I've been thinking about this section mainly because I started asking myself, "What are the important questions in computational geometry". Given all the TheoryMatters discussions of late, this question mutated itself into the (not-unrelated) question, "What are some of the grand challenges in computational geometry" ?
If you do not work on an important problem, it’s unlikely you’ll do important work. It’s perfectly obvious. Great scientists have thought through, in a careful way, a number of important problems in their field, and they keep an eye on wondering how to attack them. Let me warn you, ‘important problem’ must be phrased carefully. The three outstanding problems in physics, in a certain sense, were never worked on while I was at Bell Labs. By important I mean guaranteed a Nobel Prize and any sum of money you want to mention. We didn’t work on (1) time travel, (2) teleportation, and (3) antigravity. They are not important problems because we do not have an attack. It’s not the consequence that makes a problem important, it is that you have a reasonable attack. That is what makes a problem important. When I say that most scientists don’t work on important problems, I mean it in that sense. The average scientist, so far as I can make out, spends almost all his time working on problems which he believes will not be important and he also doesn’t believe that they will lead to important problems.
It's tricky to answer this question properly. It is quite easy to dash off a number of open problems in CG, but that is not so much the point of the second question. It's more about where the field is headed, and where we see some of the most interesting new problem areas opening up from.
Five or so years ago, the answer to this question might have been 'High Dimensional Approximate Geometry'. The core-set revolution has created a new set of techniques for dealing with approximate high-dimensional geometry, and this has fed off the developments in the theory of metric embeddings. I don't doubt that there is more work to be done here, but if we have to answer this question from the "funding perspective", i.e in terms of where geometry will be used next, I keep coming back to the issue of shape.
Some of the most important questions in biology over the next many years will involve structural analysis of proteins, and shape modelling is a key aspect of this. The work by Edelsbrunner and Mücke on alpha shapes connected ideas of shape to deeper ideas in combinatorial topology (the mathematical theory of shape, in one view), and the growing field of "computational topology" has been very active within the CG community of late.
I'm curious as to whether there are other ideas on what areas are likely to push CG forward over the next five to ten years.