(Edited to eliminate ambiguity) Where should I begin when it comes to understanding the "theory" or foundations in the field of mathematics? Mathematics is my biggest worry as its just built upon hundreds of years of literature, do I have to span that far back? (It'd be exciting nonetheless, to enter that rabbit-hole haha). It would be most beneficial if you could advise me on where to begin regarding my mathematics research paper analyzing journey, as well as how do you know when you've understood a research paper, or thesis' topic?
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5Talk to your advisors. Also ask for constructive feedback on your writing skills, always worthwhile improving them.A rural reader– A rural reader2025-06-01 02:20:38 +00:00Commented Jun 1 at 2:20
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What is a "senior in undergrad"? Isn’t "understanding the "theory" or foundations of a specific field of academia" what your university studies/courses/… already cover? Are you looking for the theory created by a field (say, mathematics), or the theory of a field itself (say, metamathemathics)?MisterMiyagi– MisterMiyagi2025-06-01 05:14:36 +00:00Commented Jun 1 at 5:14
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@MisterMiyagi , let me clarify. I am currently pursuing my bachelors degree in my fourth and final year in the U.S. This is what I mean by senior in undergrad, I hope that clears that up. Also, no, the university I currently attend does not cover or have a concrete structure on how to read a research paper, unfortunately. I'm looking for the papers that are vital to understand the key definitions and theorems (topology, algebra, geometry), that make up the field of mathematics! the key papers, or most influential ones, while also requesting guidance on how to go about understanding them.Aspen Johnson– Aspen Johnson2025-06-01 12:20:29 +00:00Commented Jun 1 at 12:20
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@Aruralreader thank you, okay I will! I just figured that it may be outside the scope of what they are specialized in and I didn't want to assume that they know ALL there is to know about their respective fields. But, you're right! They probably know where to start! Thanks :)Aspen Johnson– Aspen Johnson2025-06-01 12:26:42 +00:00Commented Jun 1 at 12:26
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1 Answer
Take a quick scan making notes as you go. Note down any unfamiliar terminology. Focus on the abstract if any.
Look up unfamiliar terminology and add the definitions to your notes.
Get an overview from both the abstract and the statements of theorems. If they are totally unclear, look at any citations and find those papers. Do this a bit recursively until things make sense.
Now, go through the proofs. Do they make sense? Talk to your advisor and, again, recursively look at the citations.
You probably don't need to go back to Euclid, but you may need to go back a ways, following citation links. A good research librarian can help get the papers cited.
Talk over your findings with your advisor if possible and, perhaps, in a study group that is dedicated to such readings. Feedback can be essential.
But making notes as you go is also essential. Build up your understanding and annotate what you have learned.
Finally focus on any insights you can get from the reading and the discussions. Make notes about those insights as they can lead to future work as well as ease the readings of future papers in that field.
Like any skill to be gained, you start small, but methodically to build up experience. Repetition and feedback.
