First, I will explain my situation.
In my University most of the careers are doing videos to explain what we do and try to attract more people to our careers.


I am in a really bad position, because the people who are in charge of the video want me to explain what a pure mathematician does and how it helps society. They want practical examples, and maybe naming some companies that work with pure mathematicians, and what they do in those companies. All this in only 5 or 10 minutes, so I think that the best I can do is give an example.


Another reason that I am in a bad position: In my University we have the career "Mathematical engineering" and they do mostly applications and some research in numerical analysis and optimization. (\*)


I know that pure mathematics is increasing its importance in society every year. 
Many people in my country think that mathematics has stagnated over time and now only engineers develop science. 
I think that the most practical thing I can do is give some examples of what we are doing with mathematics today (since 2000).


If some of you can help me, I need the following: 


1. A subject in mathematics that does not appear in (\*). Preferably dynamical systems, logic, algebraic geometry, functional analysis, p-adic analysis or partial differential equations.
2. A research topic in that subject.
3. Practical applications of that research and the institution that made the application.


Extra 1. If you know an institution (not a University) that contracts with pure mathematicians and you know what they do there, please tell me also.


Extra 2. If you have a very good short phrase explaining "what a mathematician does" or "how mathematics helps society" I will appreciate it too.


Thanks in advance.



How does the work of a pure mathematician impact society?
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](http://www.cs.cmu.edu/~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](http://fold.it/portal/)) 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](http://comptop.stanford.edu) -- 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](http://www.nature.com/srep/2013/130207/srep01236/full/srep01236.html).


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](http://www.maa.org/meetings/calendar-events/math-in-the-movies).


