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:
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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.
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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?
Best Answer
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.