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.