Fields Medal

For bringing the ideas of Hodge theory to combinatorics, the proof of the Dowling–Wilson conjecture for geometric lattices, the proof of the Heron–Rota–Welsh conjecture for matroids, the development of the theory of Lorentzian polynomials, and the proof of the strong Mason conjecture.

For contributions to analytic number theory, which have led to major advances in the understanding of the structure of prime numbers and in Diophantine approximation.解析的整数論に貢献し，素数の構造理解とディオファントス近似の理解に大きな進歩をもたらした[22]。

For the proof that theE8{\displaystyle E_{8}}lattice provides the densest packing of identical spheres in 8 dimensions, and further contributions to related extremal problems and interpolation problems in Fourier analysis.球充填問題を8次元と24次元で解決したことや，フーリエ解析における極値および補間問題への更なる貢献が評価[22]。

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For contributions to the theory ofoptimal transportand its applications inpartial differential equations,metric geometryandprobability.

For transformingarithmetic algebraic geometryoverp-adic fieldsthrough his introduction ofperfectoid spaces, with application toGalois representations, and for the development of newcohomology theories.

For his synthesis ofanalytic number theory, homogeneous dynamics,topology, andrepresentation theory, which has resolved long-standing problems in areas such as the equidistribution of arithmetic objects.

for his profound contributions todynamical systemstheory have changed the face of the field, using the powerful idea of renormalization as a unifying principle.

for developing powerful new methods in the geometry of numbers, which he applied to count rings of small rank and to bound the average rank ofelliptic curves.

for his outstanding contributions to the theory of stochastic partial differential equations, and in particular for the creation of a theory of regularity structures for such equations.

For the proof of conformal invariance ofpercolationand the planarIsing modelinstatistical physics.

For his proof of the Fundamental Lemma in the theory ofautomorphic formsthrough the introduction of new algebro-geometric methods.

For his proofs of nonlinear Landau damping and convergence to equilibrium for theBoltzmann equation.

for his contributions to geometry and his revolutionary insights into the analytical and geometric structure of theRicci flow

for his contributions bridgingprobability,representation theoryandalgebraic geometry

for his contributions to the development of stochastic Loewner evolution, the geometry of two-dimensionalBrownian motion, andconformal field theory

he defined and developed motivic cohomology and the A1-homotopy theory ofalgebraic varieties; he proved theMilnor conjectureson theK-theoryof fields

William Timothy Gowers has provided important contributions tofunctional analysis, making extensive use of methods fromcombination theory. These two fields apparently have little to do with each other, and a significant achievement of Gowers has been to combine these fruitfully.

contributions to four problems of geometry

He has made important contributions to various branches of the theory ofdynamical systems, such as the algorithmic study of polynomial equations, the study of the distribution of the points of a lattice of aLie group,hyperbolic geometry,holomorphic dynamicsand the renormalization of maps of the interval.

... such nonlinearpartial differential equationsimply do not have smooth or even C1 solutions existing after short times. ... The only option is therefore to search for some kind of "weak" solution. This undertaking is in effect to figure out how to allow for certain kinds of "physically correct" singularities and how to forbid others. ... Lions and Crandall at last broke open the problem by focusing attention onviscosity solutions, which are defined in terms of certain inequalities holding wherever the graph of the solution is touched on one side or the other by a smoothtest function

proving stability properties - dynamic stability, such as that sought for the solar system, orstructural stability, meaning persistence under parameter changes of the global properties of the system.

For his solution to the restricted Burnside problem.

for his discovery of an unexpected link between the mathematical study ofknots– a field that dates back to the 19th century – andstatistical mechanics, a form of mathematics used to study complex systems with large numbers of components.

for the proof of Hartshorne’s conjecture and his work on the classification of three-dimensionalalgebraic varieties.

proof in 1981 of the positive energy theorem ingeneral relativity

Using methods ofarithmetic algebraic geometry, he received medal primarily for his proof of theMordell Conjecture.

Developed new methods for topological analysis of four-manifolds. One of his results is a proof of the four-dimensionalPoincaré Conjecture.

Revolutionized study oftopologyin 2 and 3 dimensions, showing interplay between analysis, topology, and geometry. Contributed idea that a very large class of closed 3-manifolds carry a hyperbolic structure.

Made contributions indifferential equations, also to theCalabi conjectureinalgebraic geometry, to the positive mass conjecture ofgeneral relativity theory, and to real and complex Monge-Ampère equations.

Contributed several innovations that revised the study of multidimensionalcomplex analysisby finding correct generalizations of classical (low-dimensional) results.

Provided innovative analysis of the structure ofLie groups. His work belongs tocombinatorics,differential geometry,ergodic theory,dynamical systems, and Lie groups.

The prime architect of the higheralgebraic K-theory, a new tool that successfully employed geometric and topological methods and ideas to formulate and solve major problems in algebra, particularlyring theoryandmodule theory.

Contributed to problems of the existence and structure of varieties of moduli, varieties whose points parametrize isomorphism classes of some type of geometric object. Also made several important contributions to the theory ofalgebraic surfaces.

Generalized work ofZariskiwho had proved for dimension ≤3 the theorem concerning the resolution of singularities on analgebraic variety. Hironaka proved the results in any dimension.

Made important advances in topology, the most well-known being his proof of the topological invariance of thePontrjagin classesof thedifferentiable manifold. His work included a study of the cohomology and homotopy of Thom spaces.

Proved jointly with W. Feit that all non-cyclic finitesimple groupshave even order. The extension of this work by Thompson determined the minimal simple finite groups, that is, the simple finite groups whose proper subgroups aresolvable.

Used technique called "forcing" to prove the independence inset theoryof theaxiom of choiceand of thegeneralized continuum hypothesis. The latter problem was the first ofHilbert's problemsof the 1900 Congress.

Built on work ofWeilandZariskiand effected fundamental advances inalgebraic geometry. He introduced the idea ofK-theory(theGrothendieck groupsand rings). Revolutionizedhomological algebrain his celebrated "Tohokupaper"

Worked indifferential topologywhere he proved the generalizedPoincaré conjecturein dimension n≥5: Every closed, n-dimensional manifoldhomotopy-equivalentto then-dimensional sphereishomeomorphicto it. Introduced the method ofhandle-bodiesto solve this and related problems.

Proved that a 7-dimensionalspherecan have several differential structures; this led to the creation of the field ofdifferential topology.

In 1954 invented and developed the theory of cobordism inalgebraic topology. This classification of manifolds usedhomotopy theoryin a fundamental way and became a prime example of a general cohomology theory.

Achieved major results on thehomotopy groupsof spheres, especially in his use of the method of spectral sequences. Reformulated and extended some of the main results of complex variable theory in terms ofsheaves.

Developed generalizations of thesieve methodsofViggo Brun; achieved major results on zeros of theRiemann zeta function; gave an elementary proof of theprime number theorem(withP. Erdős), with a generalization to prime numbers in an arbitraryarithmetic progression.

Did important work of the Plateau problem which is concerned with finding minimal surfaces connecting and determined by some fixed boundary.