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.
The Empire of Mali flourished between 1200 & 1600 CE, in sub-Saharan West Africa. At one point the Empire was so wealthy that when its ruler, Mansa Musa, travelled through Egypt on his Hajj he reduced the value of gold in Egypt with the amount he gave away. Discussing the empire on In Our Time were Amira Bennison (University of Cambridge), Marie Rodet (SOAS) and Kevin MacDonald (University College, London).
Hans Holbein the Younger was one of the foremost portrait painters to work in England during the Tudor period (and perhaps ever), and it's his paintings that shape how we see the court of Henry VIII. Discussing his time at the Tudor court on In Our Time were Susan Foister (the National Gallery), John Guy (Clare College, University of Cambridge) and Maria Hayward (University of Southampton).
Perpetual motion would be a wonderful thing, if only it were possible - being able to set some machine going and then it would power itself and just carry on & on without end. Free energy from nothing! Which is, of course, why it is impossible - but this wasn't provable until relatively recently. Discussing the search for, and disproof of, perpetual motion on In Our Time were Ruth Gregory (Durham University), Frank Close (University of Oxford) and Steven Bramwell (University College London).
The Roman Republic was the subject of an In Our Time episode all the way back in 2004 - we listened to it last August while there weren't any new In Our Times airing. It's a pretty broad subject for a 45 minute programme - 500 years of history plus its rise and fall - so of necessity it was painting with fairly broad brushstrokes and looking at themes and commonalities across the centuries. Tackling it were Greg Woolf (St Andrews University), Catherine Steel (University of Glasgow) and Tom Holland (historian and author).
The Etruscans were one of the other cultures to live in Italy in the 1st Millenium BCE. They are often overlooked in favour of the Romans (who conquered them), but they were a power in their day and even ruled over Rome for a while early in its history. They were the subject of an In Our Time episode from 2011 which we listened to recently, and discussing Etruscan history and culture were Phil Perkins (Open University), David Ridgway (University of London) and Corinna Riva (University College London).
Back in the summer while In Our Time wasn't airing new episodes we dug back through the archives and found a (rare) Egyptian related one that we didn't think we'd listened to before - about Akhenaten, which aired in 2009. The experts on the programme were Richard Parkinson (British Museum), Elizabeth Frood (University of Oxford) and Kate Spence (University of Cambridge). (As it's so old affiliations of the experts have probably changed.)
The Augustan Age is the period between 27BCE and 14CE when the Emperor Augustus ruled the Roman Empire. It was discussed on In Our Time (in 2009) by Catharine Edwards (Birkbeck College, University of London), Duncan Kennedy (University of Bristol) and Mary Beard (Cambridge University). They were primarily considering the politics and arts of the Emperor Augustus's reign and how these were linked.
I'd heard of Frederick the Great before I listened to this In Our Time programme about him - I knew he was an 18th Century ruler of Prussia, and I knew he was a flautist (having seen a painting of him playing the flute). What I wasn't aware of before was that he was obsessed with being famous, and had quite serious Daddy issues. The experts who discussed him on the programme were Tim Blanning (University of Cambridge), Katrin Kohl (University of Oxford) and Thomas Biskup (University of Hull).
"Extremophiles" is a bit of a parochial term - this is the name for organisms that live happily in environments that we consider extreme. Too cold, too hot, too acid, too something to support life, in our terms. Studying the lifeforms that disagree with us on what is a good place to live has started a new field of astrobiology and a new appreciation of the possibility of life existing in the wider universe.