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.

51Kxx Distance geometry

This subtopic introduces the core ideas in distance geometry, including foundational concepts, standard methods, and the main questions used to organize the area. Typical uses include building mathematical background, framing related research problems, and supporting applications in neighboring fields where these concepts provide useful structure.

Specific topics

51K05 General theory of distance geometry

Overview

Distance geometry characterizes geometric objects through pairwise distances, often without coordinates or explicit embeddings.

Related Wikipedia Page

Wikipedia: Distance geometry

Useful Links

Key Ideas

  • distance matrices
  • embedding criteria
  • rigidity and realizability

Typical Uses

Used to reconstruct shapes and configurations from distance data alone.

Applications

  • Sensor network localization
  • Molecular conformation
  • Inverse geometry problems

References

Recommended Textbooks

51K10 Synthetic differential geometry

Overview

Synthetic differential geometry develops differential calculus in a category-theoretic setting with infinitesimals treated axiomatically.

Related Wikipedia Page

Wikipedia: Synthetic differential geometry

Useful Links

Key Ideas

  • infinitesimals
  • topos-theoretic foundations
  • smooth logic

Typical Uses

Gives a rigorous alternative foundation for differential geometry emphasizing structural and categorical methods.

Applications

  • Foundations of calculus
  • Category theory in geometry
  • Theoretical physics formalisms

References

Recommended Textbooks

51K99 None of the above

Overview

This category covers distance-geometric themes not captured by the primary classifications, including emerging variants and specialized models.

Related Wikipedia Page

Wikipedia: Distance geometry

Useful Links

Key Ideas

  • specialized metrics
  • nonstandard distance models
  • emerging methods

Typical Uses

Used to group active subareas that do not fit existing named classes.

Applications

  • Applied geometry
  • Optimization formulations
  • Computational modeling

References

Recommended Textbooks