Differential Equations

Same bump, two rules

Five equations that run the world

Why a golf ball has dimples

The seven Millennium Prize Problems

1. 1Fluid dynamicsOpen

   Navier–Stokes existence and smoothness

   $${\mathbf{u}}_0\text{smooth}\stackrel{?}{⟹}\mathbf{u}(t)\text{smooth}\forall t$$

   Start a 3-D flow smoothly. Does it stay smooth forever, or can its speed blow up to infinity somewhere?

   Computers solve these equations every day for weather and aircraft, but nobody has proved that a smooth solution always exists in three dimensions. The prize goes to a proof that it does, or to an example where it breaks.

   Where the PDE is: This is a PDE problem: it is about the Navier–Stokes equations themselves.

2. 2Quantum physicsOpen

   Yang–Mills existence and mass gap

   $$D_{\mu }{F}^{\mu \nu }=0,\Delta >0$$

   Prove that the theory behind the strong nuclear force truly exists as mathematics, and that its lightest particle has mass above zero.

   Yang–Mills theory holds quarks together inside protons, and experiments match it very well. But nobody has built it rigorously as a quantum theory in four dimensions, or proved the “mass gap” Δ: that even the lightest thing it describes has some mass.

   Where the PDE is: Classically, Yang–Mills is a set of nonlinear wave equations, D_μF^μν = 0. The prize asks for the quantum version.

3. 3TopologySolved

   Poincaré conjecture

   $$\frac{\partial g_{ij}}{\partial t}=-2R_{ij}$$

   Any closed 3-D shape where every loop can be shrunk to a point is really just a stretched 3-D sphere.

   Solved by Grigori Perelman in papers posted in 2002 and 2003; he turned down the prize in 2010. It is the only Millennium Problem solved so far. The equation shown is not the conjecture but the tool that proved it.

   Where the PDE is: Perelman used Ricci flow (shown above): a heat equation for the shape of space itself, where bumps in curvature smooth out the way hot spots do.

4. 4Number theoryOpen

   Riemann hypothesis

   $$\zeta (s)=\sum _{n=1}^{\infty }\frac{1}{n^s}=0⟹\mathrm{Re}(s)=\frac{1}{2}$$

   Prime numbers look random, but one function controls them. Do all of its important zeros sit on one straight line?

   Riemann showed in 1859 that the zeros of the zeta function ζ(s) control how the primes are spread out. More than ten trillion of the non-trivial zeros have been checked by computer, and every one lies on the line Re(s) = ½. But checking is not proving.

   Watch it happen: the Riemann Hypothesis, live →
5. 5Computer scienceOpen

   P versus NP

   $$\mathsf{P}\stackrel{?}{=}\mathsf{NP}$$

   If an answer is quick to check, is it always quick to find?

   A finished sudoku is easy to check but seems hard to solve. P is the set of problems a computer can solve quickly. NP is the set whose answers it can check quickly. Most experts think P ≠ NP. If P = NP, much of today’s online encryption could be at risk.

6. 6Number theoryOpen

   Birch and Swinnerton-Dyer conjecture

   $$y^2=x^3+ax+b:\mathrm{rank}E(\mathbb{Q})={\mathrm{ord}}_{s=1}L(E,s)$$

   For curves like y² = x³ + ax + b, can one function tell you how many points with fraction coordinates the curve has?

   Some of these elliptic curves pass through infinitely many points whose coordinates are fractions, and some through only finitely many. The conjecture says that the size of that family (its rank) equals how many times a related function L(E, s) touches zero at s = 1.

7. 7GeometryOpen

   Hodge conjecture

   $$H^{2k}(X,\mathbb{Q})\cap H^{k,k}(X)=⟨[Z]{⟩}_{\mathbb{Q}}$$

   On the nicest kinds of shapes, can every feature that looks as if it comes from algebra really be built from pieces cut out by polynomial equations?

   The most abstract of the seven. It asks whether two very different ways of describing a smooth shape agree. One is topology, which counts holes. The other is algebra, which uses polynomial equations. The conjecture says they agree on a certain class of features called Hodge classes.

Checked by machine: Lean

def RiemannHypothesis : Prop :=
  ∀ (s : ℂ) (_ : riemannZeta s = 0) (_ : ¬∃ n : ℕ, s = -2 * (n + 1)) (_ : s ≠ 1), s.re = 1 / 2

Problem statements follow the Clay Mathematics Institute’s official descriptions. The Lean code is copied from Mathlib/NumberTheory/LSeries/RiemannZeta.lean.