HighSchool Research Projects
We occasionally mentor highschool students in mathematical research projects. In such a project, the student works on a small open problem, under the close mentorship of a professor who is an active researcher. Here are some recent projects (highschool students are marked with an asterisk):

Michael N. Manta*, Triangle colorings require at least seven colors, Discrete Mathematics 344 (2021), 112411.

Michael N. Manta* and Pablo Soberón, Generalizations of the YaoYao partition theorem and the central transversal theorem, submitted. (Due to the pandemic, Michael postponed the beginning of college and did this second project with us.)

Hannah Ashbach and Kiki Pichini*, An Upper Bound for the Number of Rectangulations of a Planar Point Set, submitted.

Sarai HernandezTorres, Matthew Junge, Naina Ray*, and Nidhi Ray*, Distancedependent chaseescape on trees, submitted.

Martin Balko, Adam Sheffer, and Ruiwen Tang*, The constant of point–line incidence constructions, submitted.
Doing a project. An average project takes about a year, and some take longer. For that reason, we start projects with students who have at least a year before their college applications deadlines. The best time to start is during sophomore year.
For the first few months, the student reads material provided by the mentor and regularly meet to discuss this material. Some mentors first read a paper with the student, to see if there is a good connection, and only then the student and mentor commit to a research project.
Finally, when the student is ready, they begin to work towards solving an open problem. This is done with close guidance from the mentor. If the problem is solved, then we write a paper and submit it to a math journal. If not, that's completely fine! Math is hard and publishing a paper is not the main point. The main point is for the student to experience math research and get support for whatever they choose to do next.
This is not recommended for most people. A research project is for students who are highly enthusiastic about advanced math and want to further explore it. However, for most of these students, it is probably better to pursue math competitions or programs such as math circles (see the wonderful New York Math Circle).
Working on a math problem is not easy. There are no welldefined deadlines and it is not clear how close we are to solving the problem. Maybe we need one more hour of work, one more week, six months, or perhaps the problem is not even solvable. It becomes much easier to "procrastinate" by doing homework with a clear deadline and clear goals.
Many highschool students are extremely busy with school and extracurricular activities. This is not just one more extracurricular! A math research project requires a lot of time and mental energy, over a long period of time. If your schedule is already packed, then such a project is more likely to traumatize you about math than have a positive impact.
Additional details. We especially encourage female students to pursue math research projects. We mostly work with highschool students from the NewYork City area. However, we may make exceptions for students coming from unusual circumstances, for all of our rules.
Students who work with us are welcome to attend other math events that we run. In particular, they may attend REU events during the summer, including visits by famous scientists, math talks, and social activities. We also continue to support the students after the project ends (college applications, other science opportunities, and so on.)
Since we run this program voluntarily, while also running other programs, we can only take a few students. We apologize if we will not have room for you.
To discuss this further, you are welcome to write to Adam Sheffer at Adam.Sheffer@baruch.cuny.edu.