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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?

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I'd like to add that this helps me a lot too, because I'm pretty sure I don't understand the problems myself. –  rnmartingale Nov 8 '12 at 6:50
    
Have you seen Keith Devlin's book by any chance? –  J. M. Nov 8 '12 at 7:10
    
@J.M. I have not. It looks interesting, but the reviews make it apparent that the Millennium Problems are more or less inaccessible to laypeople, even with a whole book's worth of explanations. I guess the best response I can give my friend is, "I have no idea." –  rnmartingale Nov 8 '12 at 21:40
    
I have a further question about this site... is it too soon to accept an answer? What's the etiquette about when an answer should be accept? –  rnmartingale Nov 8 '12 at 22:21
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up vote 3 down vote accepted

I personally don't know if you can explain the Millenium problems to your friend via email, but if your friend has some knowledge of calculus, and the willingness to learn some math then here are a few books that address the Millenium problems and are meant to be popular math books:

1) The Millennium Problems: The Seven Greatest Unsolved Mathematical Puzzles Of Our Time by Keith Devlin.

2) Prime Obsession by John Derbyshire (this discusses the Riemann Hypothesis).

3) Elliptic Tales: Curves, Counting, and Number Theory by Ash and Gross (this builds up to the Birch and Swinnerton-Dyer conjecture).

4) The Poincare Conjecture: In Search of the Shape of the Universe by Donal O'Shea.

I personally read 2) when I was in school and enjoyed it a lot. I came across 3) over the past summer, and I think it is one of the best popular math books I have read so far. Warning: 3) has a fair amount of mathematics and is not easy for a beginner.

I don't think there is an easy way to learn about the Millenium problems. However, there are many problems that are easier to understand (like Fermat's Last Theorem), but which are equally deep/ interesting. You could use these problems to illustrate to your friend what mathematics is about. For example, you could take a look at the Hilbert problems. Some, but not all of them are easy to understand.

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Thanks for the third reference. I think I'm actually going to pick that one up. –  rnmartingale Nov 8 '12 at 21:42
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