Yesterday, I attended a very interesting informal talk by Roland van der Veen in which, among other things, he told me a little bit about why he cares about the Volume Conjecture. The Volume Conjecture is considered somehow to be the `big open problem’ in quantum topology. I had never understood why though (I had even asked an MO question but hadn’t really been convinced by any of the very good answers). Why, after all, should people care about any mathematical conjecture?
- Because it helps you to compute something you are interested in.
The conjecture, if true, provides us with a practical way (an algorithm?) to calculate something we care about. The Baum-Connes Conjecture, in my limited understanding, is an example of such a conjecture. The Volume Conjecture is not. SnapPea can calculate hyperbolic volumes quite efficiently, thank you very much, without any help from any NP-hard Jones polynomial calculations.
- Because it gives you a qualitative understanding of a class of objects which you care about.
The conjecture, if true, tells you that all possible objects in some class are nice and simple, and nothing “pathological” occurs. The Hodge conjecture is such a conjecture. The Poincaré Conjecture was another one. But the Volume Conjecture is nothing like that. As far as I can make out, the Volume Conjecture alone doesn’t seem to give much of a qualitative understanding of anything (certainly not of hyperbolic geometry!), except maybe for some far off corner of the Jones polynomial which nobody would otherwise care about.
- Because the conjecture sharply highlights a gap in your understanding, and suggests a mathematical journey on which you should embark.
For such conjectures, it isn’t the statement of the conjecture itself that we care deeply about per se, but the mathematical understanding which would (hopefully) have to emerge as part of any conceivable proof. An example of such a conjecture would be Fermat’s Last Theorem, in that we never really cared about integer solutions to I don’t think, but rather about gaining a qualitatively better understanding of certain classes of Diophantine equations. The Littlewood Conjecture would be a second example, and the Collatz Conjecture would be a third. It is into this class of interesting conjectures that the Volume Conjecture falls.
The main problem in quantum topology has always been “what do quantum invariants mean”? In particular, “what does the Jones polynomial mean”? The Jones polynomial was introduced through the representation theory of braid groups into Temperley-Leib algebras, which, on the face of it, has nothing at all to do with topology and everything to do with algebra. But we know empirically that the Jones polynomial in fact has everything to do with topology, because it is a super-strong invariant of knots and links. We can draw suggestive 2D diagrams to interpret the Jones polynomial by means of a skein relation, but mathematically we don’t really have any sense of a good explanation for why the Jones polynomial should have anything at all to do with dimension 3 (physically there’s Witten’s TQFT for the Jones polynomial, but that’s a different story).
The Volume Conjecture is interesting, perhaps, because it gives us a simple precise concrete statement whose proof would necessarily entail understanding (at least on some level) what exactly it is that makes the Jones polynomial an invariant of 3-dimensional objects. What, after all, could possibly be more 3-dimensional than volume?
I like Roland’s answer a lot (although I probably misrepresented it quite badly). The Volume Conjecture should be thought of not as a useful formula, nor as a classification of nice objects of some sort, but rather as an easily visible flag to run to and to grab, where the real mathematical value is going to be in the journey rather than in its eventual destination. And, if the flag is well placed, this is exactly what an interesting mathematical conjecture should be.