Mathematics Branches, Topics, and Sub-Topics

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

Search by code, branch, topic, subtopic, or a keyword from the descriptions.

14Pxx Real algebraic and analytic geometry

This subtopic studies real algebraic and analytic geometry, focusing on real varieties, semialgebraic sets, and geometric behavior over the reals.

Specific topics

14P05 Real algebraic sets

Overview

14P05 studies real algebraic sets in real algebraic and analytic geometry. It emphasizes real forms of algebraic sets, semialgebraic structure, and the topological behavior of real and analytic varieties.

Related Wikipedia Page

Real algebraic sets (Wikipedia)

Useful Links

Key Ideas

  • Canonical formulations and representative examples for real algebraic sets
  • 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

  • Real algebraic and semialgebraic geometry
  • Topology of real varieties and analytic sets
  • Applications to optimization and o-minimal geometry

References

Recommended Textbooks

14P10 Semialgebraic sets and related spaces

Overview

14P10 studies semialgebraic sets and related spaces in real algebraic and analytic geometry. It emphasizes real forms of algebraic sets, semialgebraic structure, and the topological behavior of real and analytic varieties.

Related Wikipedia Page

Semialgebraic sets and related spaces (Wikipedia)

Useful Links

Key Ideas

  • Canonical formulations and representative examples for semialgebraic sets and related spaces
  • 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

  • Real algebraic and semialgebraic geometry
  • Topology of real varieties and analytic sets
  • Applications to optimization and o-minimal geometry

References

Recommended Textbooks

14P15 Real analytic and semianalytic sets

Overview

14P15 studies real analytic and semianalytic sets in real algebraic and analytic geometry. It emphasizes real forms of algebraic sets, semialgebraic structure, and the topological behavior of real and analytic varieties.

Related Wikipedia Page

Real analytic and semianalytic sets (Wikipedia)

Useful Links

Key Ideas

  • Canonical formulations and representative examples for real analytic and semianalytic sets
  • 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

  • Real algebraic and semialgebraic geometry
  • Topology of real varieties and analytic sets
  • Applications to optimization and o-minimal geometry

References

Recommended Textbooks

14P20 Nash functions and manifolds

Overview

14P20 studies nash functions and manifolds in real algebraic and analytic geometry. It emphasizes real forms of algebraic sets, semialgebraic structure, and the topological behavior of real and analytic varieties.

Related Wikipedia Page

Nash functions and manifolds (Wikipedia)

Useful Links

Key Ideas

  • Canonical formulations and representative examples for nash functions and manifolds
  • 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

  • Real algebraic and semialgebraic geometry
  • Topology of real varieties and analytic sets
  • Applications to optimization and o-minimal geometry

References

Recommended Textbooks

14P25 Topology of real algebraic varieties

Overview

14P25 studies topology of real algebraic varieties in real algebraic and analytic geometry. It emphasizes real forms of algebraic sets, semialgebraic structure, and the topological behavior of real and analytic varieties.

Related Wikipedia Page

Topology of real algebraic varieties (Wikipedia)

Useful Links

Key Ideas

  • Canonical formulations and representative examples for topology of real algebraic varieties
  • 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

  • Real algebraic and semialgebraic geometry
  • Topology of real varieties and analytic sets
  • Applications to optimization and o-minimal geometry

References

Recommended Textbooks