Algebraic Geometry & Schemes
Grothendieck's schemes (from 1960) let geometry handle equations over any ring, even the whole numbers: a language that unites geometry and number theory and underlies Wiles's proof of Fermat's Last Theorem.
Open in the interactive tree →Building on work by Weil and Zariski, Alexander Grothendieck introduced schemes in his Elements of Algebraic Geometry (from 1960) while working at the IHES near Paris (1959-1970), and he won the Fields Medal in 1966. A scheme describes the solutions of polynomial equations over any number system in which one can add and multiply (a commutative ring), so geometric tools can be applied to equations over the whole numbers. Pierre Deligne used the theory to prove the Weil conjectures about counting solutions of equations, and the same language carried Wiles's proof of Fermat's Last Theorem and the geometric Langlands proof of 2024.
Prerequisites
- Analytic Geometry (Descartes)1637Shapes defined by polynomial equations are the objects studied.
- Number Theory & Group Theory1801Schemes bring primes and equations over the integers into geometry.
- Topology1895Topology is applied to the solution sets of equations.
- Abstract Algebra (Noether)1921-1931Commutative rings are the building blocks of schemes.
Unlocks
- Fermat's Last Theorem Proved1995Scheme theory unified geometry and number theory and led to Wiles's proof.
- Geometric Langlands Proved2024The geometric Langlands correspondence is stated and proved with algebraic geometry.
- Birch and Swinnerton-Dyer ConjectureopenElliptic curves are algebraic curves; their rational points are studied with schemes.
- Hodge ConjectureopenThe conjecture is about algebraic cycles on algebraic varieties.