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.

82Mxx Basic and computational methods

This subtopic introduces the core ideas in basic and computational methods, 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

82M10 Finite element, Rayleigh-Ritz, and Galerkin methods

Overview

This specific topic studies finite element, rayleigh-ritz, and galerkin 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: Finite element, Rayleigh-Ritz, and Galerkin 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

  • Rudin, W. (1976) Principles of Mathematical Analysis. 3rd edn. New York, NY: McGraw-Hill. — Amazon · Bookshop.org
  • Topic-focused textbook search (n.d.) Finite element, Rayleigh-Ritz, and Galerkin methods. Available via Amazon search. — Amazon · Bookshop.org

82M20 Finite difference methods

Overview

This specific topic studies finite difference 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: Finite difference method

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

  • Rudin, W. (1976) Principles of Mathematical Analysis. 3rd edn. New York, NY: McGraw-Hill. — Amazon · Bookshop.org
  • Topic-focused textbook search (n.d.) Finite difference methods. Available via Amazon search. — Amazon · Bookshop.org

82M22 Spectral methods

Overview

This specific topic studies spectral 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: Spectral method

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

  • Rudin, W. (1976) Principles of Mathematical Analysis. 3rd edn. New York, NY: McGraw-Hill. — Amazon · Bookshop.org
  • Topic-focused textbook search (n.d.) Spectral methods. Available via Amazon search. — Amazon · Bookshop.org

82M31 Monte Carlo methods

Overview

This specific topic studies monte carlo 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: Monte Carlo method

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

  • Rudin, W. (1976) Principles of Mathematical Analysis. 3rd edn. New York, NY: McGraw-Hill. — Amazon · Bookshop.org
  • Topic-focused textbook search (n.d.) Monte Carlo methods. Available via Amazon search. — Amazon · Bookshop.org

82M37 Stochastic particle methods

Overview

This specific topic studies stochastic particle 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: Stochastic particle 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

  • Rudin, W. (1976) Principles of Mathematical Analysis. 3rd edn. New York, NY: McGraw-Hill. — Amazon · Bookshop.org
  • Topic-focused textbook search (n.d.) Stochastic particle methods. Available via Amazon search. — Amazon · Bookshop.org

82M99 None of the above

Overview

This specific topic studies none of the above 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: None of the above

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

  • Rudin, W. (1976) Principles of Mathematical Analysis. 3rd edn. New York, NY: McGraw-Hill. — Amazon · Bookshop.org
  • Topic-focused textbook search (n.d.) None of the above. Available via Amazon search. — Amazon · Bookshop.org