Where's the math at?

So, I've been thinking about how to get some more math into my research. I am, after all, getting a PhD in math (applied math, yes, but still math).

I poked around looking for some ideas:

• Turns out, stable distributions, while possessing some really fine mathematical properties, are extrodinarily difficult to use in Bayesian analysis because their pdf's and are generally intractable, which makes computing posterior distributions really, really hard. As my measures come from fat-tailed stable distributions and I prefer to use Bayesian analysis, this is worth looking into.
• My problem of maximizing correlation within a block is best handled by a heuristic. The problem is most likely np-complete, and therefore not any algorithm isn't much use on a data set of any reasonable size; it would never finish. It would be nice, however, to be able to come up with some sort of bound as to how far off the heuristic is from optimal. The way to do this is to solve the same problem in what's known as the "dual space". The best non-technical description of the dual space is that it's the solution space turned inside-out. So the optimal solution in the problem space yields a converse solution in the dual space. For any sub-optimal solution, the difference between it and the optimal answer is always less than or equal to the difference between the solution and its dual. So, getting an approximation bound comes down to identifying the dual space and projecting the solution into that space. Unfortunately, Algebra isn't really my strong suit. Maybe it's worth addressing that just so I can pursue this line. (Maybe it's worth addressing that simply because math teachers really ought to know Algebra).
• For that matter, proving the problem is np-complete would be cool. It would certainly help justify all the hoops we're jumping through.
• As this is all time series data, it seems that there should be some way to leverage that. My adviser very much wants me to go down that road. I'm not exactly sure what's there. Sure, all these projections are based on actuarial models which could be reverse engineered, but that rather defeats the purpose. We ran them through the model because we wanted the time series; if all we wanted was the assumptions that generated the time series, we could just look them up directly. Still, from an estimation standpoint, all these series are very similar in the way they run off. That structure can be exploited.
• While I've been focusing mainly on cash flow projections for insurance policies (mainly because that's the data I have), there's probably some merit to generalizing this to any cash flow and looking at what assumptions matter and which ones can be relaxed.
• Warp statistics. Sorry Sabermetrics (baseball stats) nerds, I'm not talking about Wins Above Replacement Player. This is a method for transforming densities to make MCMC simulation converge. I'm not that terribly interested in MCMC techniques, except that whatever makes them converge also tends to make my sampler converge. There's some evidence, based on Warp transformations, that Stable distributions aren't really the best bet for financial modeling. OK, then what is? This one is wide open.

Anyway, the point is that the math is out there. It may not be the core of my research, but there's no reason I can't tie in some beefy math results along with my computational work.