What should a student look for in an undergraduate thesis topic/supervisor?

I am in math, and have some experience with undergrad research (from both sides), and there are lots of different ways you could have a good experience with undergrad research. What would be good for you will depend a lot on your individual situation (mathematical maturity, background, independence, etc) and your goals (do you want to get a paper out of it? just get some research experience? learn something new? help your case for grad schools?).

• For getting into a good grad school, what's most important is how well prepared you are in terms of background and how much potential you show. This is evidenced by your transcript and recommendation letters. Doing novel research or writing a paper is much less important (unless it's much more impressive than typical undergrad math research). (Note: many undergrad theses in math are just an exposition of an important topic. Some of these are very good.) So in terms of grad school, the most important thing out of this project will probably be (1) what you learn, and (2) the impression you make on your supervisor, who should write you a recommendation.

• For being able to get a paper out of it, research is never certain, but it's important that the project is well thought out beforehand. More open-ended projects may sound interesting, and teach you a lot, but may not be conducive to having a finished project in the time available. I generally find ones that involve some calcutions/computation have a higher rate of "success," in the sense you should at least be able to do the calculations, whereas projects that require you to develop some theory may just leave you completely stumped. (Though some types of theoretical projects where you can follow the method of another paper closely also have high rates of success. Conversely, try to stay away from projects of the form "look for pattern in this data.") Hopefully you can get a sense of how well thought out the project is from the description; also knowing how successful the supervisor was with previous students may suggest this person gives high-percentage or low-percentage success problems (but if you're more concerned about grad school, where the students went for grad school is more important than if they published papers).

• For learning things (which helps with grad school), a project where you have to learn some theory (even if just to do computations) is better.

• Both to make progress and to learn things, an accessible advisor (say is willing to meet you once a week or so) is better.

Apart from that, I would say put some value in your general impression from talking to the potential supervisor. If you have a chance to talk to former students of the supervisor, that could also be useful.

in addition to the hard skills mentioned above. You also want to search for someone who posses soft skills. Skills such as organization, communication, patience, coaching/teaching skills, are abilities to die for in a supervisor.

Students always look for the superstar supervisor who has all the publications and funding but lacks the tact to help a student to grow. You spend years of your life under an unbearable person for what?

Off course, if you can find someone who is a top scholar and a great teacher it's the best

Ok practical advise. Not purely focused at Math. Depending on the subfield a lot of this might be irrelevant (or alt very relevant)

• Avoid: Look out for re-occuring thesis topics within your school. Cherck last-years list of topics (and prior years), if the same topic is mentioned avoid it. If when you meet the supervisor he says "you'll carry on from the work of last years undergrad student". This is a bad sign.

  • If a topic is offered again, this means that previous students have not been able to complete it satisfactory. Which may be a sign that the topic itself is flawed. Maybe it is really really hard. Maybe it requires particular equipment that is prone to break down. Working on a "legacy project" is always harder (ask any software dev).

• Look for: supervisor who has been around the school for a long time, and is fairly senior staff. What you are looking for is Realpolitik, and having supervised many undergrad thesis before. Undergrad thesis tend to be full of bizarre and unusual requirements; because they often occupy a weird space in the curriculum; e.g. being considered officially a "coursework" unit or other odd things.

  • So you want from your supervisor an understanding of these rules; and how to loop hole them; or the capacity to pressure the higher powers to ignore them.

• Avoid: topics that are not compatible with the requirements of the marking guide. Be aware of what the marking requirement of the thesis are, and chose project to suit. Part of the aforementioned undergraduate thesis are weird is that they might not be assessed on what they look like they are; completing the projects goals perfectly might yield poor marks.

  • For example, my undergrad area was Engineering. There were a lot of "Design and Build" topics offered. Where you would design something then build it. However the marking guide is taken from the Science faculty. So it is not about designing something to do a task. It is about designing a experiment to test a hypothesis. There are no marks for building at all; even though it is the hardest part. Be on the look out for things.

• Look for: for opportunities to make multiple contacts. For applying for Post-grad you need multiple recommendation letters. One will certainly come from your undergrad thesis supervisor. But if you got to work with someone else significant in the university, then that is a great second reference. For example one of my friends had to do a lot of electron microscope work. So he also got a great contacts with (and training from) all the people in the Microscopy center. I think his second reference came from the head of that lab.

• Avoid: things that cost money. While there might be a university provided budget for your thesis; extracting money from a university can be like squeezing water from a stone. There may be a bunch of paperwork, and it may take months. However, this can be cancelled out, if the supervisor has great realpolitik -- and can thus pullout money easily.

• Look for: chances to grab useful training. I mentioned my friend who did tones of microscope work. That goes on a resume. Working on super-computers, or large clusters; another rareish marketable skill.

  • When you begin your PhD your may have to justify that you have certain skilled. Having used them in your undergrad thesis is a solid demonstration.