How to add mathematical rigor to systems research?

Question

My research focuses on wireless networking and designing various protocols. I primarily build systems. I, sometimes, analyze the performance of those systems, using techniques from probability, statistics, and algebra.

I want to improve my understanding of the mathematical underpinnings of my work. Specifically:

1. Derive and prove theorems on properties of a system and prove it;
2. Model a system.

This post differs from SE posts like this and this because I'm looking for concrete steps to incorporate math rigor into my systems research. I found the answers in those posts to be too abstract.

Other posts, such as this and this, provide concrete but scattered advice, suggesting:

1. take theory courses;
2. read math-heavy papers and learn.

For (1), what types of courses are likely to be useful? Is Great Theoretical Ideas In Computer Science a good source?

For (2), I find it difficult to lead from reading a proof to writing one of my own, or from reading about a model to creating my own model. Input is easy, output is still hard. Any good examples (papers, books, or online courses) to learn how to output theoretical results?

Besides these two lines of suggestions, any other specific suggestions on how to be a more mathematically fluent researcher for someone like me who is from systems background?

Some more context for the whole question: I'm working on my first theory-oriented paper and I have great difficulty in proving the optimality of my proposed protocol. Thus I'm asking to see if there is a general way to hone my proof skills in the short term and my general math skills in the long run.

Answer 1

You sound like me five years ago! Here's what I did to become more theoretical.

1. Go back to basics. I went back and relearned Calculus and Linear Algebra. Watch YouTube videos on different subject matters. Mathematical maturity is the key. You get this by reading lots of maths (any maths will help). I also started learning Real Analysis.
2. Search Google for books/resources on 'how mathematicians think?' Also learn how to do proofs. There are plenty of books available.
3. Start reading theoretical CS papers/books, and learn a math tool; e.g., a branch of math programming such as linear program, non-linear, etc...

Best of luck with it. Unfortunately, there is no short term solution, but the payoff is well worth the effort and frustration!

Proofing stuff in general is hard. Otherwise, any man and his dog will be able to do it! Books on how to construct proofs will help here. They will get you into the correct mindset and also provide the framework that will allow you to 'hang' your proof.