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.

82Bxx Equilibrium statistical mechanics

This subtopic introduces the core ideas in equilibrium statistical mechanics, 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

82B03 Foundations of equilibrium statistical mechanics

Overview

This specific topic studies foundations of equilibrium statistical mechanics within statistical mechanics and condensed matter physics. It focuses on core models, structural ideas, and standard analytical techniques, giving a concise entry point to the main ideas, representative examples, and the kinds of questions that are typically addressed in this part of the field.

Related Wikipedia Page

Wikipedia search: Foundations of equilibrium statistical mechanics

Useful Links

Key Ideas

  • Core definitions and standard formulations
  • Representative examples, counterexamples, and structural results
  • Connections to neighboring methods and applications

Typical Uses

This topic is commonly used to frame precise research questions, organize proofs or calculations, and choose the appropriate tools for work in the surrounding subtopic.

Applications

  • Theoretical research and classification work
  • Modeling and abstraction in adjacent mathematical fields
  • Computational or algorithmic workflows where relevant

References

Recommended Textbooks

82B05 Classical equilibrium statistical mechanics (general)

Overview

This specific topic studies classical equilibrium statistical mechanics (general) within statistical mechanics and condensed matter physics. It focuses on core models, structural ideas, and standard analytical techniques, giving a concise entry point to the main ideas, representative examples, and the kinds of questions that are typically addressed in this part of the field.

Related Wikipedia Page

Wikipedia search: Classical equilibrium statistical mechanics (general)

Useful Links

Key Ideas

  • Core definitions and standard formulations
  • Representative examples, counterexamples, and structural results
  • Connections to neighboring methods and applications

Typical Uses

This topic is commonly used to frame precise research questions, organize proofs or calculations, and choose the appropriate tools for work in the surrounding subtopic.

Applications

  • Theoretical research and classification work
  • Modeling and abstraction in adjacent mathematical fields
  • Computational or algorithmic workflows where relevant

References

Recommended Textbooks

82B10 Quantum equilibrium statistical mechanics (general)

Overview

This specific topic studies quantum equilibrium statistical mechanics (general) within statistical mechanics and condensed matter physics. It focuses on core models, structural ideas, and standard analytical techniques, giving a concise entry point to the main ideas, representative examples, and the kinds of questions that are typically addressed in this part of the field.

Related Wikipedia Page

Wikipedia search: Quantum equilibrium statistical mechanics (general)

Useful Links

Key Ideas

  • Core definitions and standard formulations
  • Representative examples, counterexamples, and structural results
  • Connections to neighboring methods and applications

Typical Uses

This topic is commonly used to frame precise research questions, organize proofs or calculations, and choose the appropriate tools for work in the surrounding subtopic.

Applications

  • Theoretical research and classification work
  • Modeling and abstraction in adjacent mathematical fields
  • Computational or algorithmic workflows where relevant

References

Recommended Textbooks

82B20 Lattice systems (Ising, dimer, Potts, etc.)

Overview

This specific topic studies lattice systems (ising, dimer, potts, etc.) within statistical mechanics and condensed matter physics. It focuses on core models, structural ideas, and standard analytical techniques, giving a concise entry point to the main ideas, representative examples, and the kinds of questions that are typically addressed in this part of the field.

Related Wikipedia Page

Wikipedia search: Lattice systems (Ising, dimer, Potts, etc.)

Useful Links

Key Ideas

  • Core definitions and standard formulations
  • Representative examples, counterexamples, and structural results
  • Connections to neighboring methods and applications

Typical Uses

This topic is commonly used to frame precise research questions, organize proofs or calculations, and choose the appropriate tools for work in the surrounding subtopic.

Applications

  • Theoretical research and classification work
  • Modeling and abstraction in adjacent mathematical fields
  • Computational or algorithmic workflows where relevant

References

Recommended Textbooks

82B23 Exactly solvable models; Bethe ansatz

Overview

This specific topic studies exactly solvable models; bethe ansatz within statistical mechanics and condensed matter physics. It focuses on core models, structural ideas, and standard analytical techniques, giving a concise entry point to the main ideas, representative examples, and the kinds of questions that are typically addressed in this part of the field.

Related Wikipedia Page

Wikipedia search: Exactly solvable models; Bethe ansatz

Useful Links

Key Ideas

  • Core definitions and standard formulations
  • Representative examples, counterexamples, and structural results
  • Connections to neighboring methods and applications

Typical Uses

This topic is commonly used to frame precise research questions, organize proofs or calculations, and choose the appropriate tools for work in the surrounding subtopic.

Applications

  • Theoretical research and classification work
  • Modeling and abstraction in adjacent mathematical fields
  • Computational or algorithmic workflows where relevant

References

Recommended Textbooks

82B25 Classical phase transitions (general)

Overview

This specific topic studies classical phase transitions (general) within statistical mechanics and condensed matter physics. It focuses on core models, structural ideas, and standard analytical techniques, giving a concise entry point to the main ideas, representative examples, and the kinds of questions that are typically addressed in this part of the field.

Related Wikipedia Page

Wikipedia search: Classical phase transitions (general)

Useful Links

Key Ideas

  • Core definitions and standard formulations
  • Representative examples, counterexamples, and structural results
  • Connections to neighboring methods and applications

Typical Uses

This topic is commonly used to frame precise research questions, organize proofs or calculations, and choose the appropriate tools for work in the surrounding subtopic.

Applications

  • Theoretical research and classification work
  • Modeling and abstraction in adjacent mathematical fields
  • Computational or algorithmic workflows where relevant

References

Recommended Textbooks

82B27 Critical phenomena

Overview

This specific topic studies critical phenomena within statistical mechanics and condensed matter physics. It focuses on core models, structural ideas, and standard analytical techniques, giving a concise entry point to the main ideas, representative examples, and the kinds of questions that are typically addressed in this part of the field.

Related Wikipedia Page

Wikipedia search: Critical phenomena

Useful Links

Key Ideas

  • Core definitions and standard formulations
  • Representative examples, counterexamples, and structural results
  • Connections to neighboring methods and applications

Typical Uses

This topic is commonly used to frame precise research questions, organize proofs or calculations, and choose the appropriate tools for work in the surrounding subtopic.

Applications

  • Theoretical research and classification work
  • Modeling and abstraction in adjacent mathematical fields
  • Computational or algorithmic workflows where relevant

References

Recommended Textbooks

82B28 Renormalization group methods

Overview

This specific topic studies renormalization group methods within statistical mechanics and condensed matter physics. It focuses on core models, structural ideas, and standard analytical techniques, giving a concise entry point to the main ideas, representative examples, and the kinds of questions that are typically addressed in this part of the field.

Related Wikipedia Page

Wikipedia search: Renormalization group methods

Useful Links

Key Ideas

  • Core definitions and standard formulations
  • Representative examples, counterexamples, and structural results
  • Connections to neighboring methods and applications

Typical Uses

This topic is commonly used to frame precise research questions, organize proofs or calculations, and choose the appropriate tools for work in the surrounding subtopic.

Applications

  • Theoretical research and classification work
  • Modeling and abstraction in adjacent mathematical fields
  • Computational or algorithmic workflows where relevant

References

Recommended Textbooks

82B41 Random walks, random surfaces, lattice animals

Overview

This specific topic studies random walks, random surfaces, lattice animals within statistical mechanics and condensed matter physics. It focuses on core models, structural ideas, and standard analytical techniques, giving a concise entry point to the main ideas, representative examples, and the kinds of questions that are typically addressed in this part of the field.

Related Wikipedia Page

Wikipedia search: Random walks, random surfaces, lattice animals

Useful Links

Key Ideas

  • Core definitions and standard formulations
  • Representative examples, counterexamples, and structural results
  • Connections to neighboring methods and applications

Typical Uses

This topic is commonly used to frame precise research questions, organize proofs or calculations, and choose the appropriate tools for work in the surrounding subtopic.

Applications

  • Theoretical research and classification work
  • Modeling and abstraction in adjacent mathematical fields
  • Computational or algorithmic workflows where relevant

References

Recommended Textbooks

82B43 Percolation

Overview

This specific topic studies percolation within statistical mechanics and condensed matter physics. It focuses on core models, structural ideas, and standard analytical techniques, giving a concise entry point to the main ideas, representative examples, and the kinds of questions that are typically addressed in this part of the field.

Related Wikipedia Page

Wikipedia search: Percolation

Useful Links

Key Ideas

  • Core definitions and standard formulations
  • Representative examples, counterexamples, and structural results
  • Connections to neighboring methods and applications

Typical Uses

This topic is commonly used to frame precise research questions, organize proofs or calculations, and choose the appropriate tools for work in the surrounding subtopic.

Applications

  • Theoretical research and classification work
  • Modeling and abstraction in adjacent mathematical fields
  • Computational or algorithmic workflows where relevant

References

Recommended Textbooks

82B44 Disordered systems (random Ising models, etc.)

Overview

This specific topic studies disordered systems (random ising models, etc.) within statistical mechanics and condensed matter physics. It focuses on core models, structural ideas, and standard analytical techniques, giving a concise entry point to the main ideas, representative examples, and the kinds of questions that are typically addressed in this part of the field.

Related Wikipedia Page

Wikipedia search: Disordered systems (random Ising models, etc.)

Useful Links

Key Ideas

  • Core definitions and standard formulations
  • Representative examples, counterexamples, and structural results
  • Connections to neighboring methods and applications

Typical Uses

This topic is commonly used to frame precise research questions, organize proofs or calculations, and choose the appropriate tools for work in the surrounding subtopic.

Applications

  • Theoretical research and classification work
  • Modeling and abstraction in adjacent mathematical fields
  • Computational or algorithmic workflows where relevant

References

Recommended Textbooks