14

For a little more context: I'm currently an undergrad (sophomore) at a small liberal arts college with a (from my experience so far) solid math program. So far, I've taken Calc I, II, III, linear algebra, and an introduction to logic class. This semester I'm taking graph theory and differential equations, an intro to statistics class, and an intro to proofs class. And outside of class, I'm (very slowly) working through an abstract algebra textbook.

What are some ways I can find questions that are accessible to me as a pretty mathematically inexperienced undergraduate? I don't even mean questions that are open or unsolved, just any questions that would be accessible that I could play around with and get a taste of working through.

David White
  • 29,779
Christian Dean
  • 243
  • 12
    You should talk to your professors about that. – Wojowu Oct 30 '22 at 20:35
  • 4
    I understand that REU is something quite common in North America - so perhaps going through https://www.nsf.gov/crssprgm/reu/list_result.jsp?unitid=5044 would help. – Vladimir Dotsenko Oct 30 '22 at 20:41
  • 1
    @Wojowu Right good point, I’m planning on doing that as well. Specifically my advisor as he has a PhD and math and has done graduate level research. – Christian Dean Oct 30 '22 at 20:49
  • Thank you @VladimirDotsenko I've never heard of that, but I'll make sure to look into it! – Christian Dean Oct 30 '22 at 21:29
  • @ChristianDean Just a few points on terminology: essentially all professors have a PhD (though exactly what "professor" signifies can vary quite wildly by country and context), and everyone with a PhD has done graduate level research (by definition). – Carl-Fredrik Nyberg Brodda Oct 30 '22 at 21:37
  • Thank you for the corrections @Carl-FredrikNybergBrodda Right, I am aware that having a PhD and having done graduate research are essentially synonymous. I think I just wanted to put emphasis on the research part, but you're right, both are definitely redundant! And thank you, I wasn't aware all that the term "professor" generally refers to those with PhD's (at least in North America). I thought having a MA or BA was sufficient. – Christian Dean Oct 30 '22 at 21:57
  • 4
    By “questions which are accessible” do you mean “questions which I can understand” or “questions where I can appreciate the challenges that experienced mathematicians see” or “questions which I might solve with hard work and good luck”? –  Oct 31 '22 at 01:42
  • Definitely REUs are a good place to start. You can also just look at result you think are beautiful and try to generalize or extend them and just keep doing it until you run into something no one’s done before. But that might a laborious route – Sidharth Ghoshal Oct 31 '22 at 02:58
  • @MattF. Probably more of 1 and 2, and less of 3. It'd be nice if it's a problem I might solve, but I can appreciate the value in wrestling with a hard question even if no answer is produced. – Christian Dean Oct 31 '22 at 05:38
  • @SidharthGhoshal Could you expand a little more on that second option? I don't quite understand what trying to generalize or extend a particular result might look like? Do you mean something like trying to extend a theorem to more general cases or trying to modify an algorithm to work for more objects? – Christian Dean Oct 31 '22 at 05:39
  • 3
    So an example might be take a function you like which is normally only defined for positive integers (ex the factorial), and try to define it for rational numbers, or if something is defined for rational numbers can be defined for arbitrary real or p-adic numbers? That's one flavor of this idea. A different flavor, if an object can be defined for functions, can a similar object be defined for functions-of-functions? Ex: we have a taylor series for functions, after a lot of work, i was able to scrap together a taylor series like construction for non-linear functions-of-functions. – Sidharth Ghoshal Oct 31 '22 at 15:07
  • 3
    Anyone can play the generalization game but the problem is as you start to play it you keep finding out that other people have done stuff, so actually finding an original result becomes very very very hard, and this is why it is better to talk to experts. But its a fun game to play and will teach you a bunch of things along the way. – Sidharth Ghoshal Oct 31 '22 at 15:08
  • @ChristianDean a professor is a specific job title, not just someone with a PhD. A PhD is necessary but not sufficient. https://en.wikipedia.org/wiki/Professor It’s worth making yourself aware of what it means — many undergrads do not know! Such is the constant life of grad students: “don’t call me professor, I am not one” for semester upon semester. – Eric Oct 31 '22 at 16:09

2 Answers2

18

Welcome to MathOverflow! I am a professor at a liberal arts college. The best place to start is by talking to your favorite math professor, or your advisor in the department. The graph theory course will have plenty of accessible problems that will give you a taste of research mathematics. Abstract algebra does, too, and I'll bet your independent study will be smoother if a faculty member is involved helping you see the "big picture", suggesting good exercises to work on, etc.

Most faculty at liberal arts colleges welcome the opportunity to work with undergraduates, because it's a lot of fun, and liberal arts colleges usually don't have graduate students. It also looks good to the university when a professor works with an undergraduate student, so don't be shy to ask your professors.

It may happen that when you talk to your favorite faculty member, they suggest doing a "directed study" with them in the spring, e.g., reading an interesting book covering material not normally taught in the curriculum (e.g., hypergraphs, or Galois theory). They might also have research problems in mind already, accessible to students. For example, I list a few on my webpage, in case students want to work with me. As a rising junior, you can also apply to REU programs (that stands for "research experience for undergraduates"), though you're more likely to get in as a rising senior. Such an experience would provide you housing and a stipend on the campus of some university, where you'd work alongside other undergraduates on a research problem suggested and supervised by a professor. It's a great way to get an idea of whether or not you like research and to improve your resume during the summer. Deadlines are often in February. The same faculty member you ask about directed studies can suggest REUs and write you a letter of recommendation. Based on your background, I think you have a good shot, even for this summer. Good luck!

David White
  • 29,779
  • 4
    Vladimir Dotsenko posted the following very helpful link while I was writing my answer. It's a list of REU programs: nsf.gov/crssprgm/reu/list_result.jsp?unitid=5044 – David White Oct 30 '22 at 20:45
  • Hi, thanks for the very detailed answer! I'm going to take some time to digest it fully. With all that in mind I'll definitely take advantage of my professor and advisor. A directed study in the spring sounds like a very fun opportunity. – Christian Dean Oct 30 '22 at 21:24
5

Exercises in abstract textbooks (the kinds where you need to prove something, not so much calculation exercises) are probably not that far away from what research mathematicians do, at least from the vantage point of an undergraduate. Bang your head against some of those for a while.

Alternatively, the motivation-theorem-proof cycle in textbooks itself is useful. Your algebra textbook is presumably of this type. When you come to a new theorem, don't start reading the proof straight away. Read the theorem, then close the book and think about it. How would you go about proving it? Is there a simpler version you can prove easily, perhaps a special case or if you assume additional preconditions? Conversely, can you find a counterexample to the theorem if you remove one of the preconditions? Can you apply the (as-yet unproven) theorem to some concrete examples? (In abstract algebra, there are many fun groups and other structures. What does a theorem in group theory say about one particular group?) Then read the proof provided and revisit your thoughts. Would you have chosen a different angle of attack? If so, would it have worked? Where in the proof do the various preconditions come in?

One very nice book for something like this is Proofs from THE BOOK by Aigner, Ziegler and Hofmann - many quite different topics, most quite accessible to an undergraduate, and beautiful proofs.

Stephan Kolassa
  • 153
  • 2
    Thanks for the answer! Most, if not all of the exercise in my Graph Theory textbook are about writings proofs rather than computation, and I've enjoyed thinking through them, so I guess I have already gotten a tiny taste of research mathematics. I'll keep in mind using the same strategy as I make my way through this abstract algebra textbook too. – Christian Dean Oct 31 '22 at 18:49