CMI 2000, The Millennium Prize Problems

Seven problems designated by the Clay Mathematics Institute on 24 May 2000, each carrying a prize of one million US dollars.

One has been resolved: the Poincaré conjecture, proved by Grigori Perelman, who declined the prize.

Listed here in the Institute's own order, which is alphabetical. The problems carry no official numbering.

Conjecture (Birch and Swinnerton-Dyer conjecture). Let $E$ be an elliptic curve defined over $Q$ . By the Mordell-Weil theorem the group $E (Q)$ of rational points is finitely generated, so

$E (Q) ≅ Z^{r} \oplus T,$

where $T$ is finite and $r$ is the rank of the curve.

Let $L (E, s)$ be the Hasse-Weil $L$ -function of $E$ .

Prove that the order of vanishing of $L (E, s)$ at $s = 1$ equals the rank $r$ .

Original: B. Birch and P. Swinnerton-Dyer, Notes on elliptic curves. II, J. Reine Angew. Math. 218 (1965), 79-108.

Prize: Clay Mathematics Institute, Birch and Swinnerton-Dyer Conjecture (Millennium Prize Problems). —

Conjecture (Hodge conjecture). Let $X$ be a non-singular complex projective variety. Its rational cohomology decomposes into Hodge components,

$H^{2 p} (X, Q) \otimes C = ⨁_{i + j = 2 p} H^{i, j} (X),$

and a Hodge class of degree $2 p$ is a class in $H^{2 p} (X, Q)$ whose image lies in $H^{p, p} (X)$ .

Prove that every Hodge class on $X$ is a rational linear combination of the cohomology classes of algebraic subvarieties of $X$ .

Original: W. V. D. Hodge, The topological invariants of algebraic varieties, Proc. Internat. Congress Math. (Cambridge, MA, 1950), vol. 1, 182-192.

Prize: Clay Mathematics Institute, Hodge Conjecture (Millennium Prize Problems). —

Problem (Navier-Stokes existence and smoothness). The motion of an incompressible fluid in $R^{3}$ is governed by the Navier-Stokes equations: for the velocity field $u (x, t) = (u_{1}, u_{2}, u_{3})$ and the pressure $p (x, t)$ ,

$\frac{\partial u _{i}}{\partial t} + \sum_{j = 1}^{3} u_{j} \frac{\partial u _{i}}{\partial x _{j}} = ν Δ u_{i} - \frac{\partial p}{\partial x _{i}} + f_{i} (x, t), i = 1, 2, 3,$

subject to incompressibility

$\nabla \cdot u = \sum_{i = 1}^{3} \frac{\partial u _{i}}{\partial x _{i}} = 0,$

where $ν > 0$ is the viscosity and $f$ an applied external force.

Given a smooth, divergence-free initial velocity field $u^{0} (x)$ on $R^{3}$ with suitable decay at infinity, and taking $f \equiv 0$ , prove that there exist smooth functions $u$ and $p$ on $R^{3} \times [0, \infty)$ with bounded energy that solve the system above – or give a counterexample.

Prize: Clay Mathematics Institute, Navier-Stokes Equation (Millennium Prize Problems). —

Survey: C. L. Fefferman, Existence and smoothness of the Navier-Stokes equation, official problem description, Clay Mathematics Institute.

Problem (P versus NP). $P$ is the class of decision problems solvable by a deterministic Turing machine in time polynomial in the length of the input. $NP$ is the class of decision problems whose yes instances admit a certificate verifiable by a deterministic Turing machine in polynomial time.

Every problem in $P$ is in $NP$ : a machine that solves a problem outright can ignore the certificate.

Determine whether the containment is strict. That is, decide

$P = ? NP .$

Original: S. A. Cook, The complexity of theorem-proving procedures, Proc. 3rd Annual ACM Symposium on Theory of Computing (1971), 151-158.

Original: L. A. Levin, Universal sequential search problems, Problemy Peredachi Informatsii 9 (1973), no. 3, 115-116.

Prize: Clay Mathematics Institute, P vs NP Problem (Millennium Prize Problems). —

Theorem (Poincaré conjecture). Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere $S^{3}$ .

Equivalently: a compact 3-dimensional manifold without boundary in which every loop can be contracted continuously to a point is topologically a 3-sphere.

Original: H. Poincaré, Cinquième complément à l'analysis situs, Rend. Circ. Mat. Palermo 18 (1904), 45-110.

Partial results: R. S. Hamilton, Three-manifolds with positive Ricci curvature, J. Differential Geom. 17 (1982), 255-306. Introduces the Ricci flow the proof is built on.

Proof: G. Perelman, The entropy formula for the Ricci flow and its geometric applications (2002). —

Proof: G. Perelman, Ricci flow with surgery on three-manifolds (2003). —

Proof: G. Perelman, Finite extinction time for the solutions to the Ricci flow on certain three-manifolds (2003). —

Prize: Clay Mathematics Institute, Poincaré Conjecture (Millennium Prize Problems). The prize was awarded in 2010 and declined. —

Conjecture (Riemann hypothesis). The Riemann zeta function is defined for $ℜ (s) > 1$ by

$ζ (s) = \sum_{n = 1}^{\infty} \frac{1}{n ^{s}}$

and extended to the whole complex plane by analytic continuation, with a single pole at $s = 1$ . It vanishes at the negative even integers $s = - 2, - 4, - 6, \dots$ ; these are the trivial zeros.

Prove that every non-trivial zero of $ζ (s)$ satisfies

$ℜ (s) = \frac{1}{2} .$

Original: B. Riemann, Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse, Monatsberichte der Berliner Akademie (1859).

Prize: Clay Mathematics Institute, Riemann Hypothesis (Millennium Prize Problems). —

Problem (Yang-Mills existence and mass gap). Prove that for any compact simple gauge group $G$ a non-trivial quantum Yang-Mills theory exists on $R^{4}$ and satisfies the Wightman axioms, and that it has a mass gap: a constant $Δ > 0$ such that every state other than the vacuum has energy at least $Δ$ .

Existence includes establishing axiomatic properties at least as strong as those cited in the standard references on constructive quantum field theory.

Original: C. N. Yang and R. L. Mills, Conservation of isotopic spin and isotopic gauge invariance, Phys. Rev. 96 (1954), 191-195.

Prize: Clay Mathematics Institute, Yang-Mills and Mass Gap (Millennium Prize Problems). —

Survey: A. Jaffe and E. Witten, Quantum Yang-Mills theory, official problem description, Clay Mathematics Institute.

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