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Nowadays, when I look at mathematics programs of study, "algebra" (at the abstract level) and "analysis" are treated as equally important.

I'm "dating" myself, but this did not appear to be true in the 1970s, when I studied math. This was just past the "space" age, and at that time, "analysis" (and its connections with calculus) was all the rage, with "algebra" (abstract algebra, algebraic topology, etc.) being relegated to the "bench."

Is it less than 40 years that algebra has become the equal of analysis importance? If so, why might that be; perhaps the greater use of "strings" and "arrays" by computers?

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    $\begingroup$ I would guess the interplay between algebra and analysis has grown in the past few decades and that inteaction has necessitated even the more pure analyst to learn a lot of algebra. Modern number theory seems to attract a lot of attention, look at the breadth of education needed there. All of this said, I wonder if education is a correct measure of importance. Other metrics would probably say pure math is useless and only applied math or heaven forbid actuarial science are relatively important. Especially if that metric is related to the replacement of education with job-training... $\endgroup$
    – James S. Cook
    Jun 21, 2018 at 1:35
  • $\begingroup$ Data analysis, DCF-NPV modeling. That's some good useful stuff. Not Moebius strips. $\endgroup$
    – guest
    Jun 21, 2018 at 4:40
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    $\begingroup$ It might just be the background in which you did your studies. I started studying math in the 1970's as well, first as an undergraduate at Michigan State and later as a graduate student at Princeton; algebra and (algebraic) topology were as important overall as analysis was. $\endgroup$
    – Lee Mosher
    Jun 22, 2018 at 19:27

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One guess might be the importance of algebraic logic for, among other things, proving that computer programs behave as desired. Algebraic logic is also quite close to topology --e.g., via category theory-- however this is close to analysis and so the two have a bit of a bridging between them.

Another reason, also from computing, is the rising trend in functional programming which is essentially executable mathematics and thus supports equational --read algebraic-- reasoning. This form of programming, among others, makes abstract algebra not only concrete but also accessible to a wide range of people with no heavy mathematical training.

For example, the algebraic and analysis notions of monoid, topology, and derivative arise quite naturally as, respectively, systems of composition, notions of computability, and as the one-hole contexts of regular data types. For more on these particular examples, see

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  • $\begingroup$ "Functional languages" and "proving programs correct" are as old as computers, and still are tiny niches in day-to-day computing. $\endgroup$
    – vonbrand
    Jul 4, 2018 at 16:51
  • $\begingroup$ Not at all! For example, Microsoft's popular C# language supports lambdas and linq-syntax as well as preferring higher order functionals in-place of for-loops and other similar low-level approaches. C# is quiet accessible and they make the functional paradigm easy to grasp. The super esoteric notion of monads from Haskell is also presented in a super friendly way as SQL-like syntax: The Linq libraries. That you may not see functional programs regularly in your life does not mean this is the same for others. $\endgroup$
    – Musa Al-hassy
    Jul 4, 2018 at 18:03
  • $\begingroup$ True. Also in C++, Python 3. But it is a very long way from being central in the language. $\endgroup$
    – vonbrand
    Jul 4, 2018 at 18:05
  • $\begingroup$ To follow up, the notion of types in programming languages is a simple form and excellent gateway into program correctness. Common "non-functional* languages like Java, C, C#, Kotlin, Go, and to some degree even Python have types! I would not call that a "tiny niche" by any stretch of the word! We use types to ensure we have meaningful expressions, which is a step into program correctness. $\endgroup$
    – Musa Al-hassy
    Jul 4, 2018 at 18:07
  • $\begingroup$ Oh, come on. Yes, in all languages objects have types. Yes, static (explicit) typing can help catching programming errors, helps in generating efficient code, and definitely makes some useful programming hard. $\endgroup$
    – vonbrand
    Jul 5, 2018 at 12:09
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  1. Theoretical Physics: since the 50ies the geometrization of theoretical physics has been costantly increasing. While in 50ies-80ies this meant mostly differential geometry/algebraic topology nowadays this means mostly algebraic geometry/category theory. And for that you need a lot of algebra.

    1. Numerical linear algebra. This field has been progressing enormously. To understand its usage you should be familiar with advanced linear algebra and quite a lot of matrix theory. Applications are countless.

    2. Graph theory. Ubiquitous in math and in applications as a very powerful tool to summarize complex informations. 30 years ago you would probably study it "by hands" while nowadays you have all sort of algebraic tools to apply to it.

Nevertheless I tend to agree with the essence of Lee Mosher's comment. All this is very much dependent on where you are and what you look at. While there certainly was an increase in interest on algebra outside math, from the inside it is very difficult to see such a defined path. Personally, for example, I'd say that the biggest success story in last 20 years of math is geometric PDE. But I may be completely wrong.

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I think a big area has been computers and crypto and such.

But even before that, I would not underestimate chemistry/physics (molecules with point symmetry affecting IR modes and crystals with space groups and such in x ray crystallography). Is a very practical application.

The sad thing is the math field has so lost track of what benefits are for practical types that it ends up being better to take a group theory for chemists class (even the miserable Cotton book) than deal with normal abstract algebra. [Same thing is of danger in happening to statistics!]

And of course, the people who first founded the field would be at home with practical types more than math theory types.

[Whole comment above is a guess, a Bayesian assessment on Friday with a beer. But still...trust your heart.]

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