I was wondering how come modern mathematicians do not seem to discover as many theorems as the older mathematicians seem to have done. Have we reached some kind of saturation limit where all commonly needed mathematics has already been discovered or is it just that the standard common textbooks do not get updated with the newly discovered theorems ? Are mathematicians of our generation only left with research topics in super specialized sub-fields ? I am wondering what made you choose your research field, what’s so beautiful about it ? I understand that this may be a little personal question and i do not mean to intrude on your privacy. I guess if you could just shed light on the field you are familiar with and why u would or would n’t recommend me to conduct research in it then i’d be great full to you. This would help me make an informed decision about picking a field.
PS:I know “commonly needed mathematics” is subjective to interpretation but i was thinking of defining that as anything one is taught in an undergraduate level.
Edit: As per the request my educational level is i have a bachelor’s degree in computer science, a graduate diploma in mathematics. I am currently a honours student and would be starting a PHD next year. I have taken mostly non-rigrous undergrad level math courses, mostly because that’s all the uni was offering at the time. These courses were on financial maths, Dynamics, ODEs, Mathematical modelling with multiple ODEs, linear algebra, basic Probability, Statistcial modelling, statistical inference, vector calculus, Time series. Out of rigrous fields i have only self studied basic abstract algebra and some basic mathematical analysis. I guess i am in mathematical infancy and being made to choose which seems very scary.
I began graduate school wanting to be an analyst. My first year I did the standard introductory algebra, analysis, and topology sequences. In the second semester, the algebra course did a lot of module theory and homological/categorical things, and I was hooked. I work mainly in module theory because I find it interesting seeing what does and doesn’t generalize from vector spaces and Abelian groups, and what I can learn about rings from modules. I enjoy what I do. I’m not at a research university so I don’t have to worry if my work isn’t fashionable or high-powered. I do it for me, in my own time. I didn’t get in the business to get rich, or famous, but to pursue what I consider some beautiful ideas, and maybe make a contribution here or there. It’s been a good ride so far.
I think mathematics is always establishing new results. In my experience, the basic graduate texts provide an introduction to the various fields. For the most part, you won’t find in them the advanced results you need to do serious research. That’s where advanced texts, papers, etc. become very valuable.