Mathematics Branches, Topics, and Sub-Topics

A structured visual guide to the major mathematical areas and their relationships.

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14Axx Foundations

This subtopic studies foundations of algebraic geometry, including the basic language and principles that organize the subject across examples and general theory.

Specific topics

14A10 Varieties and morphisms

Overview

14A10 studies varieties and morphisms in foundations of algebraic geometry. It emphasizes basic objects, morphisms, and categorical frameworks that organize modern algebraic geometry from local models to global constructions.

Related Wikipedia Page

Varieties and morphisms (Wikipedia)

Useful Links

Key Ideas

  • Canonical formulations and representative examples for varieties and morphisms
  • How structural, local, and computational viewpoints interact in the subject
  • Standard theorem patterns and invariants used in current research practice

Typical Uses

Used to organize definitions, compare equivalent formulations, and support proof-driven or computational work in algebra and algebraic geometry.

Applications

  • Foundational setup for advanced algebraic geometry
  • Comparison of geometric categories and functors
  • Interfaces with arithmetic geometry and moduli theory

References

Recommended Textbooks

14A15 Schemes and morphisms

Overview

14A15 studies schemes and morphisms in foundations of algebraic geometry. It emphasizes basic objects, morphisms, and categorical frameworks that organize modern algebraic geometry from local models to global constructions.

Related Wikipedia Page

Schemes and morphisms (Wikipedia)

Useful Links

Key Ideas

  • Canonical formulations and representative examples for schemes and morphisms
  • How structural, local, and computational viewpoints interact in the subject
  • Standard theorem patterns and invariants used in current research practice

Typical Uses

Used to organize definitions, compare equivalent formulations, and support proof-driven or computational work in algebra and algebraic geometry.

Applications

  • Foundational setup for advanced algebraic geometry
  • Comparison of geometric categories and functors
  • Interfaces with arithmetic geometry and moduli theory

References

Recommended Textbooks

14A20 Generalizations (algebraic spaces, stacks)

Overview

14A20 studies generalizations (algebraic spaces, stacks) in foundations of algebraic geometry. It emphasizes basic objects, morphisms, and categorical frameworks that organize modern algebraic geometry from local models to global constructions.

Related Wikipedia Page

Generalizations (algebraic spaces, stacks) (Wikipedia)

Useful Links

Key Ideas

  • Canonical formulations and representative examples for generalizations (algebraic spaces, stacks)
  • How structural, local, and computational viewpoints interact in the subject
  • Standard theorem patterns and invariants used in current research practice

Typical Uses

Used to organize definitions, compare equivalent formulations, and support proof-driven or computational work in algebra and algebraic geometry.

Applications

  • Foundational setup for advanced algebraic geometry
  • Comparison of geometric categories and functors
  • Interfaces with arithmetic geometry and moduli theory

References

Recommended Textbooks

14A21 Logarithmic algebraic geometry

Overview

14A21 studies logarithmic algebraic geometry in foundations of algebraic geometry. It emphasizes basic objects, morphisms, and categorical frameworks that organize modern algebraic geometry from local models to global constructions.

Related Wikipedia Page

Logarithmic algebraic geometry (Wikipedia)

Useful Links

Key Ideas

  • Canonical formulations and representative examples for logarithmic algebraic geometry
  • How structural, local, and computational viewpoints interact in the subject
  • Standard theorem patterns and invariants used in current research practice

Typical Uses

Used to organize definitions, compare equivalent formulations, and support proof-driven or computational work in algebra and algebraic geometry.

Applications

  • Foundational setup for advanced algebraic geometry
  • Comparison of geometric categories and functors
  • Interfaces with arithmetic geometry and moduli theory

References

Recommended Textbooks

14A22 Noncommutative algebraic geometry

Overview

14A22 studies noncommutative algebraic geometry in foundations of algebraic geometry. It emphasizes basic objects, morphisms, and categorical frameworks that organize modern algebraic geometry from local models to global constructions.

Related Wikipedia Page

Noncommutative algebraic geometry (Wikipedia)

Useful Links

Key Ideas

  • Canonical formulations and representative examples for noncommutative algebraic geometry
  • How structural, local, and computational viewpoints interact in the subject
  • Standard theorem patterns and invariants used in current research practice

Typical Uses

Used to organize definitions, compare equivalent formulations, and support proof-driven or computational work in algebra and algebraic geometry.

Applications

  • Foundational setup for advanced algebraic geometry
  • Comparison of geometric categories and functors
  • Interfaces with arithmetic geometry and moduli theory

References

Recommended Textbooks

14A25 Elementary questions

Overview

14A25 studies elementary questions in foundations of algebraic geometry. It emphasizes basic objects, morphisms, and categorical frameworks that organize modern algebraic geometry from local models to global constructions.

Related Wikipedia Page

Elementary questions (Wikipedia)

Useful Links

Key Ideas

  • Canonical formulations and representative examples for elementary questions
  • How structural, local, and computational viewpoints interact in the subject
  • Standard theorem patterns and invariants used in current research practice

Typical Uses

Used to organize definitions, compare equivalent formulations, and support proof-driven or computational work in algebra and algebraic geometry.

Applications

  • Foundational setup for advanced algebraic geometry
  • Comparison of geometric categories and functors
  • Interfaces with arithmetic geometry and moduli theory

References

Recommended Textbooks