Knowra Algebraic geometry Algebraic geometry Algebraic geometry studies geometric objects defined by polynomial equations using algebraic methods. It connects the shapes of their solution sets to properties of rings and fields.
Algebraic variety : A geometric object defined by polynomial equations over a field, with conventions varying between classical and modern usage. Varieties are the central geometric objects studied in classical algebraic geometry.
Scheme (mathematics) : A space locally modeled on prime spectra of commutative rings, forming a foundational object in modern algebraic geometry. Schemes extend varieties to handle nilpotents, arithmetic, and families uniformly.
René Descartes : A French philosopher and mathematician who developed analytic geometry in the seventeenth century. Coordinate geometry made polynomial equations a systematic language for describing curves.
Arithmetic geometry : The study of solutions to polynomial equations over number fields, finite fields, and related arithmetic structures. It applies geometric methods to Diophantine equations and number fields.
Differential geometry : The study of smooth spaces and geometric structures using calculus and differential methods. It studies spaces through smooth structure rather than polynomial equations.
Affine space : The coordinate space over a field whose points are tuples of field elements. Polynomial zero sets are first formed inside affine space.
Spec (mathematics) : The set of prime ideals of a commutative ring, equipped with the Zariski topology and a structure sheaf. Taking Spec turns a ring into the basic local space of scheme theory.
Bernhard Riemann : A German mathematician whose work shaped complex analysis, geometry, and the theory of Riemann surfaces. His surfaces linked algebraic functions to geometric spaces.
Elliptic curve : A smooth projective curve of genus one with a specified rational point, equipped with a group law. Elliptic curves are central to arithmetic geometry and modern cryptography.
Analytic geometry : The study of geometric objects defined locally by analytic functions, which have convergent power series expansions. Analytic equations allow broader local functions than polynomial equations.
Show all 29