I work at a Research I university where students hear about faculty conducting research all the time. Some students actually know what this means, while others have worked with faculty on their own research projects. (I’m not claiming these two sets of students overlap.) But if you’re a student, how do you find a project to work on? Even worse, if you’re a faculty and you have a student who wants to work with you, how do you find a projects which will yield a meaningful experience for the both of you?
If you’re a student, I’d suggest reading through one of the following books for problems. Richard Guy of the University of Calgary has a compilation of hundreds of wonderful problems in his text Unsolved Problems in Number Theory. The book has dozens of sections with self-contained explanations of the history and current knowledge of the problems. I’ve even written a paper based on one of the problems! Actually, this book is one in a series by Springer Publishers entitled Problem Books in Mathematics. Other books in the series are Problems in Probability by Albert Shiryaev; A Problem Book in Real Analysis by Asuman Aksoy and Mohamed Khamsi; and Problems in Geometry by Berger, Pansu, Berry, and Saint-Raymond. There are 43 different books in this series to choose from!
If you’re a faculty member looking for projects, I’d suggest looking into being a part of a summer workshop geared for such faculty to work with students. Let me discuss a couple of examples which I have been a part of myself.
The Park City Mathematics Institute (PCMI), an entity run by Princeton’s Institute for Advanced Study, will run just such a summer workshop during 2012, 2013, and 2014. Steve Cox of Rice University and Dennis Davenport of Howard University will run a one-week workshop designed to guide and develop faculty mentors of minority students who will offer REU programs primarily for these students. For more information, visit the web site http://pcmi.ias.edu/REUWorkshop/2013.
The American Institute of Mathematics (AIM), located in Palo Alto, California, has hosted a series of such workshops over the past five years or so. Organized by Leslie Hogben of Iowa State University, Roselyn Williams of Florida A&M, and Ulrica Wilson of Morehouse College, the Research Experiences for Undergraduate Faculty (REUF) bring together leading research mathematicians with faculty at undergraduate institutions who are interested in involving their students in areas of active research. Such workshops were run in 2008, 2009, 2011, and 2012. For more information, contact Brianna Dolandson, Director of Special Workshops at AIM, at email@example.com.
Whether you are a student or a faculty member, there are many opportunities to conduct research!