# What's next for me?

I'm in my last year of undergrad, and I would like to do original research for my senior thesis. I am already published in finite group theory and am looking for a new topic to study.

I have taken the graduate algebra sequence at my university, which was primarily galois theory and representation theory. I didn't find Galois theory very interesting (I guess I don't understand the motivation.) Representation theory was cool, but I must admit my intuitive grasp on modules and abstract linear algebra is not yet perfect. I've also taken real and complex analysis, combinatorics, cryptography, number theory, and a lot of physics. I pretty much unilaterally do not enjoy physics or analysis. The others were pretty neat. I performed well in all but the analysis classes.

A few of the topics I've bookmarked which seem interesting, in no particular order: algebraic graph theory, knot theory, noncommutative ring theory, module theory, lie theory, tessellations/tilings, homology, combinatorial game theory, fusion systems, algebraic combinatorics. (I have no idea what background you need for any of these, or whether I would actually like them- they just sounded like possibilities.) Do any of these seem suitable?

Given my interests and background, what would be a good area of math for me to look into next?

An ideal answer would suggest an area of math and include one or more small subtopics which could help inspire me to want to learn that area. For example, "Noncommutative ring theory is the perfect next step for you. You should explore commuting graphs."

To be clear I'm not looking for specific problems like "prove that xxx is true." I am more looking for recommendations which fit my mathematical tastes, contain a few somewhat unstudied topics where I might find some "low hanging apple" research problems, and would be reasonably accessible for someone with my background.

EDIT: To those who think I shouldn't even be asking this question, please let me reiterate what I have said in the comments. First, nobody at my school works in algebra, so I can't just ask a prof. Second, if you believe it would be better for me to study an advanced topic without trying to do original research, please let me reiterate that it is okay if I do not produce original results for the thesis. I can just write an expository paper on what I've been reading. Again, I have already done independent research, so I know from experience that it is a good motivator for me to have a topic to relate everything back to when I am exploring a new subject. An open topic is just a "carrot on a stick" to motivate my study habits. Finally, I am just looking for a bunch of suggestions- I don't have to do any of them if they aren't a good fit. Thanks for reading.

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This is maximally best asked to someone who knows you, not RandomInternetPerson. –  Mariano Suárez-Alvarez Feb 20 '13 at 8:18
@MarianoSuárez-Alvarez That's why I've provided all this lovely background information. Not many people know me mathematically- there are zero algebraists at my university. –  Samuel Handwich Feb 20 '13 at 8:26
From my experience, professors are very happy if a motivated student asks them about their future plans in math. Try to arrange meetings with professors of different fields and explain the situation to them in the same way you did it here. Especially the fact, that you will get to know the people that are working on a field that interests you is very important. Maybe you also get suggestion on certain subtopics. So as @MarianoSuárez-Alvarez said, don't ask RandomInternetPerson ;) –  sonystarmap Feb 20 '13 at 8:27
For what it’s worth, I think it a perfectly reasonable question under the circumstances that you describe; unfortunately, my interests are too far from yours for me to offer any useful assistance. –  Brian M. Scott Feb 20 '13 at 8:36
The focus for a senior thesis should be learning about an advanced topic that is interesting to you regardless of whether you think you will be able to publish in that field, and that will prepare you for your future by exposing you to many branches and new tools/techniques. –  Katie Dobbs Feb 20 '13 at 8:48

This paper http://arxiv.org/abs/1108.3202 might be of some interest to you.

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Hey, that's pretty neat. It's still finite group theory but probability isn't something I'd considered thinking about. $\operatorname{Pr}(G,H)$ seems like an interesting topic. +1 –  Samuel Handwich Feb 25 '13 at 21:36

If you enjoyed combinatorics then graph theory seems to fit your qualifications - easily accessible to somebody with your skills, many easily stated problems that are solvable (ie. possible low hanging research fruit), and it would lead to algebraic graph theory which you listed as possibly interesting.

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Do you know of any open problems that might appeal to an algebra student? I have studied graph theory. –  Samuel Handwich Feb 25 '13 at 1:24
I always felt that if there is a 57-regular Moore graph it would have some algebraic structure. There are other similar problems (and a description of that one) at the Open Problem Garden. –  Chris Hartman Feb 25 '13 at 2:03

If you enjoyed cryptography and number theory, you might be interested in post-quantum cryptography. Multivariate cryptography is one of the best candidates for a cryptographic system that will remain robust against quantum attacks.

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I took a class on multivariate cryptography actually. Do you know of any cool subtopics in there? –  Samuel Handwich Feb 25 '13 at 21:38
@SamuelHandwich If you've taken a course in it, then you probably already know more about it than I do; it's a subject that was recommended to me based on my interests, which are pretty similar to those you listed. –  6c1 Mar 5 '13 at 1:28