Worked solution: De Branges's proof of the Bieberbach conjecture via Milin's conjecture (1984)
Because Louis de Branges had previously announced other results that turned out to be flawed, his 1984 claim to have solved a 68-year-old problem was initially met with real skepticism from the mathematical community, rather than immediate acceptance.
A team at the Steklov Institute in Leningrad, where de Branges was visiting, worked through his long manuscript line by line, confirmed the argument was correct, and helped bring the proof to publication — after which mathematicians such as Carl FitzGerald and Christian Pommerenke found a shorter, cleaner path through the same ideas.
De Branges circulated his manuscript in 1984; because he had previously announced other mathematical advances that turned out to be invalid, the mathematical community did not immediately accept the claim. A team of mathematicians at the Steklov Mathematics Institute in St. Petersburg (then Leningrad), where de Branges was on sabbatical, forensically checked the proof, filled in a few gaps, and confirmed its correctness; the result was published as a 16-page paper in Acta Mathematica in 1985.
Shortly after, Carl FitzGerald and Christian Pommerenke independently found significant simplifications, publishing 'The de Branges theorem on univalent functions' (Trans. Amer. Math. Soc., 1985), which strips away extraneous functional-analytic machinery from de Branges's original argument and presents the Loewner-chain-to-Askey–Gasper route in a more streamlined, purely function-theoretic form — essentially the version of the proof outlined in Steps 3–7.
The theorem's afterlife includes further striking simplifications: Lenard Weinstein gave a four-page elementary calculus proof around 1990, and Doron Zeilberger and Shalosh B. Ekhad (Zeilberger's computer) produced a short computer-assisted verification in 1993, underscoring how completely de Branges's identification of the right monotone quantity had resolved the problem.