I am an early career researcher working in an area of "hard" analysis, but this is a fairly broad question. My technical skills are likely below par and my greatest hindrance to my research output. I often see authors in a similar point in their careers in my subfield that write on a higher technical level. I see computations, elementary technical lemmas and estimates that I would be unable to come up with and carry out if I were pursuing the project myself or if I were simply already told the statements and asked to prove them myself. I am aware that longer and more technical mathematics is not necessarily better mathematics, but I think it's undeniable that having strong technical ability is a boon. I know that the big picture is important but it's somewhat frustrating when attending a talk or discussing a paper and parts of papers are just dismissed as "technical ends". After all, they're a necessary part to doing research and strengthening this skill helps build mathematical confidence and make the process a touch more enjoyable. I also recognize that the way I've referred to technical skill might make it sound like a one size fits all tool that immediately knocks out any problem. But the notion of general technical skill feels different from that of having collected a broad toolkit. Perhaps the essence of this question is about cleverness and problem solving.
My questions have to do with two stages. How to initialize building technical skills, and how to refine and build on them throughout a career.
- For those who do very technical work, how does one lay a good foundation for building technical skills in ones research area? It does not feel like graduate coursework and prelim exams/orals provided this, nor am I sure if they're supposed to. I presume for most this base is built in graduate school and undergraduate study. Is this done by going through a foundational graduate text in the field and working through all the examples, propositions, theorems, and exercises 10 times over? Is this done by applying a similar process to research papers?
- How does one refine and improve their technical skills throughout their career? Is this skill built by brute forcing your way through new difficulties that arise and doing a million wrong computations before you find a good one? Do you teach to help reinforce fundamentals? Do you understand your technical lemmas well but have difficulty with those of other authors? I've always wondered how mathematicians with strong technical skills have reached their level.