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In undergraduate maths study, there are three main areas: analysis, algebra, and geometry. (There are of course other small topics as well, but they don't have to be learnt by every student.)

I have discovered that, in most standard undergraduate curricula, there are more analysis topics than modern algebra topics. (Many words can be put before "analysis", including "real", "complex", "functional", "harmonic", etc.) Geometry, on the other hand, only forms a very small portion of the courses taken.

Why this is the case? Why is analysis more "important" than algebra or anything else?

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    $\begingroup$ I'm not so convinced that analysis is covered so deeply in undergraduate curriculum. Some of the courses mentioned really don't cover analysis in depth, in fact, the undergraduate course is largely about the algebra or more often the linear algebra of the problem. I don't see a lot of undergraduate courses doing serious analysis. At least, I would fear attempting such a thing with typical undergraduates in the US. $\endgroup$ – James S. Cook Oct 5 at 2:44
  • $\begingroup$ @James I don't know other's case, but I really find that my techniques to attack, say, Galois theory, is not comparable to that of manipulations of measures. I mean, I collected roughly the course I took in the same year in the undergrad curriculum. Analysis was much more stressed than algebra. Other than courses listed here, I also find probability theory should be considered analytic. Things like law of large numbers, Kolmogorov's theorem of three series, etc, are essentially analytic. $\endgroup$ – Yai0Phah Oct 15 at 20:24
  • $\begingroup$ @James In my undergrad curriculum, algebra just stops at Galois theory in a one semester abstract algebra course. I find it completely insufficient. I self-learned most algebras like homological algebra, commutative algebra, etc. No undergrad course was available. By the way, no algebraic topology was covered other than fundamental groups. $\endgroup$ – Yai0Phah Oct 15 at 20:31
  • $\begingroup$ @Yai0Phah sounds like your program did a better job on the analysis side. Curricula varies from place to place depending on the available expertise and/or power of said individuals in crafting the program of study for the standard programs. Truth is that one size never really fits all, but in the age of the internet we are far less bound by our local circumstance. $\endgroup$ – James S. Cook Oct 16 at 0:50
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Analysis is useful

Physics uses analysis. Engineering uses analysis.

Continuous models are widespread everywhere, and typically they look like differential equations, if there is time involved.

Finance uses lots of analysis.

Note that a lot of modern geometry is differential geometry, which builds on a foundation of analysis.

Certainly, other parts of mathematics are also used, but they seem to not be as fundamental yet. Note also that many students learn statistics for the same reason - it is used widely.

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  • $\begingroup$ And indeed Statistics also needs a lot of analysis and linear algebra $\endgroup$ – David Oct 7 at 10:29
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(Too long for a comment and it is kind of a soft question anyways)

I'm not so sure that your assumption of an imbalance is valid. Maybe it is. But would be better if demonstrated (or at least explored) first. Otherwise, we end up finding an explanation for a phenomenon that doesn't actually exist. Plus the exploration would probably inform the answer, even if assumption shown correct. (For example perhaps there are differences in schools with/without grad programs.)

I looked at the course curriculum for USNA math and it seemed like there were about equal classes in algebra and analysis (maybe even more in algebra).

https://www.usna.edu/Academics/Majors-and-Courses/course-description/SM.php

There were two semesters of abstract algebra and three of linear algebra. (Different difficulty "tracks" also, but I'm just talking the sequence.) The formal analysis only included two semesters of real analysis and one of complex analysis. Again with a confusing multiplicity of tracks, but just the three real semester courses. No functional analysis as the institution is undergrad only.

I guess you could call the normal computational normal calculus courses, "analysis". Or ODEs, PDEs. But I'm not so sure this is valid. There's definitely a different feel for what people think of or use as calculus (solving problems) versus real analysis (theory).

In addition, if you're going to count the foundational conventional calculus semesters, then I can always bring in the four (4!) semesters of high school algebra that students typically take in the US of A.

On the actual explanation, I would say that "calculus" (and diffyQs is calculus on steroids) is useful in science and engineering. But this is a different thing than the highfalutin math major "analysis" of your question. So I think there are more likely other drivers for the math major interest in the topicm, other than physics applications. Complex analysis, has way less application than calc 1-3 and diffyscrews. (Pedants: I'm not saying there is never a single application, but not enough to make the topic a general STEM need and a required course.) Real analysis even more so not needed, not required to be taken, etc. etc. So don't blame your functional analysis courses on science and engineering.

Note also, that HIGH SCHOOL algebra is a HUGE useful topic in science and engineering...freshman chem is essentially one giant algebra word problem. (In a similar manner to how calculus and diffyQs are work horses.) But did you even think about that when discussing algebra in the context of your question?

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