Going from pure math to applied math

[Oats]

Hello,
I am a second year PhD student in pure mathematics, and I'm going through a bit of a dilemma. Up to this point I felt that I wanted to work in Mathematical Logic. To be honest, I think working in any field of math would be fun, but logic had extra cool topics like computability theory that I was interested in. Now, I understand the concept of doing mathematics for its own sake, but I am recently becoming a bit disillusioned from the whole thing, having seen a few conferences. Everything just seems so oddly specific and obscure. Papers regard some really specific problem, and I imagine that even at the end of a PhD with advanced knowledge of logic, it would still take a while before I can even get to understanding these papers, as my advisors confirm. I was considering working alternatively in Algebraic Geometry, and indeed it looks like a beautiful subject, but to my understanding, the obscurity and background knowledge problems are even more palpable there, with some problems needing years of focused study just to understand. Other fields of math, like Graph Theory, are on the other end and while still needing good background to tackle, many problems can be started on and at least understood quicker relative to other fields. This made me consider fields like graph theory, but then I started feeling a second dilemma: I wanted my work to matter more. Now yes, I understand that many research projects, even in the applied sciences and engineering, don't really establish much at all individually, but I think even that may be more personally satisfying than my situation with pure math.

In particular, I am interested in things like computability and the search for building stronger computers, computational complexity, control theory, dynamical systems, information theory, artificial intelligence and the search for building artificial consciousness (biggest interest), and have even looked at things like biomath and it also seems pretty cool.
I am pretty good at self study.
So one of my main questions is, has anyone know what it is like to switch from pure math to applied math, or anything in between?
What is it like to work in applied math vs being in pure math?
It feels weird that after having been such a pure-math person for the past couple years, and now feel like I want to do something a little different, and it's really stressful.
Does anyone just have any kind of general advice?

 

[FactChecker]

The fields you mention still seem like pure math to me. Have you considered statistics, numerical methods, or optimization techniques? And applied math should also go with some computer programming. I switched from pure math to operations research in an Industrial Engineering department. It was the best decision I ever made.

 

[Oats]

Yeah I was considering perhaps something along statistics maybe, but I don't really know what it entails on a research level. I think it's a useful skill to have regardless. I know a tiny bit of programming. I would think I'd like to stay in Academia, rather than work in industry, but I can't say I know what working in industry is like either.

 

[StatGuy2000]

I would have to disagree with @FactChecker that subjects like computational complexity, control theory, dynamical systems, information theory, and AI are research areas of pure math. All of the fields mentioned by the OP are solidly within the research purview of applied mathematics or math-related disciplines such as computer science and certain engineering disciplines (electrical, mechanical, industrial).

In fact, when @FactChecker talks about numerical methods or optimization techniques -- these are all part and parcel to the theory of computation, control theory, and dynamical systems. Much of machine learning research (which arose from AI) involve optimization methods of some sort.

At the same time, I do agree that statistics (including the closely allied field of probability theory) is an important skill to have and is worth exploring, and is highly applicable and employable (especially in the now burgeoning area of data science). And transitioning from pure math to statistics should be fairly painless.

 

[FactChecker]

I would have to disagree with @FactChecker that subjects like computational complexity, control theory, dynamical systems, information theory, and AI are research areas of pure math. All of the fields mentioned by the OP are solidly within the research purview of applied mathematics or math-related disciplines such as computer science and certain engineering disciplines (electrical, mechanical, industrial).

I am very sorry. I committed the sin of reaching a conclusion after the first paragraph (algebraic geometry and graph theory) and fired off a response without reading the second paragraph (with the subjects you mention). I agree completely that they are applied areas with a great future.