Research semester in math Admittedly, this is a soft question.
My own experience in mathematical research has been of long periods of research, mostly characterized in long "blocks" and sporadic breakthrough. 
How does this image fit with research semesters, in which many researchers (veterans, early careers, postdocs and students) gather at a single institution, e.g., ICERM, IPAM etc., for a few months? What kind of research do one usually do in these sorts of semesters? Should one get there with a well defined project a-priori? Do you find this format effective for mathematical research?
 A: Generally one applies for a short-term position to perform the (expected) final research (and writing) on a topic one has been working on for a while, for instance to explore ramifications, extensions, and so on.  It is a rare venue (like the Institute for Advanced Studies) that hires you for an extended period based more on "promise" than well-identified needed execution.
My recommendation is to do what lots of great mathematicians do:  think about several projects intermittently.  When one project seems particularly promising, or has succumbed to part of your analyses, highlight that one and apply for a short internship to explore it more fully.
Since you asked:  The benefits of such a short-term research position include:


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*The opportunity to work with others, especially experts in a field.

*Get professional exposure, meeting a wider range of mathematicians than you would otherwise.

*Opportunity to travel, especially to foreign countries (where short-term positions are easier to get than permanent positions).

*Opportunity to focus intensely on a technical problem, without teaching or administration duties.

*Temporarily solving the "two-body" problem, i.e., to follow a spouse to his/her place of employment.

