In 2000 the Clay Mathematics Institute named seven problems and attached $1,000,000 to each. A quarter-century later only one is solved. Each tab below is an honest visual teaser — not a fake resolution — that computes the real object live and then tells you plainly why the obvious picture isn't a proof. Toggle a subset to check it in an instant while brute force drowns; watch a fluid steepen toward a shock; ride the chord-and-tangent law across an elliptic curve; and see Ricci flow's baby cousin round a blob into a circle. These problems are famous precisely because seeing is not the same as proving.
P is the class of decision problems solvable by an algorithm whose running time is bounded by a polynomial in the input size — "efficient" in the theoretician's sense. NP is the class whose yes-answers come with a certificate that can be checked in polynomial time. Subset-sum is in NP: hand me the winning stones and I verify with one addition.
Clearly P ⊆ NP (if you can solve it fast, you can check it fast). The million-dollar question is whether the inclusion is strict.
Subset-sum is NP-complete: every problem in NP can be re-encoded as a subset-sum instance in polynomial time. So a fast algorithm for this puzzle would instantly give fast algorithms for all of NP — routing, scheduling, protein folding, breaking cryptography.
That universality is why the stakes are enormous and why most researchers bet P ≠ NP: if the two were equal, creativity itself would be automatable — finding a proof would be no harder than checking one.
The Riemann zeta function starts life as ζ(s)=Σ 1/nˢ for Re(s)>1 and is then analytically continued to the whole complex plane. It has "trivial" zeros at the negative even integers; everything interesting lives in the critical strip 0<Re(s)<1.
Riemann's explicit formula writes the count of primes up to x as a smooth term minus a sum over the zeros. Each zero contributes an oscillation; where the zero sits controls how big that wobble can grow.
If every non-trivial zero has real part exactly ½, those oscillations are as small as possible and the primes hug their expected count as tightly as arithmetic allows — the error in the Prime Number Theorem is squeezed to O(√x·log x).
The plotted Z(t) makes the zeros visible without complex numbers: multiply ζ(½+it) by a phase e^{iθ(t)} chosen to cancel the imaginary part. Real crossings of Z are exactly the zeros — which is how they were first hunted by hand.
The Navier–Stokes equations express Newton's second law for a fluid: the acceleration of each parcel equals pressure and viscous forces. The troublesome piece is the advection term (u·∇)u — the fluid transporting its own momentum — which is nonlinear and can sharpen features without limit.
Our 1-D model, uₜ + u uₓ = ν uₓₓ (Burgers), keeps exactly this tension: self-advection steepens, the viscous term ν uₓₓ diffuses. In 1-D the diffusion always wins and solutions stay smooth.
The Clay problem asks for a proof (or a counterexample) that the 3-D incompressible equations, starting from smooth finite-energy data, have solutions that remain smooth and finite for all time — global regularity.
Leray showed weak solutions always exist; whether they are unique and smooth is unknown. A finite-time singularity would mean some quantity — vorticity, velocity gradient — escapes to infinity, and the equations lose meaning exactly where fluids are most violent.
Yang–Mills theory is the mathematical framework behind the strong and weak nuclear forces: a "gauge field" whose values live in a non-abelian symmetry group like SU(3). Unlike electromagnetism, the field interacts with itself, which makes the theory ferociously nonlinear.
A mass gap means the lightest particle the theory can make — a "glueball" — has energy bounded away from zero. Equivalently, correlations between distant field measurements decay exponentially, with rate equal to that mass.
Our toy plots a Gaussian field with a tunable correlation length ξ=1/m. Real Yang–Mills is far richer, but the diagnostic is the same: fit C(r)∼e^{−m r} and check m>0. Every lattice-QCD experiment finds a robust gap — it's why the strong force doesn't reach across the room.
The Clay problem demands a rigorous construction of the quantum theory on ℝ⁴ and a proof that this gap is strictly positive — turning a mountain of numerical and experimental evidence into a theorem.
On an elliptic curve the rational points form an abelian group: to add P and Q, draw the line through them, take the third intersection, and reflect over the x-axis. Mordell proved this group is finitely generated — it looks like a finite part plus r copies of the integers. That r is the rank: the number of independent infinite-order generators.
Rank is subtle: some curves have rank 0 (finitely many rationals), others rank 1, 2, 3… and finding it is genuinely hard.
Separately, count the curve's solutions modulo each prime p and package the counts into an L-function L(E,s). This analytic object knows nothing, on its face, about rational points.
BSD conjectures the miracle: the order of vanishing of L(E,s) at s=1 equals the rank, and the leading coefficient encodes the finer arithmetic (regulator, Tate–Shafarevich group, torsion). Counting mod p, summed cleverly, predicts the infinitude of rational points.
On a smooth projective complex variety, the cohomology H*(X) measures its holes and higher-dimensional features. Hodge theory refines this: each cohomology group splits into pieces labelled by a type (p,q), recording how a class interacts with the complex structure.
An algebraic cycle is a formal combination of sub-varieties — sets carved out by polynomial equations. Each casts a cohomology class, and those classes always land on the diagonal, of type (p,p).
A Hodge class is a rational cohomology class that is purely of type (p,p). Every algebraic cycle gives one; the Hodge conjecture asserts the converse — every Hodge class is a rational combination of algebraic cycle classes.
For (1,1) this is the classical Lefschetz theorem. Beyond that it is wide open, with no general method to manufacture the required sub-varieties. It is a cornerstone of how algebraic geometry and topology are believed to fit together.
A space is simply connected if every loop can be continuously shrunk to a point — a sphere is, a doughnut is not (a loop around the hole is stuck). Poincaré asked: is the 3-sphere the only closed, simply-connected 3-manifold, up to deformation? For decades this resisted every direct attack.
The higher-dimensional versions (n≥5, then n=4) were settled first by Smale and Freedman; the original 3-D case was the last and hardest.
Ricci flow, introduced by Hamilton, evolves a manifold's metric so that regions of high curvature smooth out — geometry's answer to the heat equation. Left alone in 3-D it can form singularities; Perelman (2002–03) developed the estimates and surgery to push through them.
The 2-D curve-shortening flow on this stage is the honest low-dimensional shadow: same "smooth by curvature" idea, but simple enough that it always ends in a round point — letting you see the mechanism that, suitably tamed, cracked the conjecture.