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I'm a regular high schooler with an interest in mathematics, but I don't think I will study (pure) mathematics in university, because I'm leaning slightly more towards aerospace engineering and/or medicine. To quench my never-ending thirst for mathematics I just self-study topics that interest me. Yesterday I finished my Linear Algebra self-study (matrices, eigenvectors, orthonormal bases, you know the deal), a study which has brought me a lot of fulfillment.

However, I've reached a point where I don't know what to self-study anymore. I don't really want to do university calculus yet because I'm sure I'll get that regardless of my choice of career. I tried to find fields which build upon (basic) Linear Algebra but I didn't really succeed, so that's why I am here.

I want to learn more about fields which I won't encounter in regular engineering courses, but I want to ask you which would be most interesting to a person like me (i.e. a non-mathematician!). Of course you have to keep in mind that non-mathematicians probably won't be able to study advanced fields. This is what I have found as possible candidates:

  • Abstract Algebra: It builds upon Linear Algebra, and I am just extremely interested in learning what groups/rings/fields and such are. This seems to me to be the most logical choice.

  • Differential Geometry: Also builds upon Linear Algebra, and this is perhaps what I'm most interested in. I've come to understand however that the prereqs are pretty horrendous (for a high schooler). I've also read some things about Algebraic Geometry, but I don't know whether that is a notch above or below Differential Geometry (or not related at all).

And that's basically it. To sum up my question:

What fields of math would be most interesting (and realistic) for non-mathematicians with decent mathematical knowledge?

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You could probably learn some vector calculus. That will probably be useful in aerospace engineering. –  Joe Z. Feb 21 '13 at 16:47
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@JoeZeng But, as I stated, I would be more fond of fields I won't get in engineering –  Fitzgerald Feb 21 '13 at 16:47
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@JoeZeng I know that that might sounds weird but the idea behind this is that it gives us non-mathematicians topics to study which would otherwise have remained unknown by us. –  Fitzgerald Feb 21 '13 at 16:52
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Either elementary number theory or abstract algebra seem like natural choices at this point. Both fields intermingle quite well so you could potentially study both concurrently. –  EuYu Feb 21 '13 at 17:14
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Game theory perhaps? –  mrf Feb 21 '13 at 17:53
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7 Answers

I think elementary number theory and abstract algebra are probably the most natural choices for you. Both of these subjects require minimal to no calculus at the beginning and both are doable without intensive background. The two subjects intermingle well so it is plausible and sometimes even recommended to study both concurrently. Neither is likely to be the focus of an engineering curriculum but I have a feeling that you will invariably benefit from knowing either.

On another note, you could continue your linear algebra education. I don't know how far you've gotten through linear algebra, but I will assume that you've just finished dealing with some elementary properties of eigenvectors and diagonalization. If you do not yet know about invariant subspaces, direct sum decompositions of vector spaces and canonical forms (such as the rational canonical form or the Jordan canonical form) then I recommend continuing your linear algebra education until that point.

The other side of linear algebra is geometric. High dimensional geometry builds upon linear algebra. You can study Euclidean geometry, which you may have some exposure to already, and continue on to other more exotic forms of geometry such as inversive geometry or spherical geometry. These are all subjects which you may conceivably need as an engineer but are rather unlikely to be core to any standard engineering curriculum.

Combinatorics and optimization is also another branch which is accessible without calculus. Knowing linear algebra well means you will be easily introduced to linear programming. Graph theory and enumerative combinatorics are both subjects which are extremely useful to know but unlikely to be part of the core engineering curriculum.

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Differential topology is best learned after you know some calculus, but one of the prerequisites is point set topology. The intuition behind what open/closed means might be a little hard if you haven't seen real analysis before and I wouldn't recommend real analysis before calculus. But that said, there's really no prerequisites to topology other than knowing what a set is and understanding proofs.

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Given that you said "I don't really want to do university calculus yet", pretty much everything I see listed in the answers so far would seem to be inappropriate (and in reading between the lines, those answering seem to be saying this), except perhaps abstract algebra.

Instead of these sexy areas, you might want to consider things like: projective geometry, matrices and determinants (get a 60+ year old book with approximately this title -- I guarantee it'll have a lot of material you didn't see in linear algebra), three dimensional vector algebra (there are plenty of old textbooks with material you probably haven't seen), solid geometry (get a 70+ year old text), linear fractional transformations (also called Möbius transformations; this is a complex variables topic you don't need calculus for; see also the section/chapter where specific examples of conformal mappings of one region to another region are found), two- and three-dimensional rotations (including diagonalizing quadratic forms), working with trig. and exponential and hyperbolic and logarithmic functions of a complex variable (as one can find in the first chapter of most every undergraduate level complex variables text), spherical trigonometry (most every U.S. trig. text from 60 to 100 years ago has one or more chapters on this). I could easily list several other topics but I'm in a hurry at the moment.

These topics have the following advantages for you: You don't need a calculus background for them, they are topics you likely don't know much about, and they are topics that often prove useful in engineering but students rarely see them in formal coursework (any more).

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This is somewhat specific but depending on how much you know already it might be worth reading this Concise introduction to pure mathematics. Think it would streamline your learning from then on as well as it introduces several topics which occur frequently in a lot of areas of pure maths.

Also it basically gives an introduction to a wide range of topics such as number theory, group theory, combinatorics and analysis so you could do this and then be in a better place to choose what else you want to study.

We used this book in first year and I found it very useful but more importantly enjoyed working through it.

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If you go into Aerospace, you will have chance to see plenty of the following Applied math:

ODEs

PDEs

Linear and Nonlinear Control Theory

Dynamical Systems

Fluid Mechanics


All of the above have a solid pure math basis,which includes:

Real Analysis

Functional Analysis

Linear (abstract or otherwise) Algebra

Differential Geometry


But you may or may not be exposed to the rigor depending on your school and/or teachers' preferences. So if you want to study something, the above topics will be a good choice.

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Honestly, if you're not wanting to do university level calculus yet, your options are kind of limited. I don't see how you're going to do diff. geometry, for example, without it.

But limited doesn't mean "null": You could take a look at set theory and group theory.

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If you're just looking for interesting material, it's really hard to say what will interest you: obviously, even mathematicians have their own preferences. If you want to learn higher mathematics, you will almost definitely need to learn calculus, as it is an essential tool to many disciplines, even those you won't see while studying engineering. I would suggest a good background in single-variable and multi-variable calculus, ordinary differential equations, and linear algebra before attempting to study differential geometry, as it generalizes the techniques of calculus and applies them to more abstract constructions: beginning with surfaces and moving up to manifolds (and ODEs and linear algebra are simply very useful for understanding and problem solving). If you can, I would even suggest learning real analysis before studying differential geometry, although it isn't necessary, because it will make you more familiar with the proof techniques and theorems you'll see in DG. (But you can probably get by with minimal linear algebra and a decent understanding of multivariable calculus if you don't have time for the rest of my recommendations.)

Personally, I enjoy the algebra/number theory side of mathematics a bit more than the analytical side, so I have to recommend taking a number theory course or an abstract algebra course. Number theory is nice because a lot can be introduced with minimal background knowledge, but before long the problems start drawing from surprising areas of mathematics (although a first course most likely won't expose you to the applications of complex analysis in number theory). Abstract algebra is doable, but make sure you're comfortable with proof, as it is a large part of any algebra course (and any higher math course in general). Both these areas seem fitting, as a first exposure to either doesn't require too much background in other higher maths, even calculus. Be warned: these areas are probably going to have less applications to your engineering studies than some of the others, but if I'm not mistaken, this is the type of thing you're looking for.

Complex analysis is also an interesting area with many applications, both to the practical engineer and the pure mathematician. In a way, it's a bit more rigid than real analysis, because the requirements on complex functions are much more stringent: it takes more to make a respectable function here than it does in the purely real case. With complex analysis, one can solve a lot of problems that aren't able to be solved by purely real methods: for example, certain integrals that just weren't possible before become a breeze, and some algebraic problems become much more natural when you pass to a complex analogue. However, you might wind up needing to take a complex analysis course for your major anyway, so perhaps this isn't quite what you're looking for.

I would suggest waiting to look at algebraic geometry, as you need a strong background in a lot of other mathematical fields (especially abstract algebra) to really get a hold on it. I haven't taken a course yet, but I have spoken to others who have: and the consensus is that it is hard, and that if one tries to take it too early, the ideas might not sink in. If you take abstract algebra and decide that you find that interesting and want to do more, then perhaps look into a number theory course that uses abstract algebra and/or an algebraic geometry course.

As Jim said, topology is another option that doesn't have that many prerequisites, but you have to have a certain level of mathematical maturity (as you must in any upper level math course). In some sense, topology is like "rubber geometry:" it studies what properties stay the same under continuous transformations (although there's much more to it than that!). If you think you might be interested in topology, start with a course on point-set topology to get the basics, and then move on to differential or algebraic topology, depending on whether you like analysis or algebra more.

Best of luck in your mathematical pursuits!

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Thank you for this answer. It seems to me that I really should master calculus if I want to try other fields. I had no idea that mathematics in its entierity was so dependant on calculus. –  Fitzgerald Feb 21 '13 at 17:52
    
@Fitzgerald I would disagree with this answer and say that mathematics is not entirely dependent on calculus. I'd say that around half (just a very rough estimate) of pure math is dependent on calculus. If you're just learning for fun it would be very easy (although not necessarily advisable) to avoid calculus forever. –  Trevor Wilson Feb 21 '13 at 18:57
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