Category:21st-century mathematicians

In 2006, Grigori Perelman declined the Fields Medal awarded for his proof of the Poincaré conjecture, walking away from organized mathematics altogether. The episode captured something distinctive about the mathematicians grouped in this category: the work happens at the highest level of abstraction, the recognition arrives through a small set of prestigious prizes, and the personal trajectories range from the conventionally institutional to the determinedly reclusive. The figures collected here have done substantial mathematical work in the 21st century, whether they began their careers in the postwar decades or rose to prominence after 2000.

Background

The 21st century inherited a mathematical landscape reshaped by the Bourbaki project, the Grothendieck revolution in algebraic geometry, and the steady expansion of mathematics into computation, mathematical physics, and large-scale collaboration. Many figures in this category bridge eras. Michael Atiyah and Pierre Deligne, whose foundational work belongs to the second half of the 20th century, continued to influence the field through writing, lectures, and institutional leadership well into the new century. Others, such as Maryna Viazovska and Artur Avila, built their reputations almost entirely within the 21st century itself.

The institutional centers remain recognizable: the Institut des Hautes Études Scientifiques near Paris, the Institute for Advanced Study in Princeton, the Courant Institute, the Max Planck Institute for Mathematics in Bonn, the École Normale Supérieure, the universities of Cambridge and Oxford, Moscow State University, and a growing constellation of strong departments in East Asia. The Fields Medal, the Abel Prize (first awarded in 2003), the Wolf Prize, and the Breakthrough Prize in Mathematics (established in 2014) have structured public recognition. Several mathematicians in this category have received more than one of these honors.

A second background factor is geographic. The collapse of the Soviet Union dispersed a generation of mathematicians trained in the Moscow and Leningrad schools across Western Europe, Israel, and North America. The careers of Vladimir Drinfeld, Gregori Margulis, Efim Zelmanov, Maxim Kontsevich, and Stanislav Smirnov all reflect that migration, as does, in a different way, the case of Perelman, who returned to Saint Petersburg.

Notable members

The members of this category cluster around a handful of mathematical themes. In algebraic and arithmetic geometry, Pierre Deligne and Vladimir Voevodsky represent two generations of work on motives, cohomology, and the foundations linking geometry to number theory. Voevodsky's later work on univalent foundations and homotopy type theory pushed into the foundations of mathematics itself. Vladimir Drinfeld reshaped the Langlands program over function fields and introduced quantum groups, structures that recur throughout the work of others in this group.

Number theory and the Langlands program are represented by Manjul Bhargava, whose reinterpretation of Gauss composition opened new approaches to arithmetic statistics, and by Ngo Bao Chau, who proved the fundamental lemma. Jean Bourgain worked across an unusually broad range, from harmonic analysis to additive combinatorics to mathematical physics. Terence Tao has done work of similar breadth, including the Green-Tao theorem on arithmetic progressions in the primes, with later contributions in compressed sensing, partial differential equations, and analytic number theory.

Dynamical systems and ergodic theory form another cluster. Jean-Christophe Yoccoz worked on circle diffeomorphisms and small divisor problems. Artur Avila extended this tradition with results on one-dimensional dynamics, Teichmüller flow, and Schrödinger operators. Elon Lindenstrauss applied ergodic theory to questions in number theory, including the Littlewood conjecture. Gregori Margulis connected Lie groups, ergodic theory, and combinatorics through his work on lattices and expander graphs.

Geometry and topology are represented by William Thurston, whose geometrization program for 3-manifolds set the stage that Grigori Perelman would complete using Ricci flow methods developed by Richard Hamilton. Michael Atiyah contributed to index theory, gauge theory, and the long dialogue between geometry and physics.

Mathematical physics and the interaction with quantum field theory run through the work of Maxim Kontsevich, whose contributions span deformation quantization, motivic integration, and mirror symmetry, and Richard Borcherds, who proved the monstrous moonshine conjectures using vertex operator algebras. Vaughan Jones connected operator algebras to knot theory through the Jones polynomial, with consequences in low-dimensional topology and quantum computing. Stanislav Smirnov established conformal invariance for percolation and the Ising model on the planar lattice.

Group theory and discrete mathematics are represented by Efim Zelmanov, who solved the restricted Burnside problem. Analysis and stochastic partial differential equations run through the work of Martin Hairer, whose theory of regularity structures provides a framework for equations such as KPZ. Ernst Hairer, a separate figure, has worked on numerical analysis of differential equations, particularly geometric integration methods. Maryna Viazovska solved the sphere packing problem in dimension 8 in 2016, and contributed to the parallel solution in dimension 24, using modular forms in a way that revealed unexpected structure in classical lattice problems.

Recognition and the prize culture

A striking proportion of the mathematicians in this category are Fields Medalists. The award, given to mathematicians under 40 at four-yearly congresses of the International Mathematical Union, has been received by Avila, Bhargava, Borcherds, Bourgain, Drinfeld, Lindenstrauss, Margulis, Martin Hairer, Kontsevich, Ngo, Perelman, Smirnov, Tao, Viazovska, Voevodsky, and Yoccoz, among others listed here. The Abel Prize, modeled in part on the Nobel and intended to recognize career achievement rather than youthful work, has gone to Atiyah and Deligne, as well as to others in this group. The Wolf Prize and the Crafoord Prize provide additional layers of recognition. The presence of so many laureates reflects how the category was assembled: visibility through major prizes is one of the main routes by which a mathematician becomes the subject of a substantial biographical article.

Working contexts

The day-to-day work represented here takes place in a small number of settings. Research universities, dedicated research institutes, and conferences such as the ICM provide the primary venues. Collaboration patterns vary widely. Tao is known for an unusually large network of coauthors and for the public mathematical writing on his blog and on collaborative projects such as Polymath. Perelman posted his proofs to the arXiv and declined further engagement with the publication system. Long collaborations, such as that between Ernst Hairer and Gerhard Wanner on numerical methods, contrast with the solitary breakthroughs associated with Viazovska's sphere packing proof. The category thus contains both highly networked figures and isolated ones, both pure theorists and those whose work has fed directly into physics, computer science, and applied mathematics.