What I'd do with a research lab, part 2:
Attempt to generalize the Anthropic Principle to include Dimensionality
Working hypothesis: There are substantial arguments from Physics, Physical Chemistry, Biochemistry, Biology, and Entropy that intelligent life could only arise in a universe with exactly three macroscopic spacial dimensions.
If that seems overly technical, I'm taking a stab at the question, "What's so special about three dimensions? Why don't we live in four dimensions, or two?"
The Anthropic Principle is an argument about why we seem to live in a universe which seems fine-tuned for living creatures. There are 20-plus 'universal constants' (for instance, the speed of light or the strength of the electromagnetic force) which contribute to how matter, energy, and space behave, and it's been observed that if any of them were much different, life couldn't exist--nuclei couldn't form, planets couldn't form, the universe would expand too quickly and fly apart, and so forth. The Anthropic Principle argues that it's not such a big surprise that everything appears fine-tuned for life: if it were any different, we wouldn't be alive to talk about it.
It's a bit of a strange argument, and it's sometimes tough to pin down exactly what the Anthropic Principle is arguing for, especially since there are so many variants. It certainly shouldn't be taken as an excuse to throw up our hands and say, so that's why things are the way they are: the Anthropic Principle is no substitute for hard scientific knowledge about why these constants have the values they do. Used carefully, though, it's a good intuition pump for exploring the intersection between cosmology and biology.
I've sketched out an attempt to generalize the Anthropic Principle to include dimensionality: essentially, I argue that our dimensionality is very fine-tuned, and that any alternate macroscopic dimensional configuration (e.g., any configuration other than our 3 spacial dimensions + 1 time dimension) would likely not give rise to life. It's clear that we don't know all the factors and implications involved in switching our dimensionality for another, but I believe we do have enough knowledge to make certain limited but probable predictions. With these predictions in mind, I've outlined nine arguments to support my hypothesis: five are from the realm of Physics and Physical Chemistry, three from the realm of Biochemistry and Biology, and one dealing with entropy. A major goal of the project, which I have yet to attempt, is to use these arguments to try to generalize the Drake Equation across dimensionalities.
It's a little on the abstract side. And terribly difficult (some would say impossible). But I think the process of working toward this goal could highlight a lot of interesting things about life, intelligence, biology, biochemistry, physics, and cosmology.
 My frame of reference is what people call the "weak" Anthropic Principle. I disagree with the "strong" Anthropic Principle, which argues that the universe must at some point generate intelligent life such that the universe will be observed, since it gives an almost mystical, non-naturalistic primacy to observation acts by intelligent beings. Instead of tying physical laws to observation, I would rather see them tied to cosmological natural selection.
 The National Academy of Science has recently issued an analysis of (and call for further research on) the prerequisite chemical conditions for life. Ars Technica's coverage of this report is an excellent place to start for anyone who wants to get up to speed on what scientists have identified as enabling conditions for life to arise (though extending some of these chemical principles/prerequisites to other dimensionalities will not be straightforward).
 I haven't been able to find much published literature explicitly on this topic of physical and biochemical implications of alternate dimensionalities and I'm grateful to those people with whom I've discussed this.