# The Breakthrough Prize in Mathematics

The Breakthrough Prize was launched in 2012 to honour important, primarily recent, achievements in Fundamental Physics (first awards 2012), Life Sciences (first awards 2013) and Mathematics (first awards 2015):

*"All is number," taught Pythagoras. Though modern mathematics encompasses far more than numbers alone, the principle remains true. Mathematics is the universal language of nature. Mathematics is also fundamental to the growth of knowledge, as it is the scaffolding that supports all the sciences. Its relationship to physics is particularly intimate. From imaginary numbers to Hilbert spaces, what once seemed pure abstractions have turned out to underlie real physical processes. In addition, all fields in the life sciences today utilise the power of statistical and computational approaches to research. The mathematics prizes reward significant discoveries across the many branches of the subject. They were founded by Yuri Milner and are funded by grants from the foundations established by Yuri and Julia Milner.*

**2015**Richard Taylor

... for numerous breakthrough results in the theory of automorphic forms, including the Taniyama-Weil conjecture, the local Langlands conjecture for general linear groups, and the Sato-Tate conjecture.

**2015**Maxim Kontsevich

... for work making a deep impact in a vast variety of mathematical disciplines, including algebraic geometry, deformation theory, symplectic topology, homological algebra, and dynamical systems.

**2015**Terence Tao

... for numerous breakthrough contributions to harmonic analysis, combinatorics, partial differential equations, and analytic number theory.

**2015**Simon Donaldson

... for the new revolutionary invariants of four-dimensional manifolds and for the study of the relation between stability in algebraic geometry and in global differential geometry, both for bundles and for Fano varieties.

**2015**Jacob Lurie

... for his work on the foundations of higher category theory and derived algebraic geometry; for the classification of fully extended topological quantum field theories; and for providing a moduli-theoretic interpretation of elliptic cohomology.

**2016**Ian Agol

... for spectacular contributions to low dimensional topology and geometric group theory, including work on the solutions of the tameness, virtual Haken, and virtual fibering conjectures.

**2017**Jean Bourgain

... for multiple transformative contributions to analysis, combinatorics, partial differential equations, high-dimensional geometry and number theory.

**2018**James McKernan

... for transformational contributions to birational algebraic geometry, especially to the minimal model program in all dimensions.

**2018**Christopher Hacon

... for transformational contributions to birational algebraic geometry, especially to the minimal model program in all dimensions.

**2019**Vincent Lafforgue

... for ground breaking contributions to several areas of mathematics, in particular to the Langlands program in the function field case.

**2020**Alex Eskin

... for revolutionary discoveries in the dynamics and geometry of moduli spaces of Abelian differentials, including the proof of the "magic wand theorem" with Maryam Mirzakhani.

**2021**Martin Hairer

... for transformative contributions to the theory of stochastic analysis, particularly the theory of regularity structures in stochastic partial differential equations.

**2022**Takuro Mochizuki

...for monumental work leading to a breakthrough in our understanding of the theory of bundles with flat connections over algebraic varieties, including the case of irregular singularities.

**2023**Daniel A Spielman

... for breakthrough contributions to theoretical computer science and mathematics, including to spectral graph theory, the Kadison-Singer problem, numerical linear algebra, optimization, and coding theory.

**2024**Simon Brendle

... for transformative contributions to differential geometry, including sharp geometric inequalities, many results on Ricci flow and mean curvature flow and the Lawson conjecture on minimal tori in the 3-sphere.