Rationale  Name 
For playing a key role in shaping the modern form of many parts of mathematics, including topology, algebraic geometry and number theory.  
For their discovery and proof of the index theorem, bringing together topology, geometry and analysis, and their outstanding role in building new bridges between mathematics and theoretical physics.  
For their discovery and proof of the index theorem, bringing together topology, geometry and analysis, and their outstanding role in building new bridges between mathematics and theoretical physics.  
For his groundbreaking contributions to the theory and application of partial differential equations and to the computation of their solutions.  
For his profound and seminal contributions to harmonic analysis and the theory of smooth dynamical systems.  
For his fundamental contributions to probability theory and in particular for creating a unified theory of large deviation.  
For their profound achievements in algebra and in particular for shaping modern group theory.  
For their profound achievements in algebra and in particular for shaping modern group theory.  
For his revolutionary contributions to geometry.  
For his vast and lasting impact on the theory of numbers.  
For pioneering discoveries in topology, geometry, and algebra.  
For his fundamental contributions to discrete mathematics and theoretical computer science, and in recognition of the profound and lasting impact of these contributions on additive number theory and ergodic theory.  
For seminal contributions to algebraic geometry and for their transformative impact on number theory, representation theory, and related fields.  
 Rationale  Name 
For his fundamental contributions to dynamical systems, ergodic theory, and mathematical physics.  
For striking and seminal contributions to the theory of nonlinear partial differential equations and its applications to geometric analysis.  
For striking and seminal contributions to the theory of nonlinear partial differential equations and its applications to geometric analysis.  
For his stunning proof of Fermat's Last Theorem by way of the modularity conjecture for semistable elliptic curves, opening a new era in number theory.  
For his pivotal role in the development of the mathematical theory of wavelets.  
For his visionary program connecting representation theory to number theory.  
For her pioneering achievements in geometric partial differential equations, gauge theory and integrable systems, and for the fundamental impact of her work on analysis, geometry and mathematical physics.  
For pioneering the use of methods from probability and dynamics in group theory, number theory and combinatorics.  
For pioneering the use of methods from probability and dynamics in group theory, number theory and combinatorics.  
For their foundational contributions to theoretical computer science and discrete mathematics, and their leading role in shaping them into central fields of modern mathematics.  
For their foundational contributions to theoretical computer science and discrete mathematics, and their leading role in shaping them into central fields of modern mathematics.  
For his groundbreaking contributions to topology in its broadest sense, and in particular its algebraic, geometric and dynamical aspects.  
For his seminal contributions to regularity theory for nonlinear partial differential equations including freeboundary problems and the Monge–Ampère equation.  

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