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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?