P v. NP is one of the unsolved problems in computer science, essentially the question is: can all problems whose answer can be quickly checked for correctness by a computer also be quickly solved by a computer? At the moment the consensus is "no" but there is a $1million reward for anyone who finds an algorithm that works, and if someone does then current computer security measures are all compromised because encrypted passwords will become trivially crackable.
Complexity theory is a relatively new discipline, about 40 years old, that looks to model and understand the behaviour of complex systems. The systems themselves can be as diverse as the weather, crowd movements, epidemic spread or the brain. The experts talking about it on In Our Time were Ian Stewart (University of Warwick), Jeff Johnson (Open University) and Eve Mitleton-Kelly (London School of Economics).
We finished three different series over the last week so I wasn't going to write about any of the one-off programmes as well, but Heart vs Mind: What Makes Us Human? irritated me sufficiently that I wanted to say why! The premise of this film was that the presenter, David Malone, had always thought of himself as a wholly rational person but then his life had become derailed - his wife had started to suffer from severe depression and it was as if the person she had been no longer existed.
Blaise Pascal was a 17th Century Frenchman who was a scientist, mathematician & philosopher. Several of his ideas are still recognised today - either still in use (for instance some of his mathematical work) or recognised by the naming of modern things (like the programming language Pascal). Discussing him on In Our Time were David Wootton (University of York), Michael Moriarty (University of Cambridge) & Michela Massimi (University of Edinburgh).
Fermat was a 17th century lawyer who did maths in his spare time, corresponding with many other mathematicians around Europe. He had a habit of setting little challenges to his correspondents - "I can prove this, can you?". He's famous now for an annotation he made in a book - that he had found a proof that an + bn = cn has no positive integer solutions when n>2 "which this margin is too narrow to contain".