nLab mathematics

Mathematics is the part of science that constructs and studies mathematical objects used to implement mathematical models. The results of study are organized into definitions, theorems, and proofs.

Subfields of mathematics include the following:

• philosophy of mathematics

• foundations of mathematics

  • set theory, type theory, (infinity,1)-category theory

  • mathematical logic

• mathematical tools

  • proof assistants

• Topology and Geometry

  • geometry (general list), topology (general list)

  • general topology

  • algebraic topology

  • homotopy theory

  • rational homotopy theory

  • differential geometry

  • differential topology

  • algebraic geometry

  • noncommutative algebraic geometry

  • noncommutative geometry (general flavour)

  • higher geometry

  • Lie theory

• Algebra:

  • universal algebra

  • higher algebra

  • group theory (including generalizations)

  • representation theory

  • algebraic approaches to differential calculus

  • homological algebra

  • gebras (co(al)gebras, bi/Hopf-algebr-as/oids…)

• mathematical analysis

  • nonstandard analysis, functional analysis, operator algebras