What are some potential topics for an undergraduate math thesis?

[Gale]

Hello all,

I'm considering writing a senior thesis this year and I'm trying to come up with a suitable topic before presenting my ideas to one of my professors. I'm leaning towards something to do with number theory and possibly cryptography, or perhaps something more algebraic. Honestly, I took a 6 year break in the middle of my degree and the only courses I've really taken since then were "Abstract Algebra" and "Number Theory and Cryptography," so perhaps that's the reason for my preference. (Though I will be taking both topology and analysis this fall, so perhaps something related to either of those topics might be beneficial?) In general, I definitely lean towards the algebraic side of mathematics vs say, analysis or geometry, though my experience with all of them is severely limited.

I've been browsing some of the papers I've been able to find online from various schools, but I honestly don't understand many of the topics. Really, I just don't feel as comfortable with upper level maths as I probably should for a senior, (probably in large part due to the gigantic break I took in the middle of my education...) But that's part of why I'd really like to write a thesis, so that I can understand some deeper mathematics, at least in a small way. The problem is just I feel so uneducated that researching ideas is proving somewhat difficult.

Anyway, before I get too long winded, I'm was just hoping some of you could point me in the right direction. I don't have a particularly strong relationship with any of my professors right now, so I don't want to bother them until I have a more formulated plan. So if anyone has any advice about where to start looking for suitable topics or any advice on which specific topics to consider, that would be awesome. I'm really excited to start researching but right now I'm just a smidge overwhelmed. Thanks!

 

[Gale]

Okay, for example, I googled "algebraic geometry and number theory topics" and started looking at Diophantine equations because I like ancient Greek math and from the wiki, it seems to be the sort of think I like. However, I haven't actually had very much exposure to them in my math career so I'm not sure... Similarly, I came across modular forms, which seem somewhat related to my upcoming coursework as well, but the wiki somewhat scares me because I don't understand it all very well.

So I'm not sure if either of those is a suitable topic because I frankly don't have enough experience with either. And I'm not even sure my method of searching and evaluating possible topics is suitable.

 

[deekin]

I don't have any topic suggestions, however I don't think it is out of the ordinary to not understand a lot of the topic you're going to research before you start, at least as an undergrad (i.e. I don't think it is out of the ordinary that "the wiki somewhat scares" you).

For example, I wanted to do a math research project, but had no idea where to start or even an idea for a topic. I talked with a professor that I had a class with before, and we came up with the idea to do an independent study before we started the project just so I could learn a lot of the background information. Maybe something like this would work well for you?

 

[Dembadon]

You mentioned not having a deep relationship with any of your professors, but do any of them know you by name? Even if you haven't spent much time with any of them, they might still be open to help nudge you in the right direction. I had to write a paper for an honors algebra course, and even though it wasn't on the level of a thesis, it still required a lot of research in an area with which I was unfamiliar. I had not spent much time with this particular professor outside of a couple office hour visits before exams. I was pleasantly surprised to see how excited he was to help me. He said it's always nice to help enthusiastic students with things above and beyond the course material.

Now, before I get too long-winded, I just wanted to encourage you to go after one of your professors and ask if they have time to help you. It can be a bit nerve racking because it might seem like you'd be "bugging" them, but it will really help you feel not so overwhelmed to have the help of a professor. My professor was actually the one who helped me formulate a plan and find good places to look for information. Best of luck to you and keep us updated!

 

[Gale]

You mentioned not having a deep relationship with any of your professors, but do any of them know you by name? Even if you haven't spent much time with any of them, they might still be open to help nudge you in the right direction. I had to write a paper for an honors algebra course, and even though it wasn't on the level of a thesis, it still required a lot of research in an area with which I was unfamiliar. I had not spent much time with this particular professor outside of a couple office hour visits before exams. I was pleasantly surprised to see how excited he was to help me. He said it's always nice to help enthusiastic students with things above and beyond the course material.

Now, before I get too long-winded, I just wanted to encourage you to go after one of your professors and ask if they have time to help you. It can be a bit nerve racking because it might seem like you'd be "bugging" them, but it will really help you feel not so overwhelmed to have the help of a professor. My professor was actually the one who helped me formulate a plan and find good places to look for information. Best of luck to you and keep us updated!

Thanks soo much for the reply! Yes, I'm definitely am afraid I'd be bugging my professors. Sadly, I've only spent 1 year at the school because I transferred here and so I only even know two professors, both of whom I really liked and would love to work with, but one of which is on leave... The other is senior faculty and I'm definitely afraid to go talk to him because I feel like he has so much else to do. Plus, I was thinking he must already have a hundred students who want to work with him because he's really brilliant... So I was afraid to try and talk to him without a better formed plan of what I wanted to do.

I DO know that his research is in Algebraic Geometry... which doesn't really mean a whole lot to me yet. I have been thinking more about some of the interesting digressions he made in class about shapes and polyhedra, but I dunno... I think I would really like to do a thesis about something from ancient (esp Greek) mathematics that has been proven more recently with modern mathematics. Any ideas in that direction? How narrow of an idea do you think I should have before I approach my professor? And is there any advice about HOW to approach him in general? Thanks!

 

[Dembadon]

Thanks soo much for the reply! Yes, I'm definitely am afraid I'd be bugging my professors. Sadly, I've only spent 1 year at the school because I transferred here and so I only even know two professors, both of whom I really liked and would love to work with, but one of which is on leave... The other is senior faculty and I'm definitely afraid to go talk to him because I feel like he has so much else to do. Plus, I was thinking he must already have a hundred students who want to work with him because he's really brilliant... So I was afraid to try and talk to him without a better formed plan of what I wanted to do.

It's good you're wanting to be respectful of the professor's time, but also realize that I don't think it will take much time for him/her to get you going. When I went to visit my professor, it only took him about 15 minutes to give me a solid week worth of work to do, and I did not have a specific topic in mind coming into the meeting. If the professor you mentioned is popular, you can be sure he's helped many other students before you figure out a topic about which to write. So, I wouldn't stress too much about having a solid plan; the idea you've presented to us so far, Greek mathematics, should be enough to at least get a discussion going about what's being done in that area and whether it's a reasonable choice for a senior thesis. It might also happen he'll say he doesn't have time. However, he might be able to suggest someone else in the department who would be available.

I DO know that his research is in Algebraic Geometry... which doesn't really mean a whole lot to me yet. I have been thinking more about some of the interesting digressions he made in class about shapes and polyhedra, but I dunno... I think I would really like to do a thesis about something from ancient (esp Greek) mathematics that has been proven more recently with modern mathematics. Any ideas in that direction? How narrow of an idea do you think I should have before I approach my professor? And is there any advice about HOW to approach him in general? Thanks!

I'll have to defer to the more mathematically mature regarding your proposed topic, but I'll comment on how to approach the professor and hopefully it'll be helpful.

I prefer face-to-face interactions, but if you can't seem to find a way to meet him in his office, an email should be just fine. Something along the lines of:

"Hello Professor so-and-so,

My name is Gale, and I'm a mathematics major looking for some help writing my senior thesis. Would you be able to meet with me and critique/help me formulate my plan? I wish to be respectful of your time, so is there anything I should do prior to our meeting? If you're not able to advise me, do you know of anyone else in the department who might be able to do so? Thank you very much for your time.

Regards,

Gale"

Try not to be intimidated; most professors will enjoy the opportunity to help a student research mathematics. If he refers you to someone else, take it as an opportunity to meet another faculty member in the department you haven't met before, which is a good thing!

 

[jgens]

You should really ask a professor about these sorts of things. If you are interested in number theory and in Greek mathematics though, then maybe check out cyclotomic fields if you have not already. They have connections to the regular n-gons constructible by straightedge and compass and also to things like Fermat's Last Theorem.

 

[Gale]

It's good you're wanting to be respectful of the professor's time, but also realize that I don't think it will take much time for him/her to get you going. When I went to visit my professor, it only took him about 15 minutes to give me a solid week worth of work to do, and I did not have a specific topic in mind coming into the meeting. If the professor you mentioned is popular, you can be sure he's helped many other students before you figure out a topic about which to write. So, I wouldn't stress too much about having a solid plan; the idea you've presented to us so far, Greek mathematics, should be enough to at least get a discussion going about what's being done in that area and whether it's a reasonable choice for a senior thesis. It might also happen he'll say he doesn't have time. However, he might be able to suggest someone else in the department who would be available.

I'll have to defer to the more mathematically mature regarding your proposed topic, but I'll comment on how to approach the professor and hopefully it'll be helpful.

I prefer face-to-face interactions, but if you can't seem to find a way to meet him in his office, an email should be just fine. Something along the lines of:

"Hello Professor so-and-so,

My name is Gale, and I'm a mathematics major looking for some help writing my senior thesis. Would you be able to meet with me and critique/help me formulate my plan? I wish to be respectful of your time, so is there anything I should do prior to our meeting? If you're not able to advise me, do you know of anyone else in the department who might be able to do so? Thank you very much for your time.

Regards,

Gale"

Try not to be intimidated; most professors will enjoy the opportunity to help a student research mathematics. If he refers you to someone else, take it as an opportunity to meet another faculty member in the department you haven't met before, which is a good thing!

Thank you. Yes, I'm just suffering from a lack of confidence I suppose. I feel old and out of place most of the time in the math department... But that's just in my head and I want to write a thesis anyway. I reeeally appreciate the help and I'm going to use your example to write an email to him today!

You should really ask a professor about these sorts of things. If you are interested in number theory and in Greek mathematics though, then maybe check out cyclotomic fields if you have not already. They have connections to the regular n-gons constructible by straightedge and compass and also to things like Fermat's Last Theorem.

That's actually exactly the sort of thing I'd like to learn more about. We mentioned them briefly in both my number theory and algebra classes but we didn't really delve into them much. Thanks for the suggestion! At the very least, I know that my vague description of what I wanted to do is actually understandable and doable. Thank you!

 

[Dembadon]

Thank you. Yes, I'm just suffering from a lack of confidence I suppose. I feel old and out of place most of the time in the math department... But that's just in my head and I want to write a thesis anyway. I reeeally appreciate the help and I'm going to use your example to write an email to him today!

...

Excellent! Please let us know how it goes!