Mathematics Branches, Topics, and Sub-Topics

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

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58Hxx Pseudogroups and groupoids

This subtopic introduces the core ideas in pseudogroups and groupoids, 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

58H05 Pseudogroups and differentiable groupoids

Overview

Pseudogroups and differentiable groupoids. This topic studies pseudogroups and groupoids as flexible symmetry frameworks connecting geometry, analysis, and topology.

Related Wikipedia Page

Wikipedia: Groupoid

Useful Links

Key Ideas

  • local-to-global symmetry models
  • stack and foliation interactions
  • operator-algebraic and geometric realizations

Typical Uses

Used to encode partial symmetries and singular quotients in geometry and analysis.

Applications

  • Foliation theory
  • Noncommutative geometry
  • Lie groupoid methods

References

Recommended Textbooks

58H10 Cohomology of classifying spaces for pseudogroup structures

Overview

Cohomology of classifying spaces for pseudogroup structures. This topic studies pseudogroups and groupoids as flexible symmetry frameworks connecting geometry, analysis, and topology.

Related Wikipedia Page

Wikipedia: Groupoid

Useful Links

Key Ideas

  • local-to-global symmetry models
  • stack and foliation interactions
  • operator-algebraic and geometric realizations

Typical Uses

Used to encode partial symmetries and singular quotients in geometry and analysis.

Applications

  • Foliation theory
  • Noncommutative geometry
  • Lie groupoid methods

References

Recommended Textbooks

58H15 Deformations of structures on manifolds

Overview

Deformations of structures on manifolds. This topic studies pseudogroups and groupoids as flexible symmetry frameworks connecting geometry, analysis, and topology.

Related Wikipedia Page

Wikipedia: Groupoid

Useful Links

Key Ideas

  • local-to-global symmetry models
  • stack and foliation interactions
  • operator-algebraic and geometric realizations

Typical Uses

Used to encode partial symmetries and singular quotients in geometry and analysis.

Applications

  • Foliation theory
  • Noncommutative geometry
  • Lie groupoid methods

References

Recommended Textbooks