Is math intensive theoretical research in engineering sustainable?

How I got into the situation

I started out as a mechanical engineering student in mechatronics, ended up now mainly in theoretical kinematics (but what I am about to say may also apply to other fields). Since my supervisor, who graduated from UC Berkeley, is pretty much theoretical, everyone in our lab learned more math on average than the Phd students in other labs. Our research style is pretty much to use advanced mathematics to systematically study not so new problems and hopefully get new and more general results in kinematics/dynamics and control. This is not a fruitful strategy and very few survived, but luckily some years after I graduated, I made some new discovery about a paper in the 70's and got some original idea and later some solid results.

Pros

It is my personal opinion that studying advanced mathematics helps an engineer to cultivate a good research taste. Many new theoretical research papers published these days do not base themselves on rigorous math and are doomed to fade away in time. One who is equipped with sophisticated mathematical knowledge gets to the core of an engineering problem much more easily; he will be easily carried away by useless literature and shall more effectively spot the real gold. Improving one's mastery of math by the days, not in an attempt to solve millennium problems, but in an attempt to see engineering more clearly, can be a big reward and makes him wholesome.

Cons

On the other hand, a math intensive research style is not very time efficient (you are virtually taking a second bachelor and second master degree in mathematics) and one may or may not have solid results after years of hard work (I am pretty much aware of the difficulty and the huge gap between elegant mathematical theory and dirty engineering problems). This can be a disaster when the bar is raised by tons of meaningless papers created by less serious researchers. Before I got to some solid results, I already put almost half of my past ten years studying analysis, geometry and algebra (the lucky part is that I got chances to audit graduate math courses a lot, and I found a very helpful math collaborator).

A particular difficulty with the field I work in, theoretical kinematics, is that it is without a good standard on the mathematical tools that should be used to solve the problems at hand. I often find myself in a position that I need to lay the math foundation for my research (which is a much different scenario in comparison to, e.g. mathematical control theory). This not only makes the research extremely difficult (math upgrade is constantly needed), but also makes truly great results difficult to promote.

Question: Is there a way to go on? How?

Nevertheless, I want to continue my research in this way, because I am already enlightened by mathematics, and I cannot go back to ignorance, and the cost of building up the math background is simply too high, and would be a huge waste to throw it away.

My advantages

Even after I left my previous lab (which now is going from mathematical to entrepreneurial), I still kept a close relationship with people there. I could easily get access to real practical problems which prevent me from drifting away from reality...the little success for some collaboration in the past years has also proved to myself that theoretical research really contributes to solving practical problems.

I have found a mathematics professor who would willingly have a long term collaboration with me. He is usually the ice breaker (who identified the correct math tools that should be used), and I fill the gap between math and engineering. Besides, I keep contact with several math professors at my previous university, so that I could really get some solid help when I want to seriously learn some new subjects.

My worries

However, every step along the way will be more and more difficult (usually more and more sophisticated math tools are needed). I have known researchers spending thirty years on algebraic geometry without achieving anything. It is possible to go interdisciplinary (e.g. stochastic/harmonic analysis on Lie groups and manifolds for robotics), but the learning cost is not necessarily any lower. Besides, although I hesitate to admit, a researcher in engineering hardly got the same level of training a pure math undergraduate receives. This eventually slows down the learning process of advanced subjects. Also my brain usually saturates after 3 to 4 days of intensive math learning, and the cool down time is comparably long...the maximal course time I can endure is 6 hours a day. But I have seen some student taking three courses on the same day (e.g. algebraic topology/Lie algebra/modular forms), spending 3 hours on each. They don't seem to saturate at all...

I also fear that I might soon reach a point that it is impossible for me to understand the math I need to solve my engineering problems (sustainability issue). So far I can manage learning and maturing my education in algebraic topology and commutative algebra (the main issue is lack of time), but I feel that for e.g. algebraic geometry is really difficult (although engineers only work with real and complex fields).

Answers expected

• I would like someone to answer the title question.


• I would expect someone with similar situations to share his/her experience.


• I would like someone to give strategies for continuous self-study on mathematics (now I changed place and it is difficult to audit math courses)