As you can see from the title, I require advice from Applied Mathematicians. I currently have a BSC(H) degree in Mathematics. Although I have a BSC(H) mathematics degree, there are certain topics or areas of Mathematics in which I am weak. I believe I am fairly good with some areas of Mathematics that are taught up to the 2nd year, such as Calculus 1-3, Discrete Mathematics(Not rigorous discrete mathematics), Statistics. As my 3rd year started, Corona Virus took over, and well we all know how classes were conducted. However, I am not trying to make any excuses. I tried my best and self studied Real and Complex Analysis(Although I didn't study them deeply), Linear Algebra(Computational or Numerical, not that much, but I did complete Sheldon Axler's Linear Algebra Done Right. Yes, I know, it's theoretical/abstract Linear Algebra, but I did what I could.). I was also taught Numerical Analysis and Differential Geometry in the 3rd year. Although I passed both courses, I didn't understand much in Differential Geometry and I don't have any interest in it. Maybe it's for pure mathematicians? Numerical Analysis did not interest me much either, but I have a feeling that I may need to study it again if I am to study Applied Mathematics. I was also taught Mathematical Physics in which we were taught PDEs, but they really went over my head. I think I am "okay" with ODEs, but PDEs, absolutely not.
So what topics/areas of Mathematics should I study(maybe rigorously?) so that I may be able to apply(hopefully) for a MSc in Applied Mathematics? I was thinking that I should study Linear Algebra(Computational/Numerical), probably even Numerical Analysis, and also maybe even learn how to use Mathematica, and study ODEs and PDEs again? I intend on becoming a teacher of Mathematics. Maybe someone can help me out or give me any advice? Do I need to study Real, Complex Analysis? Differential Geometry? Even if there are areas/topics in Mathematics that I have not heard of yet, I am willing to study them. So please, any advice at all would be grateful. I believe I have about 5-6 months to study whatever topics are recommended to me. Will this be enough time? I will surely try my best.
Note: I am already fairly good with Linear Algebra, but not with some topics such as: Diagonalization, EVD, SVD, Bilinear/Multilinear forms, but it never hurts to study something from scratch. I think I am fairly good with ODEs as well but not with applied ODEs, such as using Linear ODEs to solve problems with mixing and stuff(was never taught those, I had to look at those myself) and some specific types of ODEs, such as Clauret or others.
Thank you.
It depends broadly with what you want to do next, since mathematics are literally applied everywhere...
If you want to do things close to engineering, you probably want to invest heavily into subjects that are used in finite elements (FEM). Those would be functional analysis, PDEs and linear algebra.
If you want to do things close to dynamical systems (Control theory, chaos, ...), you probably want to invest into ODEs, differential geometry and calculus. Also a little topology.
If you want to do things in data science (ML, ...) , I would encourage you to go for linear algebra, probability theory, statistics and optimization.
Despite all of this, I find as a mathematician that everything is useful, even the subjects that are the most theoretical. I myself am in the finite element world (so 1st category) applied to HPC, and even though it is very applied you definitely need things like topology, differential manifolds, functional analysis, group theory and much more. Having a broad understanding of every field of mathematics (even if you are not an expert) is paramount in order to get the full picture.
Cheers!