This question is intended as public service to mathematics. It is meant to benefit undergrad research students and maybe even researchers wishing to broaden their expertise and learn something away from their own subject...with the potential to quickly begin researching something in the new area.
The motivation of the question comes from an idea of David Larson, which I heard from one of his undergraduate students at a poster session. The idea is that while learning a subject, certain questions and topics are "research gateways"...or admit easy entry into research.
Clearly some subjects admit more of these research gateways than others, but nevertheless I think it would be nice to have a list of research gateways by subject area. For example, what is a nice contemporary topic or question in Algebraic Geometry that will bring one to research reasonably quickly (without having to read EGA and SGA or all of Hartshorne to get started?) This is an extreme case, but illustrates the sort of thing I'm looking for.
As I'm writing this, I notice that it is close to David Feldman's question here.
Note that the distinction from David's question is I'm looking for questions or topics that bring you to research in a subject area of mathematics with minimal preparation, but also has at least the potential to bring you to core ideas relatively quickly.
I imagine also that anyone who has such ideas is keeping them for his or her graduate students :)
Another way to judge whether your answer is a good one: If a person who ordinarily works in an adjacent field ran a reading seminar on the proposed topic, could he or she make some headway on the research question in a semester? (E.g. I think a lot about von Neumann algebras, but if I ran a seminar on topic X on C* K-theory, could I find some interesting new result?)
Anyhow...this is a hugely tall order, but it seems fun to ask...
Question: What are some current research gateways in your subject?
Note: In real research, every new thing you attack depends very sensitively on what you already know...so the current question is best targeted for undergraduate students starting with only the standard undergraduate curriculum.