Spotlight on Science: Millennium Problems

I promised a lack of biology, so here we are! The Millennium prize problems are seven problems in mathematics named by the Clay Mathematics Institute in 2000 (hence the name). There’s a possible prize of ONE MILLION DOLLARS if you solve any of them as well, and only one has been solved so far. The Poincaré conjecture (I’ll get on to explaining all of them soon, don’t worry) was solved in 2010 but the guy that solved it, Grigori Perelman, rejected the prize.

So here are the seven problems (YOUR CHANCE TO WIN ONE MILLION DOLLARS):

1. P vs NP

Okay the basic idea of P vs NP is “if you can check the answer to a problem quickly, can you find that answer quickly?” If you can check a problem quickly, it’s NP; so, for example, if you were given a HUGE set of numbers, and asked whether there was a subset of 5 numbers that added up to 0, that’s really easy to check. It’s less easy to find though (if it was easy to find it’d be P). The P vs NP problem focusses on proving that P equals NP OR that P does not equal NP.

Actually, here’s a good example of an NP-complete problem:

Randall Munroe, xkcd.com

2. The Riemann Hypothesis

Back when I was much more mathematically minded than I am now, the Riemann hypothesis was my favourite millennium problem (I was frantically crushing on a boy whose favourite problem was P vs NP, which was a major sticking point for me). There’s a fancy equation that can be graphed in the complex plane

okay so the square root of minus one is an imaginary number (i) and when numbers are made up of a bit of i and a bit of real numbers, they’re complex. Think a vodka and orange, where the real number is the nice refreshing vitamin-heavy orange and the imaginary might not give you a headache now but just… don’t think about it too hard, okay?

Right so the Riemann-zeta function is graphed in the complex plain. There are some points where it crosses zero that are really easy to prove, and they’re called “trivial”, but there are some that are trickier, and they are “non-trivial” – we know a bunch of them lie between zero and one, and the Riemann hypothesis says they are ALL on 1/2 (for the real bits)

It’s got pretty cool implications in where prime numbers are – numbers are weird, guys, and there’s a lot of them, and where prime numbers are is like seeing the tops of mountains but not knowing how the rest of the mountain range falls out.

3. Yang-Mills Existence and mass gap

also known as “let’s make electromagnetism MORE BADASS”. The Yang-Mills theory (make a theory, get your name everywhere, instant fame) is very quantum physics’y, so I’m just going to leave it alone.

4. How do fluids?

Uh, the official name is “Navier–Stokes existence and smoothness” but it’s basically How Do Fluids? There are a bunch of equations about how fluids move from the 19th century and we still don’t really know what’s up with that. The MILLION DOLLARS will be yours if you can make a mathematical theory that tells us how these equations work, and so, how fluids do.

5. The Birch and Swinnerton-Dyer conjecture

This actually ties really well into P vs NP (she says boldly) as it centres around working out whether you can find out how many solutions there are to a given problem. The problem is something about elliptic curves, which I care very little about unless they are defining the motion of the planets.

Ellipses have two centres, because they’re like slightly melted circles (think Dali clocks), and the distance apart of these two centers is used to determine the “eccentricity” of an ellipse. The eccentricity of Pluto’s orbit was a factor in giving it the side-eye with this “planet” status. As a side note, Earth has a centre about 5 million kilometers to the … uh, left of the sun, which means the Southern Hemisphere has summers closer to the Sun (and winters further away). The effect of this is pretty negligible compared to the tilt of the Earth’s axis, but I’ll get nerdy about astronomy at some other time.

6+7. No thank you.

The other two problems are the Poincaré conjecture (solved, so obviously no good to anyone now), and the Hodge conjecture, which I don’t understand.

Woo! Maths nerd, done! Sorry if that was a bit dense, I just really enjoy the Millenium problems. There’ll be something a bit lighter and sweeter next week.

Sophia, who spent a long time determined to do a maths degree.

I get that this might be a bit dull/mathsy for some people, but next week’s article is going to be on PLANETS and ALIENS and I’ll try find an excuse to talk about DINOSAURS.

One thought on “Spotlight on Science: Millennium Problems

  1. I LIEK MATHS 2 April 21, 2013 / 10:21 am

    A little bit more:
    P=NP:
    It’s basically assumed that P != NP. A lot of cryptography relies on NP problems being hard to solve. If you found that P = NP then you can break into banks, set off missiles, all those good things. But don’t worry, P!=NP, probably. We wouldn’t base all our security on it if we weren’t really sure, right?
    In fact, run-times complexities are so dramatic that they made a movie about it:
    http://en.wikipedia.org/wiki/Travelling_Salesman_(2012_film)

    Riemann Zeta:
    My favourite thing about this is that in his early career, Bernhard Riemann made this hypothesis in basically the most important Number Theory paper ever. Then decided that he didn’t like Number Theory and stuck to Topology, rendering all future Number Theorists invalid.

    Yang-Mills:
    The Standard Model is based on a mathematical model, which is based on a framework made by Yang and Mills. Currently it can only be solved theoretically for trivial cases, but computer models and experiment show certain properties(e.g. quarks are never seen outside nucleus) and it would be nice to know the theory is sound. The prize is for showing that these equations give a positive “mass gap”, meaning it predicts a lightest particle, and that the particle has positive mass.

    Hodge Conjecture:
    http://www.claymath.org/millennium/Hodge_Conjecture/ has a description which is understandable. Trying to find properties of shapes is a hard problem, especially once you start having more than 3 dimensions. So Mathematicians solved this by finding the properties of two shapes glued together were, and used this to find general rules for combining shapes into more dimensional, weirder shapes. But in doing so, there’s awkward bits in the theories that don’t intuitively correspond to geometric ideas. So the Hodge Conjecture is to show that some of the nonsense bits are actually a bunch of geometric thingies in disguise.

    I really like Gregori Perelman. He got offered a Fields Medal(Maths Nobel Prize) and this and turned them both down because he didn’t want money or fame. In fact because he didn’t explicitly solve the Poincaré conjecture(since it was trivial from his work), two people wrote a paper claiming to have solved it to try and claim the prize. Perelman was so upset by this and the attention he was getting that he quit mathematics and became a recluse in his mother’s basement.
    It’s really easy to say “ooh, a million dollars”, but it’s always a much better thing to say “ooh, an interesting problem”. Anyone who tries to solve these problems for the money is doing it for the wrong reason. They should change it to: solve problem number 1 and I’ll give you problem number 2.

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