Here are two recent speakers from our departments lecture series on applied mathematics (the Wing Lectures at U. Rochester). We've had a lot of great lecturers, but these two stick out to me as having an impact on society.

Adrien Treuille -- he works in computer graphics, and his research purely in this areas includes algorithms realistic modeling of crowds, and real time fluid mechanics (e.g. with basically no delay, they can add digital trails of flames behind a race cars on TV). Also, he can construct initial data to make (digital) smoke form specified shapes at specified times.

But, what's even cooler, is that he's collaborated with biologists on an interactive game (called FoldIt) for finding the optimal conformation of proteins. Players get points for moving the protein into better conformations. Humans are way better at this than computers, and they've actually published papers based on conformations discovered by players; in one case, the answer they found had eluded scientists working in the field for at least 10 years! They call this "crowd source science". They've also created another game for engineering shapes with RNA, and the players of that game have discovered things as well.

Gunnar Carlsson -- he is an algebraic topologist who originally worked in K-theory, but shifted to using algebraic topology to understand data. Specifically he was one of (the?) pioneers of persistent homology, which is a way of using topology to understand data. The really great thing about it is that it discovers structure for you--instead of fitting the data to a model, persistent homology discovers the model for you. For instance, they used PH to find the most commonly occurring 9-bit patterns in black and white images, which in theory would allow better image compression that JPEG (but in practical, JPEG is highly optimized, so it would take a lot of work to benefit from this discovery). In another example, they analyzed genomic information from cancer patients and linked it up with survival information; PH discovered the threshold for the expression of a certain gene at which survival drops significantly.

Besides discovering structure in data, PH is also able to incorporate data collected at different times and from different sources (perhaps a reflection that the method is "metric free" --I'm not sure about that). A good reference to check is their paper in Nature.

Also, Tony DeRose from Pixar gave a really great talk on the methods they've developed in computer graphics. I don't remember the details so well (maybe because I was distracted by the entertaining presentation :) so here's a link to a good article.