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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]。

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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.