Search Results

For solving longstanding problems in the probabilistic theory of phase transitions in statistical physics, especially in dimensions three and four.

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]。

14

For the proof of the boundedness ofFano varietiesand for contributions to theminimal model program.

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 her outstanding contributions to the dynamics and geometry ofRiemann surfacesand theirmoduli spaces.

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 his results on measure rigidity inergodic theory, and their applications tonumber theory.

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 topartial differential equations,combinatorics,harmonic analysisand additive number theory

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

Laurent Lafforgue has been awarded the Fields Medal for his proof of theLanglands correspondencefor thefull linear groupsGLr(r≥1) overfunction fields.

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

for his work on the introduction of vertex algebras, the proof of the Moonshine conjecture and for his discovery of a new class of automorphic infinite products

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.

Bourgain's work touches on several central topics of mathematical analysis: the geometry ofBanach spaces, convexity in high dimensions,harmonic analysis,ergodic theory, and finally, nonlinearpartial differential equationsfrommathematical physics.

... 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.