Many mathematicians prefer not to work alone. It is basically for the same reason why people who go mountain climbing don't go alone; having somebody nearby, as you explore an unknown and difficult territory, can help you avoid pitfalls and desaster, or find a brilliant shortcut. Plus, in the occasional breaks from intense thinking that must occur to preserve mental sanity, you have someone around to chat with.
I usually prefer to choose my coworkers among people I am close to anyway. In fact, I wrote my first joint paper with WS, who at that time was just a Wonderful Boyfriend. The next joint paper was written with FCo, my (so far only) female coauthor, a close friend who had been a student with me, on a topic which was very close to both our theses. I am very proud of that work, because it is the first of my papers where I didn't just prove the theorems but significantly contributed to the statements. All my papers so far had been suggestions of my advisor, except for the joint work with WS which was based on an idea of WS.
You may guess the next logical step: I wrote a paper with Ununderstandable Coauthor UnCo, a close friend of WS who had been a student with him. With time, I think that UnCo and I have also become friends. I am not sure because I still find him quite ununderstable, though.
A the same time, I started a collaboration with another friend who was had been a student with me: I'll refer to him as ImCo, for impressive coauthor, since when he applied for a job at the university where I was working, a colleague told me how he had been so impressed by this particular candidate. Except he used a word in my language which is much stronger than impressive, and which made everybody laugh. For the record, ImCo did turn down our offer, and has since only worked in Impressive Universities.
Excluding PD's and S's, my youngest coauthor is NeCo, my nervous coauthor. He's an extremely smart person who tends to worry overmuch about any mistake he might possibly make. When he's not talking about himself, which tends to be a very gloomy topic, he's a very pleasant person to spend time with; he also drinks precisely the correct amount so that, when having dinner together, we can order haf a liter of wine. NeCo works in a different country, but not too far away, so hopefully we'll see a lot of each other in the future.
I'll keep updating this page, so as to have all the coauthors together.
A grad student fellowship
4 hours ago