Background:
I am in the final year of a bachelor's degree program in Data Science with the University of London. I am interested in exploring the possiblity of going to graduate school for pure mathematics, most likely at the master's level in Canada. (I live in Canada.)
All of my studies with the University of London are done remotely and with almost no support from my institution, so I don't really have anyone to ask for any kind of advice. I have recently had some correspondence with the math department chair at a top Canadian university and was told that students must have supervisors in order to be admitted, and that I would significantly improve my chances of admission if I had some idea of my research interests and who my supervisor would be.
To provide some context, I am currently about 130 pages into Analysis I by Amann and Escher, 20 pages into Algebra by Lang, and 20 pages into Topology: A Categorical Approach by Bradley et al. In other words, I am currently working on what I consider to be relatively foundational mathematics, so the idea of mathematical research seems very far off to me.
I do not know precisely what kind of research I would be interested in. As mentioned, I have no one to ask for advice about this at my university, which is why I am asking about this here.
I am not sure exactly what I will end up doing for a career---I think academia is probably quite unlikely for me---so I would like to try to keep my options open as much as possible. To my mind, part of keeping my options open is to try to learn math that is broadly useful as opposed to niche. If you permit the metaphor of mathematics knowledge as being tools in a toolbag, I am more interested in tools that are more commonly used and/or more versatile as opposed to highly specialized tools that are inapplicable outside of a narrow set of circumstances.
End of background
To that end, I am trying to figure out what the most "general" areas of math are, as opposed to niche areas of math. I do not quite know what I am looking for, but I will try to give some examples:
-Without citing any sources, I would say that calculus and linear algebra are probably tremendously versatile, although perhaps not advanced enough to be considered research interests (?)
-If you look at the list of applications of differential geometry on Wikipedia (https://en.wikipedia.org/wiki/Differential_geometry#Applications) you will see it's quite long and varied.
-Wikipedia says that representation theory is "pervasive across fields of mathematics"; also, p. xi of A Tour of Representation Theory by Lorenz says that "representation [theory] enjoys the additional benefit of having applications in myriad contexts other than algebra, ranging from number theory, geometry and combinatorics to probability and statistics [58], general physics [198], quantum field theory [210], the study of molecules in chemistry [49] and, more recently, machine learning [126]."
-At 12:30 in the video https://www.youtube.com/watch?v=vOboYNh_5is, Ravi Vakil seems to suggest that algebraic geometry is in some sense the "unification" of number theory, geometry, algebra and topology.
-Edward Frenkel has said that the Langlands program is "a kind of grand unified theory of mathematics."
Something that, as far as I know, is relatively niche and thus an example of something I'm not looking for would be algebraic number theory. (See for example: https://mathoverflow.net/questions/24971/practical-applications-of-algebraic-number-theory).
I appreciate any help.
TL;DR I am looking into grad school and have been told that I should try to figure out what my research interests are. I don't have any particular interests. I'm trying to figure out which areas of math are the most broadly useful.
