What are some research problems that fit as a good candidate for undergraduate research?

Question

I'm in my junior year in college and I hope thag I can do some original research in while being in college. Sadly, my departement is not large and most professors are elements of one of the following types:

1.They are available but they have not done research for a long time.

2. They are doing good research but either they don't come much to the uni or have a view against undergraduate research.

So it seems that it would be difficult to get involved in research with some faculty member in my college. Nevertheless, I want to do some research.

I'd be satisifed if the problems are in any topic: algebra, analysis, combinatorics, graph theory, logic etc ...

More Importantly, I have the interest and passion to study and work through the required background if I don't fully have it as long as I find an interesting problem that I worlk work on.

It would be even better if you think that the problem is suited to undergraduate research.

I know that it is hard to indicate some problems without knowing the background of the student, but generally, as an indicatiin, you assume that the student has a good background in algebra (groups, rings, fielda, modules,universal algebra, category theory), in logic (FOL, completness, incompletness, set theory,forcing, boolean algebras) etc ...

If needed I can provide more details regarding my background in a specific area if needed.

Answer 1

I'm still an undergrad, and I started my own research in "generalised inverse limits" (GILs) in November last year. This is a subfield of topology (although any course in metric spaces and inner product spaces would introduce the underlying concepts).

So, you said you've done some analysis. Within analysis you have topology, and within topology you have continuum theory. A "continuum" is a compact, connected metric space. (If you don't know about compactness and connectedness it's likely you need one more undergrad course offered at your university.) Within continuum theory you have inverse limits, which is essentially a "tool" for studying continua. However, a very recent development (circa 10 years ago) is generalised inverse limits.

Note that $2^Y$ denotes the set of closed subsets of $Y$. Let $\bf X$ be a sequence of closed sets. Let $\bf F$ be a sequence of functions, so that $F_i:X_{i+1}\rightarrow 2^{X_i}$. This means each function maps a "point" to a "subset". Then the generalised inverse limit is defined as: $\{x \in \Pi_i X_i: x_j \in F_j(x_{j+1}), j \in \mathbb{N}\}$.

It took me about a month to two months to get around what the definitions all mean and to understand important concepts studied with inverse limits (e.g. indecomposable continua). However, after lots of playing around I found many questions to investigate until eventually I found something which hasn't been answered yet. I've been working on it for a while now and I believe I'll be able to publish later this year! Exciting!

Your starting point: Ingram's "An Introduction to Inverse Limits with Set Valued Functions".

At the moment there have only been entirely analytic approaches to this field of maths, so considering your interests (more algebra based?) you may be able to bring something new into it. (Inverse limits are also studied in algebra so it should definitely be possible to combine the ideas.)

Good luck! Sorry I haven't given you an actual "question", but I started in the same boat as you and found my own question after a few months of self learning. I think it's best if you do the same - it'll be a great feeling when you eventually ask a question on your own and realise nobody has ever done it before! - irrespective of whether or not you look into GILs.

Answer 2

I think it's important for undergrad math majors to have some positive research experiences. This is contrary to the conventional wisdom, but in my opinion, the measure of whether research is "valuable" is not "the paper got accepted", but that the researcher gained some sort of deeper insight. I didn't go into math because I had some childhood dream of having unread papers rotting on library shelves, but because solving problems was delightful. I've advised a number of undergrads doing research projects. I try to find problems for them to work on which will lead them to deeper math. We can take some problem from the integers and export it to the Gaussian integers and suddenly the student is learning about field extensions, for example. Does he get his result published? Maybe not, or maybe only in an "undergraduate" journal. Does he delight in the process? Yep.

That doesn't answer your question, but I wanted, first, to counter some of the other comments. Toward your question: There are sources of unsolved problems. One is Richard Guy's Unsolved Problems in Number Theory or UPINT, as they call it. There's probably a copy in your math library, and it's available on Amazon. You can also google "unsolved problems in math" and find a few websites (although most of these I find unsatisfying.)

If you get your hands on a copy of UPINT, just start browsing and find some problems that push your buttons. Do some calculations. Get some tiny partial results. (It's likely that a research mathematician could have gotten the same results in 5 minutes that took you 6 weeks, but so what. You're digging with a spoon and he's got a bulldozer. Your spoon will get bigger.) Write up neatly what you find and then (hopefully) you can interest some professor to give you some guidance.

Anyway "UPINT" is my answer to your question.

Answer 3

I know this is not what you want to hear, but when I was in school the following recipe worked for me:

• Find an interesting open problem. (One of these worked for me http://www.math.ucsd.edu/~erdosproblems/)
• Find a recent survey on the problem. Read it.
• Get the papers referenced by the survey, read them.

You'll have the rest of your life to look at the problem if you still care, and in the meantime you will have started learning how to read research level math.