What made you choose your research field?

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.

Solution 1:

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.

Solution 2:

I agree with Martin's assessment. I seems that most of the "low hanging fruit" has been picked. It is rare to come across a recent result that is both accessible and important (in areas like "Euclidean geometry", "highschool algebra", or "freshmen calculus" etc.). This isn't too surprising given that the areas which are presented to the population as a whole (i.e. non-math majors) have been studied for hundreds of years and thus contain mostly very old results.

Counter-examples to the idea "everything useful is old": Consider fields like numerical analysis and algorithmics (i.e. fields related to computation). Many algorithms and numerical techniques are decades not centuries old. Another good counter-example is statistics. Many statistical methods are relatively "new".

There is certainly another component of inertia. It does take a long time for new results to filter on down from research level mathematics to grad school to undergrad etc. For example, it's taken a couple hundred years for calculus to find its way into most (US) highschools. I imagine over time we'll see more and more statistics show up secondary education.

Is the current generation somehow doomed to studying obscure subfields whose results have limited reach? No. It's only a matter of time some new field emerges with immediate astounding applications. I can't tell you what that might be. I imagine such a field will appear quickly without much warning.

To answer your other (rather) different question: "How did I choose my subfield and what is beautiful about it?" My subfield is the representation theory of infinite dimensional Lie algebras and vertex operator algebras. Like many other mathematicians I think many random factors played into my choice. I very nearly went into algebraic topology, but while I was trying to decide on topology or algebra, the adviser I had in mind for topology took on another student. Thinking he would be too busy to work with me, I decided on Lie algebras. Although I was already drawn to this area. Lie theory is filled with beautiful connections to all sorts of different areas of math. A Lie group has a group structure and a manifold structure, so in studying Lie groups you get to play with group theory and manifold theory (so essentially modern algebra, differential equations, analysis, geometry -- all rolled up into one area). There also was something very alluring about Lie algebras. They're non-associative and at first quite mysterious. Then after getting to know them better -- they're everywhere! I also loved the connection between infinite dimensional Lie algebras and string theory. This leads to vertex operator algebras which then lead back into sporadic groups and all other manner of interesting (seemly disconnected) mathematics. So I guess in the end it was the connections (often quite surprising connections) to other areas of mathematics which drew me to this area.