01 · The idea
In everyday language
Instead of studying one curve, a moduli space organises many curves into a single geometric object. Certain intersection numbers on the moduli spaces of stable marked curves encode how natural geometric conditions meet. Witten predicted that a generating function collecting these numbers satisfies equations from the KdV hierarchy. Kontsevich proved the prediction using a matrix integral and combinatorial descriptions by ribbon graphs. The proof translates geometric integrals into a form where the hidden recurrence becomes visible. Its subject is a precisely defined family of intersection numbers, not a claim that every possible curve-counting problem has been solved.
Why it matters
The theorem helped establish a powerful exchange between moduli-space geometry, integrable systems and mathematical physics, showing how combinatorial models can make difficult geometric invariants accessible.
