Cohen project? Is that like the surname DNA? Could you explain?
There is a certain bit of DNA shared by most Jews with variations of the last name “Cohen” (Cohn, Coen, etc), which derives from the Hebrew word for priest. It’s assumed that they share this bit of DNA because of their common descent from the priestly tribe of Levites.
I know there was a project originally set up to use test for this marker, in order to determine which populations around the world claiming Jewish descent are actually related to the ancient tribes. If, for example, some tribe in Africa says they are part of the diaspora, they can be tested, and if some of them have the “Cohen” marker, then they have strong evidence for their claim. It sounds like they might have expanded their mission since I last read about them, to tracing ancestry and not just checking for that marker.
The others posters pretty much told the story, but, long story short most of the males who share the name “Cohen” or a version thereof -— and would, at least according to tradition and the Torah -— be ancestors of Aaron of the Tribe of Levi -— do, indeed, have a common middle eastern male ancestor during the time in question.
Obviously, it’s not proof that the common ancestor was, indeed, Aaron, but is certainly very consistent with family histories.