Analysis & PDEs2026
From colliding particles to a gas equation
How can reversible collisions of countless tiny particles produce the statistical equation used to describe the evolution of a gas?
The story1936–2026 · Mathematics, illustrated
New ways
to see the
invisible.
Meet 68 mathematicians who changed how we understand shape, number and possibility. Explore their ideas through clear explanations and original illustrated guides.
Explore the awards ↓21Award editions
68Remarkable people
1936–2026Ninety years of discovery
From the first medals in 1936 to the 2026 edition. Each medalist has an individual story; subject groups are our editorial guide.
An archive of lasting ideas
The awards
Showing 12 of 68
Newest first · scroll to exploreFields Medal
2026
4 medalists
Explore this edition ↗Geometry & topology2026
Counting curves when the counting space is singular
Symplectic geometry extracts information from families of curves, but those families can be too singular for ordinary counting to work.
The storyNumber theory2026
Tame geometry reveals hidden arithmetic structure
An idea from mathematical logic helps distinguish exceptional arithmetic patterns from the ordinary points of complicated geometric parameter spaces.
The storyAnalysis & PDEs2026
How much space do all directions require?
A collection containing a unit line segment in every direction can have zero volume, yet its fractal dimension remains constrained.
The storyFields Medal
2022
4 medalists
Explore this edition ↗Probability & dynamics2022
When local randomness becomes collective order
Random connections or microscopic spins can suddenly organize at a critical threshold, creating large-scale behavior invisible in any single component.
The storyCombinatorics2022
Geometry hidden inside counting sequences
Numbers obtained by counting finite configurations often rise and fall regularly; geometric ideas explain why certain patterns cannot fluctuate arbitrarily.
The storyNumber theory2022
Finding clusters in the sparse prime numbers
Primes become rarer as numbers grow, yet carefully designed sieve weights reveal infinitely many intervals containing several of them together.
The storyGeometry & topology2022
A perfect certificate for eight-dimensional sphere packing
The E8 lattice looks extraordinarily efficient, but proving optimality requires ruling out every competing arrangement, including irregular and nonperiodic packings.
The storyFields Medal
2018
4 medalists
Explore this edition ↗Algebraic geometry2018
Putting limits on families of algebraic shapes
Polynomial equations define endlessly many spaces; birational geometry searches for building blocks and asks how complicated those blocks can become.
The storyAnalysis & PDEs2018
The geometry of moving mass at minimum cost
Optimal transport compares distributions by moving mass between them, turning a question about cheapest routes into deep information about geometric regularity.
The storyAlgebraic geometry2018
A bridge between two kinds of arithmetic geometry
Perfectoid spaces let mathematicians transfer difficult geometric questions between characteristic zero and characteristic p, where different tools become available.
The storyNumber theory2018
Watching arithmetic objects spread through geometric spaces
Number-theoretic objects can become points or trajectories in a geometric space, allowing dynamics to reveal patterns that direct calculation hides.
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