Lemma Set

A field guide · September 2026

Where certainty
ends.

Six questions stand at the edge of established mathematics. Each is precise enough to fit on a page and deep enough to have resisted generations of work.

In 2000, the Clay Mathematics Institute named seven Millennium Prize Problems and assigned one million dollars to each. Poincaré is resolved. Five remain unsolved; an announced Navier–Stokes solution is now moving through the Institute’s deliberately unhurried evaluation process.

01 Number theory

The Riemann
Hypothesis

ζ(s) = 0, 0 < Re(s) < 1  ⇒?  Re(s) = ½

Prime numbers arrive irregularly, yet their large-scale frequency follows a remarkably stable law. The bridge between those two facts is the Riemann zeta function. For complex numbers with real part greater than one it begins as the convergent series ζ(s) = 1 + 2−s + 3−s + ···, and analytic continuation extends it to nearly the whole complex plane. Its zeros encode fine corrections to the average distribution of primes.

The hypothesis says that every nontrivial zero has real part exactly one half. “Nontrivial” excludes the known zeros at negative even integers. The claim does not predict the next prime; it constrains how far the prime-counting function can deviate from its main trend. Thousands of results are known conditionally on this single vertical alignment, sharpening prime-counting estimates and illuminating multiplicative functions, spectral analogies, and number-theoretic error terms.

Extensive computation has found zeros on the critical line, but checking any finite collection cannot settle a statement about infinitely many zeros. A proof must explain the rigidity; a disproof needs only one zero off the line. The Clay Mathematics Institute continues to list the hypothesis as unsolved and links the official formulation by Enrico Bombieri.

Clay Mathematics Institute · official problem page ↗
A spectral order
behind the primes?
02 Theoretical computation

P versus NP

P = NP  ?

A decision problem belongs to P when a deterministic algorithm can solve every instance in time bounded by a polynomial in the input length. It belongs to NP when a proposed “yes” answer comes with a certificate that can be checked in polynomial time. Because solving a question also gives a way to verify its answer, P is contained in NP. Whether the two classes are equal is unknown.

The issue becomes sharp through NP-completeness. Problems such as Boolean satisfiability can represent every problem in NP through efficient reductions. A polynomial-time method for one NP-complete problem would therefore give polynomial-time methods for all of them. Conversely, proving that no such method exists would separate P from NP. The distinction concerns worst-case growth, not whether a particular instance is manageable or whether heuristics work well in practice.

Either outcome would reshape computation. Equality could make many search, scheduling, design, and proof-discovery tasks tractable in principle, while also undermining widely used cryptographic assumptions—though an impractically large polynomial could limit immediate effects. Inequality would formally establish that efficient verification can exceed efficient discovery. The Clay Mathematics Institute attributes the independent 1971 formulations to Stephen Cook and Leonid Levin and provides Cook’s official description.

Clay Mathematics Institute · official problem page ↗
Is finding
as easy as checking?
04 Algebraic geometry

The Hodge
Conjecture

H2p(X, ℚ) ∩ Hp,p(X)  =?  algebraic cycles

A smooth projective algebraic variety can be studied in two languages. Algebraic geometry describes subsets cut out by polynomial equations; topology records global features such as holes through cohomology classes. Hodge theory further decomposes complex cohomology into types Hp,q. Classes of algebraic subvarieties land in the balanced part Hp,p, after taking their fundamental classes and rational linear combinations.

The conjecture asks for the converse: on a smooth projective complex variety, is every rational cohomology class of type (p,p) a rational combination of classes of algebraic cycles of codimension p? The rational qualifier matters. The corresponding statement with integral coefficients is false, and removing projectivity changes the terrain. In low codimension and several special families the answer is known, but the general problem remains beyond current methods.

A positive answer would say that certain topological features detected analytically are always built from genuinely algebraic pieces. This would tighten the relation between equations and shape and clarify the structure of motives, periods, and algebraic cycles. The obstacle is that Hodge theory produces classes through analysis, while algebraic cycles are rigid geometric objects; there is no general procedure for turning one into the other. The Institute’s official description is by Pierre Deligne.

Clay Mathematics Institute · official problem page ↗
When does topology
come from equations?
05 Arithmetic geometry

Birch &
Swinnerton–Dyer

ords=1L(E,s) = rank E(ℚ)

An elliptic curve over the rationals is a smooth cubic curve with a chosen rational point, often written y² = x³ + ax + b with nonzero discriminant. Its rational points form a finitely generated abelian group. The number of independent infinite-order generators is the rank, a basic arithmetic quantity that is surprisingly difficult to determine from the equation.

The curve also has an L-function assembled from counts of its reductions modulo primes. The Birch and Swinnerton-Dyer conjecture says that the order to which L(E,s) vanishes at s = 1 equals the rank of E(ℚ). Its refined form predicts the leading Taylor coefficient using the regulator, real period, local Tamagawa factors, torsion subgroup, and the Tate–Shafarevich group. Thus information gathered prime by prime would determine the global supply of rational points.

The conjecture grew from computer experiments by Bryan Birch and Peter Swinnerton-Dyer in the 1960s. Major cases are known: results connecting modular forms, Heegner points, and Euler systems establish important rank-zero and rank-one situations under hypotheses. Yet arbitrary rank and the full leading-coefficient formula remain open. A proof would organize central questions about Diophantine equations; a counterexample would expose a break in one of arithmetic geometry’s most productive correspondences. The Institute provides Andrew Wiles’s official account.

Clay Mathematics Institute · official problem page ↗
Local counts,
global rational points.
06 Mathematical physics

Yang–Mills
& the Mass Gap

Δ = inf(spec H \ {0}) > 0

Yang–Mills theory generalizes electromagnetism using a noncommutative symmetry group. Classically, a connection on a principal bundle defines a curvature field, and the Yang–Mills equations select critical points of an energy built from that curvature. After quantization, the framework underlies the Standard Model’s description of the strong and electroweak interactions. Its physical success is extraordinary; its four-dimensional mathematical construction is incomplete.

The Millennium problem asks for a nontrivial quantum Yang–Mills theory on ℝ⁴ for every compact simple gauge group, satisfying axioms at least as strong as those specified in the official statement, and for proof of a positive mass gap Δ. A mass gap means that the vacuum has zero energy while every excitation has energy bounded below by a fixed positive amount. It helps explain why correlations decay rapidly and why the strong interaction does not produce observable massless carriers.

Perturbative calculations describe high-energy behavior, and lattice computations give compelling numerical evidence, but neither supplies the required continuum construction and proof. The central difficulty is controlling a nonlinear quantum field across all length scales while preserving locality, symmetry, and positivity. A solution would place a foundational physical theory on rigorous ground and likely create new mathematics linking probability, analysis, geometry, and representation theory. The Institute’s official formulation is by Arthur Jaffe and Edward Witten.

Clay Mathematics Institute · official problem page ↗
Why does empty space
have an energy threshold?