What gets you a million dollars?

1 down (by Perelman), 6 to go. The seven millennium problems. Hardest way (debatable) to generate a million dollars.

  1. Birch and Swinnerton-Dyer Conjecture: "Imagine you've just been on a first date with someone. Now, based on how that date went, you're trying to predict how many more dates you two will have. Let's say there's a unique "vibe meter" that measures the chemistry between two people on a date."

    • This conjecture relates the number of rational points on an elliptic curve to the behavior of the curve near the origin.
    • The Birch and Swinnerton-Dyer conjecture relates the rank of the abelian group of points over a number field of an elliptic curve  to the order of the zero of the associated-function at \(s = 1\).

  2. Hodge Conjecture: "Think of a jigsaw puzzle. Some pieces are unique and special. This problem asks if certain mathematical shapes can always be broken down into a combination of these special pieces."

    • Given a projective complex manifold, every Hodge class on it is a linear combination with rational coefficients of the cohomology classes of complex subvarieties of it.

  3. Navier–Stokes Existence and Smoothness: "Fluids can be calm, but sometimes they throw tantrums and become turbulent. Can we predict when and how these tantrums occur?"

    Letting \( u_i \) be the \( i \)th component of the velocity field (which is a vector field), \( p \) be the pressure field (which is a scalar), \( \rho \) be the density of the fluid, \( \nu \) be the kinematic viscosity, and \( f_i \) be the \( i \)th component of the force field, the equations are:

    \[ \frac{\partial v_i}{\partial t} + \sum_{j=1}^{n} u_j \frac{\partial u_i}{\partial x_j} = -\frac{1}{\rho} \frac{\partial p}{\partial x_i} + \nu \sum_{j=1}^{n} \frac{\partial^2 v_i}{\partial x_j^2} + f_i. \]

  4. P vs NP Problem: "Solving a riddle is hard, but checking an answer is easy. But what if computers could solve as easily as they check? This would change the world and not always in a good way!"

    \[ P \stackrel{?}{=} NP \]

  5. Riemann Hypothesis: "There's a mysterious music note that every mathematician wants to hear. It's believed all these notes line up perfectly in a row. Proving this could unlock the secrets of prime numbers."

    States that all nontrivial (the trivial roots are when \(s=-2, -4, -6, …\)) zeros of the Riemann zeta function have real part \({1}/{2}\).

    \[ \zeta (s)=\sum _{n=1}^{\infty }{\frac {1}{n^{s}}}={\frac {1}{1^{s}}}+{\frac {1}{2^{s}}}+{\frac {1}{3^{s}}}+\cdots \]
    \[ \sum _{n=1}^{\infty }{\frac {1}{n^{s}}}=\prod _{p{\text{ prime}}}{\frac {1}{1-p^{-s}}} = {\frac {1}{1-2^{-s}}}\cdot {\frac {1}{1-3^{-s}}}\cdot {\frac {1}{1-5^{-s}}}\cdots {\frac {1}{1-p^{-s}}}\cdots \]

  6. Yang–Mills Existence and Mass Gap: "Particles are the universe's aunties; some carry forces like they're passing on the latest gossip. Even though we thought some of them shouldn't weigh anything, turns out they've got some substance!" “It's like trying to weigh a ghost. The equations say these particles should have no mass, but in real life, they seem to have weight. Did they sneak in some cosmic donuts?”

    \[ \text{I HAVE NO FUCKING CLUE} :) \]