Branches of Mathematics
Algebraic Geometry
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Algebraic geometry is a branch of mathematics that uses abstract algebraic techniques, mainly from commutative algebra, to solve geometrical problems. It chiefly studies algebraic varieties, the geometric shapes that form the solution sets of systems of polynomial equations.
Facts
Central QuestionWhat geometric shape does the solution set of a system of polynomial equations form, and what can algebraic techniques reveal about that shape's intrinsic properties that geometry alone cannot? 1 Key DebateWhether the classical style of algebraic geometry, studying varieties as concrete geometric loci of solutions over the complex numbers in the manner of the Italian school, or Grothendieck's scheme-theoretic reformulation, recasting varieties as functors of points over any commutative ring, is the right foundation for the subject. Grothendieck's schemes won out as the working foundation for research from the 1960s on, prized for their generality and their functorial precision, but the classical, more geometrically visualizable approach remains how the subject is usually first taught and pictured. 1 Cross-Tradition Connections
Sources
1. Algebraic Geometry (Wikipedia)
Wikimedia Foundationlead paragraphQuote, lead paragraph
Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometrical problems.
1. Algebraic Geometry (Wikipedia)
Wikimedia Foundationbody text on Grothendieck's schemesQuote, body text on Grothendieck's schemes
Most remarkably, in the early 1960s, algebraic varieties were subsumed into Alexander Grothendieck's concept of a scheme.
Alexander Grothendieck (Wikipedia)
WikipediaIncludes: Alexander Grothendieck, Lead sectionQuote, Includes: Alexander Grothendieck, Lead section
His research extended the scope of the field and added elements of commutative algebra, homological algebra, sheaf theory, and category theory to its foundations, while his so-called "relative" perspective led to revolutionary advances in many areas of pure mathematics.
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