There seems to actually be real momentum behind this attempt as reviving Less Wrong. One of the oldest issues on LW has been the lack of content. For this reason, I thought that it might be worthwhile opening a thread where people can suggest how we can expand the scope of what people write about in order for us to have sufficient content.
Does anyone have any ideas about which areas of rationality are underexplored? Please only list one area per comment.
Thanks for the detailed reply, sen. I don't follow everything you said, but I'll take a look at your recommendations and see after that.
Unfortunately, the subject is rather disjoint. Most fluid dynamicists would have no idea that group theory is relevant. My impression is that some mathematicians have interpreted what fluid dynamicists have done for a long time in terms of group theory, and extended their methods. Fluid dynamicists call the approach "dimensional analysis" if you reduce the number of input parameters or "similarity analysis" if you reduce the number independent variables of a differential equation (more on the latter later)
The goal generally is dimension reduction. For example, if you are to perform a simple factorial experiment with 3 variables and you want to sample 8 different values of each variable, you have 8^3 = 512 samples to make, and that's not even considering doing multiple trials. But, if you can determine a coordinate transformation which reduces those 3 variables to 1, then you only have 8 samples to make.
The Buckingham Pi theorem allows you to determine how many dimensionless variables are needed to fully specify the problem if you start with dimensional quantities. (If everything is dimensionless to begin with, there's no benefit from this technique, but other techniques might have benefit.)
For a long list of examples of the dimensionless quantities, see Wikipedia. The Reynolds number is the most well known of these. (Also, contrary to common understanding, the Reynolds number doesn't really say anything about "how turbulent" a flow is, rather, it would be better thought of as a way to characterize instability of a flow. There are multiple ways to measure "how turbulent" a flow is.)
For a "similarity variable", I'm not sure what the best place to point them out would be. Here's one example, though: If you take the 1D unbounded heat equation and change coordinates to \eta = x / \sqrt{\alpha t} (\alpha is the thermal diffusivity), you'll find the PDE is reduced to an ODE, and solution should be much easier now. The derivation of the reduction to an ODE is not on Wikipedia, but it is very straightforward.
Dimensional analysis is really only taught to engineers working on fluid mechanics and heat transfer. I am continually surprised by how few people are aware of it. It should be part of the undergraduate curriculum for any degree in physics. Statisticians, particularly those who work in experimental design, also should know it. Here's an interesting video of a talk with an application of dimensional analysis to experimental design. As I recall, one of the questions asked after the talk related the approach to Lie groups.
For an engineering viewpoint, I'd recommend Langhaar's book. This book does not discuss similarity variables, however. For something bridging the more mathematical and engineering viewpoints I have one recommendation. I haven't looked at this book, but it's one of the few I could find which discusses both the Buckingham Pi theorem and Lie groups. For something purely on the group theory side, see Olver's book.
Anyhow, I asked about this because I get the impression from some physicists that there's more to applications of group theory to building models than what I've seen.
This is an important realization. The Buckingham Pi theorem doesn't tell you which dimensionless variables are "valid" or "useful", just the number of them needed to fully specify the problem. Whether or not a dimensionless number is "valid" or "useful" depends on what you are interested in.
Edit: Fixed some typos.
Regarding the Buckingham Pi Theorem (BPT), I think I can double my recommendation that you try to understand the Method of Lagrange Multipliers (MLM) visually. I'll try to explain in the following paragraph knowing that it won't make much sense on first reading.
For the Method of Lagrange Multipliers, suppose you have some number of equations in n variables. Consider the n-dimensional space containing the set of all solutions to those equations. The set of solutions describes a k-dimensional manifold (meaning the surface of the manifold forms a k-dimensiona... (read more)