How to efficiently find articles/theory/books on topics I want to learn more about

Question

Thank you for taking the time to open this post.

My question is as follows: how do I efficiently find papers/books/talks/any academic resources that have been published about topics that I find myself wanting to learn more about in the course of research?

Context: I have always liked trying to figure out math for myself, and I often have ideas that feel like they should have a well-established theory behind them. For example, I'm working on a project involving an infinite dimensional vector space, and I'm interested in thinking about the lattice of subspaces of this vector space as a fractal; in this case, the issue has been finding a suitably algebraic/category theoretic resource on such an idea. I guess in this case it's specifically hard because fractals are relatively new objects of study in pure math, so maybe the resource I'm wishing for isn't available because there hasn't been enough time for it to materialize. Still, I have a vice of being overzealous to "reinvent the wheel" that I'm trying to break, and I'm sure something out there would be helpful.

It's just hard to justify trying to go through a book like Gerald Edgar's Measure, Topology, and Geometry when I don't really know if it will be relevant to the specific questions and topics I want to know about. I'm sure it would help to read it, or at least some chapters of it, but I'm worried that I waste too much time getting unnecessarily involved in background reading, and, though this is slightly off-topic, I find myself not knowing how to find a compromise between reading a textbook book cover to cover and not reading it at all.

Another aspect that makes this difficult to me is that when I have these kinds of ideas about an overarching theory I see a potential for and/or a need to utilize, I have articulated my ideas in such specific ways that I don't know what the correct generalization of my context would be or should be, especially vis-a-vis the above problem about specificity of prior learning materials.

What I'm asking for: Could someone with experience in research can shed light on how they deal with this, or what they think I should do? Maybe there are some databases or repositories I'm not aware of. Typically I search google, wikipedia, this website (although I should probably post more), and more recently I've been looking through my university's library database. Should I be going through arXiv? nLab? JStor? academic journals? Should I ask professors I know whose areas of expertise overlap? What is the best way of going about finding more information about ideas I come up with that I suspect are well-established?

Answer 1

University

When you are studying at a university, either as an undergraduate or graduate student, the most obvious (and also most "classical") approach is to ask a professor, or specifically your advisor. Even if they can't help you specifically, they will definitely give you directions where to find more about what you are asking and also what other people at the university or maybe even other universities you can ask.

The same applies when you are a lecturer, professor etc. working at the university. Ask your colleagues.

When you are not studying or working at a university, or your university simply does not represent the topics you are working on, fortunately there are other approaches as well. (I have used all of them.)

Emails

Don't be too shy to write professors an email who worked in the field of your interest. But please do your research first! Professors usually publish their email address on their university website and are usually open to help, but they will not bother answering trivial questions. Which professors? You need to find some papers published in the field of your interest, check their authors, then with the help of asking some professors you can find even more papers. It's a recursive process.

Actually, everything that I describe here is a recursive process. Each step brings you more insights into what has been done before.

MSE

Ask on this site if a concept is already known. Ask for relevant literature and use the tag reference request. Look through the questions tagged with this tag to get some inspiration how it can be done. But please remember that MSE is not a place to publish your work. So you should keep the request short, self-contained, but also complete in the sense that everyone understands the question in full detail.

The same holds for mathoverflow. Here is an example of mine.

Citations

Every paper that you find is actually the start of a long process, if you want, to find more papers about the subject. Namely, there are two things which can be repeated as often as necessary.

1. Check all the references mentioned in the paper (in case their titles sound promising). Then you have more papers.
2. Conversely, for every paper that you find, use Google Scholar to find all citations of that paper.

This way, you will find more and more material. To get an overview and also to make a rough filtering, it is sufficient to read the abstracts.

Learn the basics

Needless to say, you need to learn the basics. (I would not write this to every person but I have to since I am not familiar with your level of mathematical education, and even less so of potential readers of this post.) When you have just learned about vector spaces and matrices and think you have proven a new fancy result saying that an invertible matrix has a non-zero determinant: maybe you need to learn more basics first to really understand what can be new or is just standard stuff.

Answer 2

AI search engines may be of use. The one I use is Perplexity, but I am not claiming that it is better than others. If you enter a search term, it gives you a paragraph long summary of the topic, along with selected references.

For example, I entered the phrase "lattice structure of infinite dimensional vector spaces" to discover that this is a standard, fairly accessible topic, with a bunch of good references.

I then added "and fractals" to the phrase, and didn't get anything useful, suggesting that this was not a real area of research. But it did propose the search term "how are fractals used in the study of infinite-dimensional vector spaces", which then produced the following answer:

The use of fractals in the study of infinite-dimensional vector spaces is a topic of ongoing research. Fractals have been explored in the depths of Hilbert spaces, specifically l2, to investigate interconnected fractals, as discussed in a research paper titled "Interconnected Fractals in Infinite-Dimensional Hilbert Spaces" 1. Fractals, which provide a measure of the space-filling capacity of a pattern and characterize complex geometric forms, can offer insights into the intricate structures that may arise in infinite-dimensional spaces. However, the specific applications and implications of fractals in the study of infinite-dimensional vector spaces are still an active area of investigation, and their full potential in this context is yet to be fully realized.

It includes a link to the paper in question, along with further references. But of course, my point is not to say anything about fractals, but to illustrate how these kinds of search engines can very quickly give you an overview of relevant areas and the state of the art. For that kind of thing, it seems to me vastly superior than anything google produces. In this case, I quickly learned that the topic of lattices and infinite dimensional vector spaces is standard, useful to know about, and would be accessible to me if I wanted to learn more. On the other hand, the use of fractals in that area appears to be at best highly specialist. For one person, that may indicate a potentially interesting research topic; for another, too advanced or niche to be worth going into.