Looking for interdisciplinary research topics

Question

If this question is inappropriate for this SE group, I will remove it.

TL;DR: Are there any areas that combine (1) statistics/ML, (2) computation theory, (3) set/type theory, and (4) real/functional analysis? I apologize if I did not accurately categorize the domains listed.

I am a CS+math undergraduate, and my introductory real analysis, computation theory courses have piqued my interests (especially constructing proofs and set theory).

I would like to start hoarding papers and books for reading during Thanksgiving break. The only field that combines...

1. theoretical statistics+ML
2. computation+algorithm
3. programming languages (type theory)
4. math analysis

... that I could think of was AI, but I am unfamiliar with the specific subtopics within AI and would like some direction/recommendation(i.e. what questions require use of the topics above to answer?).

Generally speaking, I would like to learn more heavy duty analysis, combinatorics, theoretical statistics while remaining in the computational sciences (if possible).

Thank you very much for your time and help.

Answer 1

I would suggest reading Russell and Norvig to give you a better feel for what is out there. I think if you throw out type theory, you'll have a lot of interesting options in AI and ML. One of the important applications of real analysis is linear programming and optimization of continuous functions. Much of what you do in ML is stochastic optimization, where you maximize expected payoffs and such. I suspect you'll find game theory and microeconomic theory to be of interest here. Algorithmic game theory is quite hot nowadays. One important problem is computing Nash equilibria, which are fixed points. So you get complexity theory, algorithms, analysis, and some applied combinatorics. The Noam Nisan text is a good starting point here. The Econ department's graduate sequence in Microeconomic Theory is good to take next year, and it is accessible to someone with senior analysis under his or her belt.

The type theory and programming language material is more central to the foundations of computer science. I might look at textbooks in this specific domain. This is outside my area of expertise, so I cannot advise you on specific resources. Though this material is certainly not covered in Sipser's text.

Answer 2

Combinatorics has applications in physics, combinatorics of feynman diagrams, quantum computers. Real analysis has applications in chaos theory, quantum or classical chaos theory. Like Poincare recurrence theorem.