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.

19Exx K-theory in geometry and topology

This subtopic studies K-theory in geometry and topology, where vector bundles, characteristic classes, and topological invariants interact.

Specific topics

19E08 $K$-theory of schemes

Overview

19E08 studies k-theory of vector bundles in K-theory in geometry and topology. It applies K-theoretic ideas to vector bundles, manifolds, and topological invariants, connecting algebraic and geometric viewpoints.

Related Wikipedia Page

K-theory of vector bundles (Wikipedia)

Useful Links

Key Ideas

  • Canonical formulations and representative examples for k-theory of vector bundles
  • How structural, local, and computational viewpoints interact in the subject
  • Standard theorem patterns and invariants used in current research practice

Typical Uses

Used to classify vector bundles, study generalized cohomology, and analyze geometric invariants.

Applications

  • Vector bundle and topological K-theory
  • Geometric applications of K-theory
  • Connections with topology and cohomology

References

Recommended Textbooks

19E15 Algebraic cycles and motivic cohomology

Overview

19E15 studies topological k-theory and generalized cohomology in K-theory in geometry and topology. It applies K-theoretic ideas to vector bundles, manifolds, and topological invariants, connecting algebraic and geometric viewpoints.

Related Wikipedia Page

Topological K-theory and generalized cohomology (Wikipedia)

Useful Links

Key Ideas

  • Canonical formulations and representative examples for topological k-theory and generalized cohomology
  • How structural, local, and computational viewpoints interact in the subject
  • Standard theorem patterns and invariants used in current research practice

Typical Uses

Used to classify vector bundles, study generalized cohomology, and analyze geometric invariants.

Applications

  • Vector bundle and topological K-theory
  • Geometric applications of K-theory
  • Connections with topology and cohomology

References

Recommended Textbooks

19E20 Relations of algebraic $K$-theory with cohomology theories

Overview

19E20 studies applications of k-theory in geometry and topology in K-theory in geometry and topology. It applies K-theoretic ideas to vector bundles, manifolds, and topological invariants, connecting algebraic and geometric viewpoints.

Related Wikipedia Page

Applications of K-theory in geometry and topology (Wikipedia)

Useful Links

Key Ideas

  • Canonical formulations and representative examples for applications of k-theory in geometry and topology
  • How structural, local, and computational viewpoints interact in the subject
  • Standard theorem patterns and invariants used in current research practice

Typical Uses

Used to classify vector bundles, study generalized cohomology, and analyze geometric invariants.

Applications

  • Vector bundle and topological K-theory
  • Geometric applications of K-theory
  • Connections with topology and cohomology

References

Recommended Textbooks