Algebraic geometry

Proved the general quintic unsolvable by radicals, founded the theory of elliptic and abelian functions, and gave his name to commutative groups.

algebra, analysis, elliptic functions

Inventor of ideal numbers, who proved Fermat's Last Theorem for regular primes and found the quartic surface with sixteen nodes.

algebraic number theory, analysis, algebraic geometry

Prolific Victorian algebraist who founded matrix algebra, gave the first abstract definition of a group, and built classical invariant theory.

algebra, invariant theory, group theory

The king of invariant theory, who proved constructively that the covariants of binary forms are finitely generated.

invariant theory, algebra, algebraic geometry

Founder of the algebraic and birational theory of curves and surfaces, known for the AF+BG theorem, Brill–Noether theory and Noether's formula.

algebraic geometry, algebraic functions, birational geometry

French analyst known for the Picard theorems on entire functions, Picard iteration, Picard–Vessiot theory and the Picard group, and permanent secretary of the Académie des Sciences from 1917 to 1941.

complex analysis, differential equations, algebraic geometry

Engineer turned mathematician who brought topology into algebraic geometry, proved the Lefschetz fixed-point theorem, and founded an American school of nonlinear differential equations.

algebraic geometry, algebraic topology, differential equations

Rebuilt algebraic geometry on the foundations of commutative algebra, introducing the Zariski topology, the Main Theorem on normal varieties and the first algebraic resolutions of singularities.

algebraic geometry, commutative algebra, topology of plane curves

Author of Moderne Algebra, which gave abstract algebra its modern form, and of foundational work in algebraic geometry, Ramsey theory and the history of ancient science.

abstract algebra, algebraic geometry, combinatorics

Founder of Bourbaki who proved the Riemann hypothesis for curves over finite fields and formulated the Weil conjectures.

number theory, algebraic geometry, arithmetic geometry

Founding member of Bourbaki who introduced ideles into class field theory and constructed the Chevalley groups, simple groups attached to Lie algebras over every field.

class field theory, lie groups, algebraic groups

French mathematician who transformed algebraic topology, algebraic geometry and number theory, youngest Fields medallist and first Abel laureate.

algebraic topology, algebraic geometry, number theory

Rebuilt algebraic geometry on the theory of schemes, sheaves and topoi, creating the tools behind the proof of the Weil conjectures and much of modern number theory.

algebraic geometry, homological algebra, functional analysis

Founded geometric invariant theory and the modern theory of moduli spaces of curves and abelian varieties, then turned to the mathematics of vision and pattern theory.

algebraic geometry, moduli theory, geometric invariant theory

Belgian mathematician who proved the Riemann hypothesis over finite fields and created the theory of weights, mixed Hodge structures and Deligne–Mumford stacks.

algebraic geometry, number theory, hodge theory

Proved the Langlands correspondence for GL(2) over function fields with Drinfeld modules and shtukas, and created quantum groups, associators and, with Beilinson, the geometric Langlands framework of opers and chiral algebras.

number theory, algebraic geometry, representation theory

Fields Medallist who turned ideas from quantum field theory and string theory into mathematics, proving Witten's conjecture and founding homological mirror symmetry and deformation quantization of Poisson manifolds.

algebraic geometry, mathematical physics, symplectic geometry