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.

51Bxx Nonlinear incidence geometry

This subtopic introduces the core ideas in nonlinear incidence 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

51B05 General theory of nonlinear incidence geometry

Overview

This branch studies incidence structures that are not governed by linear projective axioms, including Möbius and Laguerre-type geometries.

Related Wikipedia Page

Wikipedia: Incidence geometry

Useful Links

Key Ideas

  • nonlinear incidence
  • generalized geometries
  • axiom systems

Typical Uses

Useful for broadening the classical theory of projective geometry.

Applications

  • Synthetic geometry
  • Transformation geometry
  • Design theory

References

Recommended Textbooks

51B10 Möbius geometries

Overview

Möbius geometry studies transformations preserving generalized circles and related conformal structures.

Related Wikipedia Page

Wikipedia: Möbius geometry

Useful Links

Key Ideas

  • circles
  • inversions
  • conformal maps

Typical Uses

Used in complex analysis and the geometry of circles.

Applications

  • Circle geometry
  • Conformal mapping
  • Computer graphics

References

Recommended Textbooks

51B15 Laguerre geometries

Overview

Laguerre geometry studies circles and oriented lines, emphasizing tangency and contact relations rather than metric distance alone.

Related Wikipedia Page

Wikipedia: Laguerre geometry

Useful Links

Key Ideas

  • oriented circles
  • contact geometry
  • tangency

Typical Uses

Appears in classical geometry, optics, and computational geometry.

Applications

  • Circle packings
  • Optics
  • Robotics

References

Recommended Textbooks

51B20 Minkowski geometries in nonlinear incidence geometry

Overview

Minkowski geometries encode distance and incidence through normed or convex structures, extending classical Euclidean ideas.

Related Wikipedia Page

Wikipedia: Minkowski geometry

Useful Links

Key Ideas

  • normed spaces
  • convex unit balls
  • distance geometry

Typical Uses

Relevant in convex geometry and metric geometry.

Applications

  • Normed spaces
  • Optimization
  • Geometric analysis

References

Recommended Textbooks

51B25 Lie geometries in nonlinear incidence geometry

Overview

Lie geometries exploit incidence and transformation structures associated with Lie groups and their homogeneous spaces.

Related Wikipedia Page

Wikipedia: Lie geometry

Useful Links

Key Ideas

  • Lie groups
  • homogeneous spaces
  • symmetry groups

Typical Uses

Central in classical geometry, differential geometry, and geometric group theory.

Applications

  • Symmetry analysis
  • Differential geometry
  • Physics

References

Recommended Textbooks

51B99 None of the above

Overview

This code is used for nonlinear incidence geometry topics that do not fit more specific classifications but still belong to the broad geometric framework of incidence structures.

Related Wikipedia Page

Wikipedia: Incidence geometry

Useful Links

Key Ideas

  • miscellaneous nonlinear geometry
  • general incidence structures
  • broad classification

Typical Uses

Serves as a catch-all for related geometric topics that are not otherwise categorized.

Applications

  • Geometry research
  • Generalized incidence problems
  • Survey work

References

Recommended Textbooks