astrotemp said:
It really depends for research careers... I'm doing my PhD now in a theoretical area of astrophysics (black holes, simulations, theory) and did my undergrad in math, and honestly most of the math I do each day is basic. Note that I did all of my "graduate coursework" in undergrad, and the postdocs and professors use the exact same theory I do.
We do use some higher end stuff and there's plenty you can only find in research papers, but it's not typically that difficult, it just uses language and relies on a history you're not taught in school. And you don't do the complicated stuff that often - I only have to code up the GRMHD equations once and then they're done, and all the changes I make to those will be simple power laws.
If I were to plot the frequency with which I used each area of mathematics in daily work, the winner by far would be basic trigonometry. The higher level stuff is definitely needed for the occasional hard theory you have to work with, but you use basic math skills way more often.
The real difficulty isn't in knowing hard math. That's actually a trivially easy thing to achieve, you just have to stick your nose in a book and study. The hard part in research is using all levels of mathematical knowledge to do proper research and discover new things. You can't just know random math-y things, you have to know what maths is most appropriate and how you can tweak/approximate things. For a lot of things, you don't want anything more complicated than a power law.
This seems to be quite an accurate representation for almost all typical research, i.e.
intradisciplinary research.
Myself, I mostly do atypical research, i.e.
interdisciplinary research, specifically in dynamical systems theory in three interlocking ways, namely:
- theoretical research utilizing mathematical physics and applied mathematics methods in order to discover new methods or theory,
- experimental research utilizing statistical physics, numerical analysis, machine learning and computational methods in order to fit models to empirical data
- applied research through direct mathematical modelling of naturally occurring dynamical systems (i.e. physical, financial, sociological, political, physiological, etc systems).
Having a very broad knowledge base of different fields in mathematics helps enormously, especially when these fields show up unexpectedly in another completely unrelated context. This reoccurrence of a piece of mathematics is something which happens way more frequently than is often realized by researchers who didn't start off in (theoretical) physics; for example, Riemannian geometry learned for GR rearing its head in machine learning.
To illustrate some of this better using the Riemannian geometry example: in the actual practice of machine learning, researchers unfamiliar with Riemannian geometry seem to take a completely different attitude w.r.t. it showing up in their work than researchers who are already familiar with it. This difference in attitude is not merely trivial in that they try to learn it properly (NB: they usually don't because it isn't strictly necessary).
This difference in attitude has real consequences both for the direction and results of their research by giving those unfamiliar less methodological broadness by being able to tweak the methodology less than those intimately familiar, typically leading to research that is easier to understand and reproduce but overall less ambitious and/or generalizable.
On the other hand, a researcher who is familiar with Riemannian geometry - or more generally has more mathematical broadness as a researcher - can take advantage of this and more easily produce groundbreaking results in this field, simply by quickly regurgitating old knowledge already learned in a different context and so end up completely changing the perspective of the field.
