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.

68Qxx Theory of computing

This subtopic introduces the core ideas in theory of computing, 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

68Q01 General topics in theory of computing

Overview

This topic examines general topics in theory of computing within theory of computing, complexity, and formal models of computation, focusing on the conceptual models, algorithmic structures, and evaluative criteria used in modern research and practice. It provides a practical bridge between formal theory and deployable computational methods.

Related Wikipedia Page

Theoretical computer science (Wikipedia)

Useful Links

Key Ideas

  • Formal models of computation and language recognition
  • Complexity classes, reductions, and hardness landscapes
  • Algorithmic limits and provable resource bounds

Typical Uses

Used to classify problem difficulty, compare computational models, and establish tractability boundaries.

Applications

  • Algorithm design and analysis
  • Formal verification and model checking
  • Learning theory and randomized computation

References

Recommended Textbooks

68Q04 Classical models of computation (Turing machines, etc.)

Overview

This topic examines classical models of computation (turing machines, etc.) within theory of computing, complexity, and formal models of computation, focusing on the conceptual models, algorithmic structures, and evaluative criteria used in modern research and practice. It provides a practical bridge between formal theory and deployable computational methods.

Related Wikipedia Page

Theoretical computer science (Wikipedia)

Useful Links

Key Ideas

  • Formal models of computation and language recognition
  • Complexity classes, reductions, and hardness landscapes
  • Algorithmic limits and provable resource bounds

Typical Uses

Used to classify problem difficulty, compare computational models, and establish tractability boundaries.

Applications

  • Algorithm design and analysis
  • Formal verification and model checking
  • Learning theory and randomized computation

References

Recommended Textbooks

68Q05 Models of computation (nondeterministic, parallel, interactive, probabilistic, etc.)

Overview

This topic examines models of computation (nondeterministic, parallel, interactive, probabilistic, etc.) within theory of computing, complexity, and formal models of computation, focusing on the conceptual models, algorithmic structures, and evaluative criteria used in modern research and practice. It provides a practical bridge between formal theory and deployable computational methods.

Related Wikipedia Page

Theoretical computer science (Wikipedia)

Useful Links

Key Ideas

  • Formal models of computation and language recognition
  • Complexity classes, reductions, and hardness landscapes
  • Algorithmic limits and provable resource bounds

Typical Uses

Used to classify problem difficulty, compare computational models, and establish tractability boundaries.

Applications

  • Algorithm design and analysis
  • Formal verification and model checking
  • Learning theory and randomized computation

References

Recommended Textbooks

68Q06 Networks and circuits as models of computation

Overview

This topic examines networks and circuits as models of computation within theory of computing, complexity, and formal models of computation, focusing on the conceptual models, algorithmic structures, and evaluative criteria used in modern research and practice. It provides a practical bridge between formal theory and deployable computational methods.

Related Wikipedia Page

Theoretical computer science (Wikipedia)

Useful Links

Key Ideas

  • Formal models of computation and language recognition
  • Complexity classes, reductions, and hardness landscapes
  • Algorithmic limits and provable resource bounds

Typical Uses

Used to classify problem difficulty, compare computational models, and establish tractability boundaries.

Applications

  • Algorithm design and analysis
  • Formal verification and model checking
  • Learning theory and randomized computation

References

Recommended Textbooks

68Q07 Biologically inspired models of computation

Overview

This topic examines biologically inspired models of computation within theory of computing, complexity, and formal models of computation, focusing on the conceptual models, algorithmic structures, and evaluative criteria used in modern research and practice. It provides a practical bridge between formal theory and deployable computational methods.

Related Wikipedia Page

Theoretical computer science (Wikipedia)

Useful Links

Key Ideas

  • Formal models of computation and language recognition
  • Complexity classes, reductions, and hardness landscapes
  • Algorithmic limits and provable resource bounds

Typical Uses

Used to classify problem difficulty, compare computational models, and establish tractability boundaries.

Applications

  • Algorithm design and analysis
  • Formal verification and model checking
  • Learning theory and randomized computation

References

Recommended Textbooks

68Q09 Other models of computation

Overview

This topic examines other models of computation within theory of computing, complexity, and formal models of computation, focusing on the conceptual models, algorithmic structures, and evaluative criteria used in modern research and practice. It provides a practical bridge between formal theory and deployable computational methods.

Related Wikipedia Page

Theoretical computer science (Wikipedia)

Useful Links

Key Ideas

  • Formal models of computation and language recognition
  • Complexity classes, reductions, and hardness landscapes
  • Algorithmic limits and provable resource bounds

Typical Uses

Used to classify problem difficulty, compare computational models, and establish tractability boundaries.

Applications

  • Algorithm design and analysis
  • Formal verification and model checking
  • Learning theory and randomized computation

References

Recommended Textbooks

68Q10 Modes of computation (nondeterministic, parallel, etc.)

Overview

This topic examines modes of computation (nondeterministic, parallel, etc.) within theory of computing, complexity, and formal models of computation, focusing on the conceptual models, algorithmic structures, and evaluative criteria used in modern research and practice. It provides a practical bridge between formal theory and deployable computational methods.

Related Wikipedia Page

Theoretical computer science (Wikipedia)

Useful Links

Key Ideas

  • Formal models of computation and language recognition
  • Complexity classes, reductions, and hardness landscapes
  • Algorithmic limits and provable resource bounds

Typical Uses

Used to classify problem difficulty, compare computational models, and establish tractability boundaries.

Applications

  • Algorithm design and analysis
  • Formal verification and model checking
  • Learning theory and randomized computation

References

Recommended Textbooks

68Q15 Complexity classes

Overview

This topic examines complexity classes within theory of computing, complexity, and formal models of computation, focusing on the conceptual models, algorithmic structures, and evaluative criteria used in modern research and practice. It provides a practical bridge between formal theory and deployable computational methods.

Related Wikipedia Page

Theoretical computer science (Wikipedia)

Useful Links

Key Ideas

  • Formal models of computation and language recognition
  • Complexity classes, reductions, and hardness landscapes
  • Algorithmic limits and provable resource bounds

Typical Uses

Used to classify problem difficulty, compare computational models, and establish tractability boundaries.

Applications

  • Algorithm design and analysis
  • Formal verification and model checking
  • Learning theory and randomized computation

References

Recommended Textbooks

68Q17 Computational difficulty of problems (lower bounds, completeness, difficulty of approximation, etc.)

Overview

This topic examines computational difficulty of problems (lower bounds, completeness, difficulty of approximation, etc.) within theory of computing, complexity, and formal models of computation, focusing on the conceptual models, algorithmic structures, and evaluative criteria used in modern research and practice. It provides a practical bridge between formal theory and deployable computational methods.

Related Wikipedia Page

Theoretical computer science (Wikipedia)

Useful Links

Key Ideas

  • Formal models of computation and language recognition
  • Complexity classes, reductions, and hardness landscapes
  • Algorithmic limits and provable resource bounds

Typical Uses

Used to classify problem difficulty, compare computational models, and establish tractability boundaries.

Applications

  • Algorithm design and analysis
  • Formal verification and model checking
  • Learning theory and randomized computation

References

Recommended Textbooks

68Q19 Descriptive complexity and finite models

Overview

This topic examines descriptive complexity and finite models within theory of computing, complexity, and formal models of computation, focusing on the conceptual models, algorithmic structures, and evaluative criteria used in modern research and practice. It provides a practical bridge between formal theory and deployable computational methods.

Related Wikipedia Page

Theoretical computer science (Wikipedia)

Useful Links

Key Ideas

  • Formal models of computation and language recognition
  • Complexity classes, reductions, and hardness landscapes
  • Algorithmic limits and provable resource bounds

Typical Uses

Used to classify problem difficulty, compare computational models, and establish tractability boundaries.

Applications

  • Algorithm design and analysis
  • Formal verification and model checking
  • Learning theory and randomized computation

References

Recommended Textbooks

68Q25 Analysis of algorithms and problem complexity

Overview

This topic examines analysis of algorithms and problem complexity within theory of computing, complexity, and formal models of computation, focusing on the conceptual models, algorithmic structures, and evaluative criteria used in modern research and practice. It provides a practical bridge between formal theory and deployable computational methods.

Related Wikipedia Page

Theoretical computer science (Wikipedia)

Useful Links

Key Ideas

  • Formal models of computation and language recognition
  • Complexity classes, reductions, and hardness landscapes
  • Algorithmic limits and provable resource bounds

Typical Uses

Used to classify problem difficulty, compare computational models, and establish tractability boundaries.

Applications

  • Algorithm design and analysis
  • Formal verification and model checking
  • Learning theory and randomized computation

References

Recommended Textbooks

68Q27 Parameterized complexity and exact algorithms

Overview

This topic examines parameterized complexity and exact algorithms within theory of computing, complexity, and formal models of computation, focusing on the conceptual models, algorithmic structures, and evaluative criteria used in modern research and practice. It provides a practical bridge between formal theory and deployable computational methods.

Related Wikipedia Page

Theoretical computer science (Wikipedia)

Useful Links

Key Ideas

  • Formal models of computation and language recognition
  • Complexity classes, reductions, and hardness landscapes
  • Algorithmic limits and provable resource bounds

Typical Uses

Used to classify problem difficulty, compare computational models, and establish tractability boundaries.

Applications

  • Algorithm design and analysis
  • Formal verification and model checking
  • Learning theory and randomized computation

References

Recommended Textbooks

68Q30 Algorithmic information theory (Kolmogorov complexity, etc.)

Overview

This topic examines algorithmic information theory (kolmogorov complexity, etc.) within theory of computing, complexity, and formal models of computation, focusing on the conceptual models, algorithmic structures, and evaluative criteria used in modern research and practice. It provides a practical bridge between formal theory and deployable computational methods.

Related Wikipedia Page

Theoretical computer science (Wikipedia)

Useful Links

Key Ideas

  • Formal models of computation and language recognition
  • Complexity classes, reductions, and hardness landscapes
  • Algorithmic limits and provable resource bounds

Typical Uses

Used to classify problem difficulty, compare computational models, and establish tractability boundaries.

Applications

  • Algorithm design and analysis
  • Formal verification and model checking
  • Learning theory and randomized computation

References

Recommended Textbooks

68Q32 Computational learning theory

Overview

This topic examines computational learning theory within theory of computing, complexity, and formal models of computation, focusing on the conceptual models, algorithmic structures, and evaluative criteria used in modern research and practice. It provides a practical bridge between formal theory and deployable computational methods.

Related Wikipedia Page

Theoretical computer science (Wikipedia)

Useful Links

Key Ideas

  • Formal models of computation and language recognition
  • Complexity classes, reductions, and hardness landscapes
  • Algorithmic limits and provable resource bounds

Typical Uses

Used to classify problem difficulty, compare computational models, and establish tractability boundaries.

Applications

  • Algorithm design and analysis
  • Formal verification and model checking
  • Learning theory and randomized computation

References

Recommended Textbooks

68Q42 Grammars and rewriting systems

Overview

This topic examines grammars and rewriting systems within theory of computing, complexity, and formal models of computation, focusing on the conceptual models, algorithmic structures, and evaluative criteria used in modern research and practice. It provides a practical bridge between formal theory and deployable computational methods.

Related Wikipedia Page

Theoretical computer science (Wikipedia)

Useful Links

Key Ideas

  • Formal models of computation and language recognition
  • Complexity classes, reductions, and hardness landscapes
  • Algorithmic limits and provable resource bounds

Typical Uses

Used to classify problem difficulty, compare computational models, and establish tractability boundaries.

Applications

  • Algorithm design and analysis
  • Formal verification and model checking
  • Learning theory and randomized computation

References

Recommended Textbooks

68Q45 Formal languages and automata theory

Overview

This topic examines formal languages and automata theory within theory of computing, complexity, and formal models of computation, focusing on the conceptual models, algorithmic structures, and evaluative criteria used in modern research and practice. It provides a practical bridge between formal theory and deployable computational methods.

Related Wikipedia Page

Theoretical computer science (Wikipedia)

Useful Links

Key Ideas

  • Formal models of computation and language recognition
  • Complexity classes, reductions, and hardness landscapes
  • Algorithmic limits and provable resource bounds

Typical Uses

Used to classify problem difficulty, compare computational models, and establish tractability boundaries.

Applications

  • Algorithm design and analysis
  • Formal verification and model checking
  • Learning theory and randomized computation

References

Recommended Textbooks

68Q55 Semantics

Overview

This topic examines semantics within theory of computing, complexity, and formal models of computation, focusing on the conceptual models, algorithmic structures, and evaluative criteria used in modern research and practice. It provides a practical bridge between formal theory and deployable computational methods.

Related Wikipedia Page

Theoretical computer science (Wikipedia)

Useful Links

Key Ideas

  • Formal models of computation and language recognition
  • Complexity classes, reductions, and hardness landscapes
  • Algorithmic limits and provable resource bounds

Typical Uses

Used to classify problem difficulty, compare computational models, and establish tractability boundaries.

Applications

  • Algorithm design and analysis
  • Formal verification and model checking
  • Learning theory and randomized computation

References

Recommended Textbooks

68Q60 Specification and verification (program logics, model checking, etc.)

Overview

This topic examines specification and verification (program logics, model checking, etc.) within theory of computing, complexity, and formal models of computation, focusing on the conceptual models, algorithmic structures, and evaluative criteria used in modern research and practice. It provides a practical bridge between formal theory and deployable computational methods.

Related Wikipedia Page

Theoretical computer science (Wikipedia)

Useful Links

Key Ideas

  • Formal models of computation and language recognition
  • Complexity classes, reductions, and hardness landscapes
  • Algorithmic limits and provable resource bounds

Typical Uses

Used to classify problem difficulty, compare computational models, and establish tractability boundaries.

Applications

  • Algorithm design and analysis
  • Formal verification and model checking
  • Learning theory and randomized computation

References

Recommended Textbooks

68Q70 Algebraic theory of languages and automata

Overview

This topic examines algebraic theory of languages and automata within theory of computing, complexity, and formal models of computation, focusing on the conceptual models, algorithmic structures, and evaluative criteria used in modern research and practice. It provides a practical bridge between formal theory and deployable computational methods.

Related Wikipedia Page

Theoretical computer science (Wikipedia)

Useful Links

Key Ideas

  • Formal models of computation and language recognition
  • Complexity classes, reductions, and hardness landscapes
  • Algorithmic limits and provable resource bounds

Typical Uses

Used to classify problem difficulty, compare computational models, and establish tractability boundaries.

Applications

  • Algorithm design and analysis
  • Formal verification and model checking
  • Learning theory and randomized computation

References

Recommended Textbooks

68Q80 Cellular automata

Overview

This topic examines cellular automata within theory of computing, complexity, and formal models of computation, focusing on the conceptual models, algorithmic structures, and evaluative criteria used in modern research and practice. It provides a practical bridge between formal theory and deployable computational methods.

Related Wikipedia Page

Theoretical computer science (Wikipedia)

Useful Links

Key Ideas

  • Formal models of computation and language recognition
  • Complexity classes, reductions, and hardness landscapes
  • Algorithmic limits and provable resource bounds

Typical Uses

Used to classify problem difficulty, compare computational models, and establish tractability boundaries.

Applications

  • Algorithm design and analysis
  • Formal verification and model checking
  • Learning theory and randomized computation

References

Recommended Textbooks

68Q85 Models and methods for concurrent and distributed computing

Overview

This topic examines models and methods for concurrent and distributed computing within theory of computing, complexity, and formal models of computation, focusing on the conceptual models, algorithmic structures, and evaluative criteria used in modern research and practice. It provides a practical bridge between formal theory and deployable computational methods.

Related Wikipedia Page

Theoretical computer science (Wikipedia)

Useful Links

Key Ideas

  • Formal models of computation and language recognition
  • Complexity classes, reductions, and hardness landscapes
  • Algorithmic limits and provable resource bounds

Typical Uses

Used to classify problem difficulty, compare computational models, and establish tractability boundaries.

Applications

  • Algorithm design and analysis
  • Formal verification and model checking
  • Learning theory and randomized computation

References

Recommended Textbooks

68Q87 Probability in computer science

Overview

This topic examines probability in computer science within theory of computing, complexity, and formal models of computation, focusing on the conceptual models, algorithmic structures, and evaluative criteria used in modern research and practice. It provides a practical bridge between formal theory and deployable computational methods.

Related Wikipedia Page

Theoretical computer science (Wikipedia)

Useful Links

Key Ideas

  • Formal models of computation and language recognition
  • Complexity classes, reductions, and hardness landscapes
  • Algorithmic limits and provable resource bounds

Typical Uses

Used to classify problem difficulty, compare computational models, and establish tractability boundaries.

Applications

  • Algorithm design and analysis
  • Formal verification and model checking
  • Learning theory and randomized computation

References

Recommended Textbooks

68Q99 None of the above

Overview

This topic examines none of the above within theory of computing, complexity, and formal models of computation, focusing on the conceptual models, algorithmic structures, and evaluative criteria used in modern research and practice. It provides a practical bridge between formal theory and deployable computational methods.

Related Wikipedia Page

Theoretical computer science (Wikipedia)

Useful Links

Key Ideas

  • Formal models of computation and language recognition
  • Complexity classes, reductions, and hardness landscapes
  • Algorithmic limits and provable resource bounds

Typical Uses

Used to classify problem difficulty, compare computational models, and establish tractability boundaries.

Applications

  • Algorithm design and analysis
  • Formal verification and model checking
  • Learning theory and randomized computation

References

Recommended Textbooks