Algebraic Geometry

Introduction

Algebraic Geometry is a branch of mathematics that combines abstract algebra, especially commutative algebra, with geometry. It can be seen as the study of solutions of systems of algebraic equations.

History

Algebraic Geometry has its roots in the study of algebraic curves, which were studied by the ancient Greeks. The modern study of Algebraic Geometry began with the work of Descartes and Fermat in the 17th century.

Fundamentals

The fundamental objects of study in Algebraic Geometry are algebraic varieties, which are the common solutions of systems of polynomial equations. These varieties are equipped with a topology, the Zariski topology, and a sheaf of rings, the structure sheaf, making them into locally ringed spaces and allowing one to define regular functions and morphisms of varieties.

Modern Algebraic Geometry

Modern Algebraic Geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical problems about these sets of algebraic equations.

Schemes

The language of schemes, introduced by Grothendieck, allows one to flexibly use techniques from both algebra and topology to study these spaces.

Applications

Algebraic Geometry has many applications, including in number theory, physics, and coding theory. It also has many connections with other areas of mathematics, such as topology, complex analysis, and representation theory.