This was asked a while ago, and never received an answer but did receive some helpful comments. Since future visitors might have the same question, I'll try to answer.
First, an REU is a "research experience for undergraduates" and seems to mostly happen in the USA (indeed, many of them are funded by the NSF so are not available to international students). The idea is that the student is paid, given housing, and joins a research group where a professor guides them in some research problem. They exist for both pure and applied math.
Next, the OP appears to be more interested in pure math, but sees an option for an applied math REU and is not sure whether to take it or to do something else with the summer. Normally, it's not really possible to do a reading course with a professor during the summer (though it's a great idea to do one during the regular semester, if you can convince a professor to say yes, and even better if you can do one related to one of the standard PhD qualifying exam courses) so let's rule that option out. There are plenty of ways an undergraduate can spend their summer, ranging from a research experience, to an internship (e.g., the National Security Agency's Director's Summer Program, or internships with Google, JP Morgan Chase, etc.), to a normal job, or just reading for pleasure, or programming for pleasure (and putting the result on GitHub at the end of the summer), or traveling, etc. I think sometimes students have the idea that they absolutely must have something to show for themselves every summer but I don't think that's really accurate, especially for the early summers during undergrad. As a rising senior, it's definitely good to do something that last summer. If you intend to go to grad school, an REU is a great option. If you intend to get a job, then it's better to do an internship with a company, as those often culminate with a job offer if they like your work.
Since the OP seems to be interested in research, let me comment that:
- Even if you did a pure math REU, it's very unlikely that the topic would be related to your future research.
- The main purpose of an REU is to figure out if you enjoy research, to get some experience doing research in a team setting, and to sharpen your learning, writing, and presentation skills. An REU in applied math is just as good as one in pure math for this purpose.
- The OP suggests the applied math REU would be an opportunity to learn about "dynamical systems, networks, stochastic modeling, operations research, and data analysis." That's an amazing opportunity! As an undergraduate, it's wise to explore broadly. You might find that you fall in love with one of these topics and prefer to do applied math or data science instead of pure math. If so, you'll probably have better job prospects. Even if you decide to stick to the pure math path, knowing a bit about these areas will make it a lot easier to find interesting applications for your work, to get people interested in your work (because you can connect it to some applied area), to write the "broader impacts" section of a grant, to give good talks, to teach (knowing a bit about those topics can make you a stronger applicant for a tenure track job), to advise students, etc. You might even find yourself supervising undergrad research someday related to this summer, especially because it's often easier to supervise research on applied topics than pure topics, because more students are able to do applied research, because less coursework is required to get to "research level."
- The OP suggests it's important to publish the results from the REU, but in my experience most REU projects don't lead to a publication anyway, whether in pure or applied math. And the type of research you can do as an undergrad in 8 weeks is not going to be stuff you're particularly proud of later on in life. From the point of grad school admissions, just having some research experience (whether an REU or with a professor during the semester) and the experience of actually writing up your results, matters much more than whether or not you got it published in some undergrad journal.
Related to the last point, I would expect that research in mathematical modeling is actually easier to publish than pure math research. That's certainly been my experience when I supervise students. The reason is that, with a pure math project, they spend comparatively more time learning the topic, because the frontier of research is further away compared to what they've learned in the undergraduate curriculum. Whereas, with an applied math project, they tend to get to the research question sooner, and more quickly produce results (e.g., interesting graphics that tell a story with the data, various statistical tests yielding p-values, etc.), kind of building up a portfolio of results they can wrap up in a paper later, rather than hitting their head against a wall and being unsure if they will ever successfully prove the pure math result I asked them to prove. Lastly, there are plenty of publication venues for applied math research, including SIAM, and it also seems easier for undergrads to publish in professional journals (where professor's publish, albeit lower ranked journals) compared to pure math.
There are several other posts related to undergrad research:
These days it seems to be harder and harder to get accepted to an REU, so I encourage any undergrads reading this to apply to more than you think you need to, and to consider both pure and applied REUs. As I've tried to argue above: the fundamental skill set you get from an REU is highly transferrable and it won't matter a ton if the topic is pure or applied, as long as you go into the experience with an open mind. Plus, there are lots of benefits of exposure to applied math, even if you become a pure mathematician.
