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I'm re-writing a question I posted yesterday, put in a way as general as possible which may hopefully be of use to others in the future in a similar position.

I would like to pursue a career as an applied mathematician. I'm interested in mathematical modelling of real world systems in areas such as biology and physics. And I'm generally interested in using deep mathematics to solve problems.

I want to keep my options as wide as possible, and want to make sure my mathematical training allows for this.

I'm currently studying for my Bachelor's degree in CS and Physics. After my Bachelor's I will pursue a Master's in applied mathematics. I'm debating whether to switch to a Math major for my Bachelor's, to get more rigorous math training.

My question (which I think is generalizable to many people), is about the kind of mathematical training one should attain to succeed in applied math, and keeping one's future options wide.

Is highly-rigorous mathematical training important for this field? Is one mathematically limited in their abilities, if their major is a non-math-STEM-field (such as CS and Physics)?

Or is the level of rigor in a typical non-Math-STEM-major serious enough for wide horizons in applied mathematics, and leaves room to fill in the gaps during the Master's?

If I might, let me focus my area of concern a bit:

I'm able to take a few extra math classes in my current non-math STEM Bachelor's as needed. I can also always fill in a few specific courses in the future, before my Master's. So I'm not that concerned with missing 2-3 math topics in the Bachelor's.

My specific question is along the lines of: Is the high-rigor sustained throughout a math Bachelor valuable for the aspiring applied mathematician?

Do I end up limited mathematically, if I "skip" this level of mathematical depth and hardship, and stick with mostly the CS/Physics/Engineering level of math understanding for most of my Bachelor's?

I just came across a post on this site which talked about Mathematical Maturity. This is exactly what I fear I might be missing if I don't go the Math-major route:

... fearlessness in the face of symbols: the ability to read and understand notation, to introduce clear and useful notation when appropriate (and not otherwise!), and a general facility of expression in the terse—but crisp and exact—language that mathematicians use to communicate ideas. [...] And also: The capacity to handle increasingly abstract ideas.

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    I suspect that your question is leaning a little in the direction of a possible false dichotomy. It isn't necessarily all or nothing - either non-rigorous courses with applications or a full-on, say, "pure" mathematics Bachelor's degree with a lot of rigorous courses. Courses themselves will vary and you might be able to take a rigorous mathematics course or two alongside your other courses.
    – J W
    Commented 2 days ago

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As one data point, I'm a professor of mathematics. My undergraduate education (in Germany in the 1990s, comparable today to a Masters degree) is in physics. That did include a solid background in applied mathematics topics, such as analysis, partial differential equations, and linear algebra. I then got a PhD in mathematics. I have quite a number of friends in the computational mathematics community who have similar backgrounds.

Personally, having an education in something other than mathematics has given me the ability to talk to scientists from many applied disciplines (outside mathematics) in the language they speak, and then translate their problems into the language of mathematics where I can solve them.

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  • Thank you for the answer. This is indeed encouraging. At the possible risk of over-analyzing this, do you remember whether there was a delta between the basic math classes you took, and the equivalent ones that math students took? For example, in my university, I took Linear Algebra for CS Students, which has a different level of rigor than the equivalent course for the math students. My concern is less about "missing" a few math topics - because I can learn them later on before my Master's. It's more about whether my somewhat lesser basis of mathematical training will be limiting later on.
    – Aviv Cohn
    Commented 2 days ago
  • I just came across a post somewhere on this site, which describes exactly what I'm concerned I might be missing by not picking the Math-major route. "Mathematical Maturity": "The capacity to handle increasingly abstract ideas, ... fearlessness in the face of symbols: the ability to read and understand notation, to introduce clear and useful notation when appropriate (and not otherwise!), and a general facility of expression in the terse—but crisp and exact—language that mathematicians use to communicate ideas." Do you feel that a Physics major sufficiently grants these skills?
    – Aviv Cohn
    Commented 2 days ago
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    @AvivCohn: Your question is impossible to answer without more knowledge. I suspect a good physics major imparts these skills to some degree. Then again, many math majors at my university graduate without these skills, and judging by our graduate students and their recommendation letters, there are universities where no math majors graduate with these skills.
    – Alexander Woo
    Commented 2 days ago
  • @AvivCohn My math courses were pretty good, but there was a delta. The biggest delta, however, is that I have no background at all in pure mathematics. I'm a professor of mathematics, but I never took a course in number theory, algebra, combinatorics, topology, geometry, or algebraic geometry.
    – Wolfgang Bangerth
    Commented yesterday
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Applied mathematics research spans a wide range of activities.

Some applied mathematicians focus on developing new mathematical methods, usually with the intent that they be used in applications outside of mathematics; their work may sometimes be difficult to distinguish from the work of theoretical mathematicians.

Some applied mathematicians focus on applying known mathematical methods to questions outside of mathematics; particularly in cases where the applicability of these methods to the particular domain is already understood, their work may be difficult to distinguish from that of (more mathematically minded) researchers in those fields.

Most applied mathematicians are some mixture of both.

The more developing mathematical methods are part of your work, the more important it is for you to have an undergraduate background in theoretical mathematics.

Do note that, at some point, using known methods of known applicability becomes no longer research because you are no longer creating new knowledge but applying it in known ways.

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I think there's no easy answer to your question. On the one hand, it's probably unwise to learn applied mathematics in a vacuum. You should have something to apply mathematics to. So, studying physics, biology, economics, engineering or some other discipline is to be recommended, especially if you are interested in mathematical modeling. There are some, though, who focus on solution techniques for differential and other equations (applied and numerical analysis) and/or approximation theory, concerning themslves less with modeling as such. This can require anything from good skills in programming and mathematical software to strong theoretical knowledge (e.g. in functional analysis) to prove theorems in applied/numerical analysis.

So, if you want to keep your options open, then find a way to take at least some real analysis during your undergraduate studies. And check what is offered at the Master's level. In some countries, functional analysis, for instance, is usually taught at advanced Bachelor level; in others, it is a first-year Master's topic.

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  • Thank you for the insight friend. Do you mind mentioning your background? Also, please see the short edit I made at the bottom of the question. It refines what I'm concerned about here.
    – Aviv Cohn
    Commented 2 days ago
  • @AvivCohn: mathematics in the 1990s in New Zealand, taking pretty much all the recommended applied mathematics courses, with supporting calculus, linear algebra, real and complex analysis, statistics, computer science and physics. These days, teaching engineering students mathematics with an applied focus.
    – J W
    Commented 2 days ago

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