The Fields Medal

1. The Pre-War Era: 1936–1950

• 1936 (Oslo): Lars Ahlfors (complex analysis), Jesse Douglas (Plateau problem).
• 1950 (Cambridge MA): Laurent Schwartz (theory of distributions), Atle Selberg (elementary proof of the prime number theorem).

The Second World War prevented an ICM between 1936 and 1950. Schwartz’s distributions and Selberg’s sieve methods were defining mid-century achievements.

2. The Age of Grothendieck: 1950s–70s

• 1954 (Amsterdam): Kunihiko Kodaira (complex manifolds), Jean-Pierre Serre (homotopy of spheres, sheaf cohomology). Serre, at 27, remains the youngest medallist.
• 1958 (Edinburgh): Klaus Roth (rational approximation), René Thom (cobordism theory).
• 1962 (Stockholm): Lars Hörmander (PDEs), John Milnor (exotic 7-spheres — smooth structures on S⁷).
• 1966 (Moscow): Michael Atiyah (K-theory, index theorem), Paul Cohen (independence of CH), Alexander Grothendieck (algebraic geometry, schemes), Stephen Smale (Poincaré in dim ≥ 5).
• 1970 (Nice): Alan Baker (transcendence), Heisuke Hironaka (resolution of singularities in characteristic 0), Sergei Novikov (cobordism, foliations), John Thompson (finite simple groups).
• 1974 (Vancouver): Enrico Bombieri (analytic NT), David Mumford (algebraic surfaces).
• 1978 (Helsinki): Pierre Deligne (Weil conjectures), Charles Fefferman (Fourier analysis), Grigori Margulis (Lie groups), Daniel Quillen (algebraic K-theory).

3. The Witten Era and the Geometric Revolution: 1980s–90s

• 1982 (Warsaw): Alain Connes (von Neumann algebras), William Thurston (geometrisation of 3-manifolds), Shing-Tung Yau (Calabi conjecture).
• 1986 (Berkeley): Simon Donaldson (4-manifolds via gauge theory), Gerd Faltings (Mordell conjecture), Michael Freedman (topological 4-manifolds).
• 1990 (Kyoto): Vladimir Drinfeld (quantum groups), Vaughan Jones (knot polynomials), Shigefumi Mori (3-fold birational geometry), Edward Witten (the only physicist to receive a Fields Medal — for topological field theory).
• 1994 (Zürich): Jean Bourgain, Pierre-Louis Lions, Jean-Christophe Yoccoz, Efim Zelmanov.
• 1998 (Berlin): Richard Borcherds (Monstrous Moonshine), Tim Gowers (Banach spaces), Maxim Kontsevich (knot invariants & deformation quantisation), Curtis McMullen (renormalisation).

4. The Modern Era: 2000–present

• 2002 (Beijing): Laurent Lafforgue (Langlands for function fields), Vladimir Voevodsky (motivic cohomology).
• 2006 (Madrid): Andrei Okounkov, Grigori Perelman (Poincaré conjecture — declined the medal), Terence Tao, Wendelin Werner.
• 2010 (Hyderabad): Elon Lindenstrauss (ergodic theory), Ngô Báo Châu (fundamental lemma), Stanislav Smirnov (percolation, conformal invariance), Cédric Villani (transport theory).
• 2014 (Seoul): Artur Avila (dynamical systems), Manjul Bhargava (NT, ranks of curves), Martin Hairer (regularity structures for SPDEs), Maryam Mirzakhani (moduli of Riemann surfaces — the first woman, the first Iranian).
• 2018 (Rio de Janeiro): Caucher Birkar (birational geometry), Alessio Figalli (optimal transport), Peter Scholze (perfectoid spaces), Akshay Venkatesh (analytic NT).
• 2022 (St. Petersburg / virtual): Hugo Duminil-Copin (statistical physics), June Huh (combinatorics & algebraic geometry), James Maynard (analytic NT), Maryna Viazovska (sphere packing in dimensions 8 and 24 — the second woman).

5. Patterns & Notes

• Two women have received the Fields Medal: Mirzakhani (2014) and Viazovska (2022).
• One declined: Perelman (2006), citing dissatisfaction with the mathematical community’s judgement of work and his own ethical objections.
• Algebraic geometry dominates: Grothendieck’s revolution and its continuation through Deligne, Faltings, Lafforgue, Voevodsky, Birkar, Scholze.
• Geometry & topology: Milnor, Thurston, Donaldson, Freedman, Perelman, Mirzakhani.
• Combinatorics has emerged: Tao, Gowers, Bhargava, Maynard, Huh.