As some comments suggests, it's not clear what you want from the professor. If there's no clarity from your end, it's very unlikely the professor can help you (even if you get a chance to meet them). I recall my thesis adviser saying something like, "I receive >100 emails/day from various students requesting to work with me". This was in 2001-2002.
I "hustled" few professors and got some face time around 2001 time frame. But I was out of undergrad (not high school as you requested). These are the things I recommend.
0) Have clear goals on what you want from the professor (and what you can offer too). I showed interest and competence in doing research and I wanted funding from them.
1) I worked extensively in couple of areas (Genetic Algorithms, Simulated Annealing, Finite Element Methods, Optimization of Manufacturing Processes) and published conference and journal papers before requesting a meeting. I am not suggesting you do this. But you need to show you have done some work to warrant face time with professors.
2) After you have done 0 and 1, email them, call them and somehow "stalk" them "respectfully". To meet the professor I did Master's thesis, I wrote to him few times and called him. But finally, I waited in front of his office few hours every day and finally got a chance to meet him. I eventually got funding and completed my degree with him.
3) Alternately, if you can attend some conference (figure out a way not to pay but attend the conference) and meet them there. That's a big plus, IMO.
4) Write to the professor's PhD students and start communicating with them. Learn from them and do your research and build some projects/ideas and use it to meet the professor. This is a "back door" entry in to professor's lab. But PhD students are quite helpful and can guide you in to building a "portfolio" of projects.
What I would like is to be around and ideally involved in new ideas, research, and discovery. Even if that means getting coffee and hours inside during the summer.
As a high school student, I'm not sure what I can offer any ideas there?
I also get the impression from your post that I need to get more specific and engage graduate students that are working on projects. Should I possibly be asking the graduate students to work with them as opposed to their professor?
Firstly, please identify areas of your interest. Even a bit broader area is alright to begin with
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Secondly, you can start out reading technical publications in those areas. Getting in to the habit of reading research papers takes time. Slowly you have to reach a stage to remove fluff from the paper and draw out the essence in ~10 minutes. It will also help make your mind fertile for new ideas.
Thirdly, graduate students, particularly PhD students, are helpful and can guide you. Remember, some of them are preparing for a future in academics. So, they are tuned in to guiding new students. But you need to do your work prior to approaching them. I also feel, you'll get more returns for your invested time if you meet PhD students.
I'd like to get involved in AI and I have already found a research group at this university along with a list of graduate students working on projects and their professor.
I'll begin reading through some of their research and getting used to reading papers as you suggested. Should I go through their professor to get to the students or contact them directly?
Does the university offer any events within your field of interest open to you as a high school pupil? If so: attend! Once you're there, you can try to talk to one of the professors. (Hint: if you know in advance who is likely to be there, do your homework. What is he currently working on? What's his position? etc.) Alternately, get a common acquaintance to introduce you.
If you can manage that, you've established a bridgehead to move on from. In my experience professors are a lot more open to people they have met in person, even briefly, than to complete strangers writing to them out of the blue. Show that you are really fascinated by the topic (I'm assuming you are - don't fake it), and have taken time to understand as much of it as you can.
Don't forget that professors are humans too. Some are amazing people, incredibly kind and supportive, others are the opposite. If it doesn't work out with one professor, try another one. Then move on to their PhD & grad students.
Be advised that you are asking for quite a lot, so be humble in your requests. Don't be disappointed if you are turned down. In all honesty, that is the most likely outcome. (Having said that, though: where there is a will, there is a way. Although the way may not lead to where you thought it would.) Perhaps you should start a little smaller: ask for a tour of the lab, or a one week internship.
And finally: no matter what the outcome, keep your head up! :-) Here's one of my favourite quotes that has often motivated me:
"It has long since come to my attention that people of accomplishment rarely sat back and let things happen to them. They went out and happened to things." - Elinor Smith
Any method of reaching out is good, but if class is in session profs usually have posted office hours (hours where you can visit them and talk about the class). In my experience (long ago) they tend not to get a lot of visitors, so if you can, doing it in person might not be a bad idea. Of course being in high-school limits the time available to do this..
Just email them and introduce yourself. Professors are people too. But just like people, they may be too busy or might not get back to you, or they may be very happy to hear from someone young who is interested in their work. You have nothing to lose but a few minutes of your time writing an email and maybe a little disappointment. Good Luck,
"Work with" how? You mean you want to learn from that professor in a non-classroom setting? Or you want to just get to know the professor before you start classes there? Or...?
Kind of hard to say unless you're more specific with what you want to be doing.
Generally, try to find someone to make an introduction on your behalf. That will go over a lot better, IMO.
Professors don't do much of the actual research work themselves. They are the ideas person, giving their grad students direction, and applying to grants to fund them.
So, what will probably happen is that they will direct you to one of their students, if they think you can give him a hand on his project.
Are you looking for a summer internship kind of thing?
Anyways, just wanted to give you a heads up. Do go ahead and ask! And if one prof isn't interested, ask another one. All profs are quite different. If you know someone in the department (grad or undergrad student), you could ask them for suggestions as to which prof would be open to having a HS level intern.
If that is not possible, just write an email and suggest you can talk further on the phone or in person. Just ask and if they say no, go to the next.