I plan to study a topic which is at the intersection of several fields. I am not aware of a graduate degree in this topic, so I will self-study it after graduating. I will receive my undergraduate degree in the Spring for math and computer sciences.
My goals for the self-study are to either publish, or to make enough progress that if I joined a team my work could solve problems.
I would like to enter into an advising/consulting relationship with one or more professors in the related fields. In exchange, I would offer programming consulting or work. Is this type of relationship common, and is there a better way to frame it? What else can I offer professors in exchange for research guidance?