed are either upper division electives or graduate level.

[susanrigetti]

I suspect that people forget that undergraduate programs don’t really cover very much. This true not just for math, but for pretty much every other major. I mean, think about how little of physics you learn if you only take undergraduate courses!

[adastra22]

Yeah I was a physics major and honestly it felt like we barely scratched the surface. An undergraduate degree in physics is sufficient to be a high school physics teacher, but not to be a physicist of any sort.

[Py-o7]

What I had in mind was something more or less in line with https://math.mit.edu/academics/undergrad/major/course18/pure...

though I am a little surprised that they have 1 course of differential equations in there instead of complex analysis as a required topic, as I think the latter is a better pure math topic. But it's MIT, so be it. Whether directly or indirectly, many of us learned to view math the MIT way by patiently working through foundational books like Artin and Munkres.

That said, my mention of non-introductory algebra topics probably is more of a personal idiosyncracy/interest.