Millennium Prize Problems

The seven problems named by the Clay Mathematics Institute in 2000, and the one that has been settled.

7 questions. Reveal each answer when you have committed to one.

1. How many problems did the Clay Mathematics Institute name as Millennium Prize Problems in 2000?

   • A10
   • B5
   • C6
   • D7
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   D. 7

   Seven were named in May 2000, each carrying a one million dollar prize. Note that six is also a true answer to a differently worded question, since six of the seven are still open, which is why this one names the announcement. More on poincaré conjecture.

2. How many of them remain unsolved?

   • A6
   • B7
   • C0
   • D5
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   A. 6

   Six remain open. Only the Poincaré conjecture has been settled. More on poincaré conjecture.

3. Which Millennium Prize Problem has been solved?

   • AThe Hodge conjecture
   • BThe Poincaré conjecture
   • CP versus NP
   • DThe Riemann hypothesis
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   B. The Poincaré conjecture

   Perelman proved it in preprints of 2002 and 2003 using Hamilton’s Ricci flow, and verification was complete by 2006. More on poincaré conjecture.

4. Who proved the Poincaré conjecture?

   • AYitang Zhang
   • BAndrew Wiles
   • CGrigori Perelman
   • DTerence Tao
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   C. Grigori Perelman

   Wiles proved Fermat’s Last Theorem, and Zhang proved that infinitely many prime pairs differ by a bounded amount. Neither is this problem. More on poincaré conjecture.

5. What became of the Millennium Prize for that proof?

   • AIt was split between two mathematicians
   • BIt is still unclaimed pending review
   • CIt was never awarded
   • DIt was awarded and declined
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   D. It was awarded and declined

   The Clay Institute awarded the prize in 2010 and Perelman declined it. The problem is solved and the prize was awarded; it was simply refused. More on poincaré conjecture.

6. Which Millennium problem concerns the equations of fluid motion?

   • ANavier–Stokes existence and smoothness
   • BThe Hodge conjecture
   • CBirch and Swinnerton-Dyer
   • DYang–Mills existence and mass gap
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   A. Navier–Stokes existence and smoothness

   The problem asks whether smooth solutions always exist for the three-dimensional Navier–Stokes equations, or whether they can break down in finite time. More on manifold.

7. The Birch and Swinnerton-Dyer conjecture concerns

   • Aprime gaps
   • Belliptic curves
   • Cknot invariants
   • Dgraph colouring
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   B. elliptic curves

   It relates the number of rational points on an elliptic curve to the behaviour of an associated L-function, linking an arithmetic count to an analytic object. More on elliptic curve.