Though not an undergraduate student , I just wanted to know where can I find hard new types of problems regarding the problems in graduate level mathematics. As per my information , standard books from renowned authors and publications are the only sources of problems but still I want to know some sources where I can get difficult problems which can be useful for acquiring some skill set. Also I wanted to know where can I find all kinds of research and open problem? I know that Wikipedia gives a list of problems but I still doubt that there are many problems on some authentic math-research websites. Can anyone provide any information regarding this? Thank you.
As per my information , standard books from renowned authors and publications are the only sources of problems but still I want to know some sources where I can get difficult problems which can be useful for acquiring some skill set.
I don't really understand this first question. If you want difficult problems that are useful for acquiring some skill set, wouldn't standard textbooks be the optimal resource for that? What is a textbook, if not a collection of information and exercises that the author believes is optimal to acquire some skill set? (If I'm misinterpreting this question, then perhaps you can edit your post to clarify it and leave a comment to let me know.)
where can I find all kinds of research and open problem? I know that Wikipedia gives a list of problems but I still doubt that there are many problems on some authentic math-research websites.
This question I understand and am happy to answer, though I need to preface my answer with a couple disclaimers:
It's really unlikely that you'll be able to rigorously solve any of these problems without a thorough background in both undergraduate and graduate-level math. Even graduate students generally need to spend some time specializing before they're in a position where they can make a contribution towards the "edge" of humanity's collective mathematical knowledge.
I'm not a professional research mathematician. I just had a phase during undergrad when I was really interested in learning more about what kinds of problems are "open" in math today -- in particular, the "typical" kinds of open problems, not just genius-level Millennium Prize problems and the like. (I was curious to see what the "edge" of math looks like nowadays, since today's math goes so deep that it can be hard to know just how deep.)
Anyway, here's my answer.
The site http://www.openproblemgarden.org/ gets mentioned a lot for these kinds of questions, but it's heavily slanted towards graph theory and isn't exactly comprehensive and up-to-date.
I think you'll have better luck searching Google Scholar with the query
"unsolved open problems in <<insert specific field of math here>>" and putting some date constraint to limit the search to fairly recent results.
For example, you might search "unsolved open problems in combinatorics" and limit the results since 2020. (Of course, this is just a starting point, and you'll want to play around with the date range and the specificity of the query to get better results.)
Note that if you identify an open problem that you're interested in learning more about, you'll also need to do follow-up searches to learn more about the work that has been done on that problem, especially any work that has been done since the "open problem" article that you're reading was published (who knows, the problem may have been solved since then, or some other kind of progress may have been made).
Nowadays, you will be able to find many research mathematicians' websites with lecture notes for advanced courses, many with exercises/examples, many with solutions/discussions. I myself certainly put lots of things on-line.
If you haven't studied much graduate-level mathematics, I'd anticipate that the main obstacle for you would be understanding the words used in the very formulation of the questions. And it's not just about "parsing definitions"... very often the definition itself is unintelligible without knowing lots of stuff... but, awkwardly, most definitions do not acknowledge or explain the necessary background to understand them! :)
In my own experience, the most productive mental viewpoint is to think about simply working-to-understand various things, rather than problem-solving. Sometimes, you'll happen to come to an understanding that no one had previously. Maybe not describable as "solving a problem". :)