This felt like it was written by a physicist or engineer.
Too much emphasis on differential equations and not enough on things like topology, functional analysis and/or non-introductory parts of algebra like say representation theory.
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.
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…
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.
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.).
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.
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.
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!
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.
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.
As someone with a keen interest in learning Engineering part time, I found your write ups really helpful though! I enjoy learning math but like to have an angle towards a practical and useful application. It keeps me a little more motivated than pure math learning. With ADHD the concept of being able to build cooler things always keeps me going. But somewhere along the way of learning purely theoretical things for too long my brain just loses interest (not enough reward), even though I enjoy it in the moment it is hard to get to the starting line and take the first step after a while :)
As someone studying teaching math, I found it interesting to compare her suggestions:
- Calc I-IV
- Intro to proofs
- Linear algebra
- (Abstract) algebra I-II
- Real analysis
- Complex analysis
- Ordinary differential equations
- Partial differential equations
- (Others)
to my program plan:
- A couple teaching courses (including one for roughly grades 5-8)
- Calc I-IV
- Statistics (one without calc, one with)
- Linear algebra
- Discrete
- Geometry
- Number theory
- History of math (apparently not just a history class, I haven't taken it yet)
- Abstract algebra and into to topology
Too much emphasis on differential equations and not enough on things like topology, functional analysis and/or non-introductory parts of algebra like say representation theory.