When I was a graduate student in math, I felt very lonely because I had to be inside my own head day after day for prolonged period of time. It became very depressing and isolating. Whenever I would discuss the problem with my advisor, I would get excited again but I can only meet him once every two weeks. I love learning about math and teaching it but not sure if the research career is right for me due to this isolation issue. Are there groups who perform math research together? I remember seeing one of my PhD colleague doing a joint paper with another visitor. They would work in front of the white board for three hours every afternoon. But I don't think that situation is common. Thanks for reading this.
Some mathematicians realize that they do not like (or do not do well) working alone, and therefore almost always work jointly on problems with another mathematician. You may be one of those "social" mathematicians. [Other mathematicians work better alone, and feel a bit uncomfortable working in real-time with others.]
In the past, social mathematicians would have to be in a large enough department so that others to work with were in the same building. Or have enough seniority and/or funding to perpetually invite collaborators to come for a semester. But nowadays with Skype and essentially free telephone service, it should be possible even if you are physically isolated.