First of all I explain my situation: On my University most of the careers are doing videos to explain what we do and try to atract more people to our careers.
I'm in a really bad position, because the people who are in charge of the video want me to explain what a pure mathematician do and how it help society (with practical examples and maybe naming some companies that work with pure mathematician and what they do in that companies) in only 5 or 10 mintues, so I think that the best that I can do is make a example.
Other reason that have me in a bad position: In my University we have the career "Mathematical engineering" and they do most applications and some research in numerical analysis, optimization. (*)
I know that pure mathematics are 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:
A topic in mathematic that not appear in (*). Preferably dinamical systems, logic, algebraic geometry, functional analysis, $p$-adic analysis or partial differential equations.
A research in that topic.
Practical applications of that research and the institution that make the application.
Extra 1. If you know a institution (not a University) that contract pure mathematicians and you know what they do there you can tell me too.
Extra 2. If you have a very good short phrase explaining "what a mathematician do" or "how mathematics help society" i will regard it too.
Thanks in advance.