I've been playing with topological methods for data analysis recently and I think there are some fruitful things happening. Seems like there are some ideas emerging from theory to practice which might be useful (Betti curves, multiparameter persistence, etc) but they're not quite there yet.

Another idea that's been intriguing me lately is applied sheaf theory. Robinson, Ghrist, and Curry are the only people I see working on this but I don't know what I'm not seeing. The "big idea" is taking local data and seeing if it patches together to a global coherent whole or not. Sometimes it doesn't (old Russian example: arbitrage in currency exchange networks). Or sometimes it's about using interpolation to fill in missing data, if you know that it's something for which there is a global function (temperatures across ocean surfaces), or providing a probability distribution for the missing parts. Category theory has something to offer here as well.

Anyone know more about any of these things?