I am a high school student who is interested in mathematics. As such I have been emailing several professors from prominent universities. I have been running into trouble when trying to figure out what their research is from their websites (it's either way too specific and jargon-filled, or so general I can't find anything good to ask them.) Would it be a good idea to just plainly say: "I see you research 'area in math.' Would you mind telling me what specifically your research entails?"?
I advise you to put yourself in the shoes of the professors reading your emails. Why would they answer you? What's in it for them? They explain mathematics for a living, so your request is a bit like asking a barber you aren't friends with to cut your hair for free.
It's great that you want to learn about math research, but with due respect, you can't understand the specifics of most math research. Not because you aren't smart enough, but because it's written in a language that takes years to learn. It is (or can be) possible to communicate some of the ideas behind a piece of research math to nonexperts, but this is a difficult thing to do, and most researchers aren't inclined to try. This is unfortunate, and it would be nice if there were more resources out there for laymen who want to get a sense of what math research is all about, but emailing professors out of the blue is not the way to go about this.
Some professors are open to interested/enthusiastic students. However, they are busy people. If you want them to spend their time with you, you have to give them a starting point that gives them something specific to start from, not some generic "tell me what you do".
Better to check out what you would be interested in (e.g. robotics, quantum physics, topology, or whatever), read up on that, and then find a friendly lecturer at a local college who covers that material. If there are none, then you could contact a prof, but be specific with your question, and do not waste their time.
With a little bit of poking around the Internet, you can find videos of lectures given by prominent mathematicians freely available online. For example, here is a video of Manjul Bhargava giving his Fields Medal lecture, where he gave an explanation of his work to a general mathematical audience. As another example, the Arizona Winter School is an instructional conference for graduate students, and they have posted videos of all their lectures for a long time. Most of them are quite good.
If you just want to get an idea of what leading researchers are working on, I think that watching videos might be more enlightening (and more entertaining) than trying to read papers.
In general, please don't send unsolicited e-mails to professors -- especially if you are choosing the recipients because they (or their universities) are famous. If you develop a particular interest in, and some understanding of, a subject area in contemporary research, then it may become okay to send unsolicited e-mails in certain circumstances. Also, if you want to seek a research mentor from a local university, then e-mails might get a positive response, especially if this university is not famous, and/or if you have a math teacher who is willing to write first and say you are exceptionally talented.
One famous mathematician told me that he gets a huge number of e-mails such as yours. He feels bad ignoring them, but he gets so many that if he tried to answer all of them it would leave him no time for research.
Have a look at the chair's web site. The textual content is often outdated, but the list of publications is usually very up-to-date. You can also use academic search engines, such as citeseer or Google Scholar, to search for that professor's name. The abstracts and introductions of papers usually detail the field of research.
If available, you can also read the list of supervised student theses. These are frequently about current research topics at that chair.
Look at lecture material, slides and scripts, if they put it online. Lectures are often about the fundamentals of the research field of that chair. This way it is easier to judge whether you have the necessary foundation to understand the research topics.
If you can visit that university without too much hassle and have the time, you could also just visit a lecture by that professor. This makes it way easier to get in contact in person.
As a suggestion, when it comes to math, I would highly advise looking up other resources. A few things I could recommend:
- Slide decks: How they communicate their work to peers. Or, even better, the slide decks that they use to present their work to grant sponsors who aren't necessarily experts in their field.
- Websites: These should summarize their goals and general field of work, and should also be more accessible.
- MS Student Theses: Masters' theses from their students will often be more accessible, since they often cover a lot of background that established experts take for granted and don't bother covering in much detail. Or, in other words, they're worse papers, but the redundant/useless information could be useful to you.
- Wikis: Honestly, lots of commonly-used advanced math (at least for stats and machine learning) is on the web. While it won't show you the novel aspects that a given professor is working on, it will help fill in the baseline information. Otherwise, it would be like asking a professor who designs steering wheels what he does, but without even knowing about the existence of cars.
- MOOCs: Try out EdX or other free online courses that are for introductions to advanced math that are interesting. Some of these are geared very much toward people in your position.
These are often vastly more accessible than the published papers, which tend to be for a narrow audience. For example, during grad school I once leafed through an optimization algorithm paper off and on for a few days to grasp the novelty of the approach. After a bit, I finally figured out from the math and supporting papers that the prior approach used a shrinking n-dimensional ellipse, while the new one was a shrinking rectangle. Which was not stated outright in the text anywhere in the ~15 page manuscript. A single slide with 2D or 3D depiction of each approach would have made the concept very clear.
Finally, it seems like you would be much better off looking into high-school research experiences where you'd visit a lab for a few weeks. This would give you a much better idea about what to pay attention to. MOOCs are also an option for this.