- What did Kunihiko Kodaira's work on harmonic integrals connect differential geometry to?
- Why was Serre's Fields Medal at age twenty-seven considered a record, and how long did that record stand?
- Why did the 1954 medals go to mathematicians who applied algebraic tools to problems that had resisted purely geometric methods?
Algebraic Geometry
The geometry of solutions to polynomial equations.
- How did Alexander Grothendieck's use of schemes and sheaves change the foundations of algebraic geometry?
- What does it mean to 'resolve singularities' in algebraic geometry, and why was Heisuke Hironaka's proof so significant?
- What are moduli spaces, and how did David Mumford's work on them organize algebraic geometry?
- What field did David Mumford move into after leaving pure mathematics, and how did it connect to his earlier work?
- What are the Weil conjectures, and why did completing their proof require an entirely new mathematical language?
- What is noncommutative geometry, and how did Alain Connes extend geometric thinking to settings where coordinates do not commute?
- What are Calabi-Yau manifolds, and why did they turn out to matter for string theory?
- What are quantum groups, and how did Vladimir Drinfeld's construction connect representation theory, number theory, and quantum mechanics?
- What did Shigefumi Mori's minimal model program achieve for classifying three-dimensional algebraic varieties?
- What is the Langlands correspondence, and what did Laurent Lafforgue prove about it for function fields?
- What connections did Andrei Okounkov find between probability theory and algebraic geometry?
- What did Maryam Mirzakhani's work on Riemann surfaces and moduli spaces actually establish?
- What did Caucher Birkar contribute to the classification of higher-dimensional algebraic varieties?
- What are perfectoid spaces, and how did Peter Scholze's framework connect number theory and algebraic geometry?
- What happened to Caucher Birkar's medal right after the award ceremony?
- What are D-modules, and how do they let differential equations be studied using algebraic geometry?
- What are crystal bases, and how do they make calculations in representation theory of quantum groups more tractable?
- Why was Kashiwara's 2025 Abel Prize notable as a first for Japanese mathematics?
- What is the Langlands program, and what fields of mathematics does it try to unify?
- How does the geometric Langlands conjecture differ from the classical Langlands program in number fields?
- What did Gaitsgory's proof establish about the correspondence between geometric objects and representations?
- What is a singularity on an algebraic variety, and why does it break the usual tools of analysis?
- What did Hironaka prove in 1964 about resolving singularities over fields of characteristic zero?
- Why was Hironaka's proof considered so technically difficult that mathematicians needed years to verify it?
- How did the resolution of singularities problem become harder in positive characteristic, and what did Hironaka contribute there?