Is it common for an undergraduate thesis in pure mathematics to prove something new?

What do undergraduate students in mathematics do for their thesis, if they have done one, besides expository or applied math?

I was thinking that the kind of research they do is something applied, say using math in social sciences or a problem in one of the less rigorous natural sciences, or discussing such a problem (that's what expository is, right?).

To me it seems something non-expository or non-applied is an original contribution to mathematics, something that PhD students do.

I attended some pure math undergraduate thesis presentations. I was quite surprised: Did they prove anything new? Never bothered to ask due to fear of looking stupid. Would it be out of the ordinary to expect an undergraduate proves something new? If they did not prove anything new, what the heck are they talking about?

It seems like if it's not new, they are giving a lecture. If it's new, that seems like a PhD-level accomplishment.

I mean, do math undergraduates frequently prove new things?

Answer

I'm going to disagree with Oswald. In my experience, undergraduate students do not often prove new things in pure math. I wouldn't even say master's theses often contain new results. There are a few main reasons for this.

Firstly, pure mathematics operates at a level that is not very accessible for most undergraduates, even those doing research. Undergraduates doing research are often well out of their depth and holding on for dear life. This can mostly be attributed to just not having enough time to get up to speed with what is considered modern mathematics. Most courses in mathematics at the undergraduate level are about math from 50-100 years ago (if not older).

Secondly, undergraduates do not often have the mathematical experience to know what the right plan of attack is when faced with an abstract and new problem and they may not know how to check their work thoroughly to make sure there are no major oversights or blunders. A lot of mathematics involves lateral thinking and it takes a lot of time to build those connections. The hardest part of a pure math PhD (in my opinion) is learning how to attack a problem no one has considered before. Standard techniques that others used may not be useful at all to you for one reason or another. An undergraduate won't have the creativity to navigate this kind of issue because the kind of creativity that is needed comes with a lot of experience. Even when an undergraduate student thinks they've proved something, the nuances of their argument likely will not be apparent to them. (This is especially true when it comes to functional analytic/measure theoretic arguments - the devil is in the details.) Thus a proposed proof may not even be close to being right.

Lastly, not many undergraduates in pure math do research because the gap they have to overcome between coursework and modern mathematics is pretty substantial. Those that make contributions in pure math are those that are very, very talented and have very thorough backgrounds (backgrounds that rival master's/PhD students).

Undergraduates in pure math are not expected to make contributions. That is not what research is about for them. Introducing an undergraduate to research serves a couple of different purposes: it introduces them to more advanced topics and it gives them a taste of what research is like so that they can make an informed decision about whether or not graduate school is right for them. As such, the theses are more like surveys of a specialized topic in mathematics. There is a lot of independent learning involved and there may be some unique examples, insights, and connections contained therein. They may not be presenting "original" work, but poster sessions are there to present what they've learned regardless of whether or not it was original. So yes, it is kind of like a lecture. They are undergraduates and far from being experts in their field.

Note that I am not saying that no undergraduate ever produces new results in pure math (there are some high school students that are better than most PhDs), but it is not a common occurrence and is not expected or considered the norm.