I'm writing an email to a friend, who asked me to explain what higher level mathematics is like. Since I'm still an undergraduate student, I referred him to this Wikipedia page, saying that I assume a large part of mathematics investigates the phenomena around problems like these.
He asked me to explain them in a way that would make sense to an English major. As you might imagine, while the $P\neq NP$ problem isn't too hard to explain, things like the Birch and Swinnerton-Dyer conjecture aren't accessible to laymen.
The Birch and Swinnerton-Dyer conjecture deals with a certain type of equation, those defining elliptic curves over the rational numbers. The conjecture is that there is a simple way to tell whether such equations have a finite or infinite number of rational solutions. Hilbert's tenth problem dealt with a more general type of equation, and in that case it was proven that there is no way to decide whether a given equation even has any solutions.
Yeah, an English major won't get this. I'm not even sure if I get this.
So, if it can be done, how would you explain the seven (six unsolved) Millennium Prize Problems to a layman?