Higher algebra studies the algebraic structures that arise in higher mathematics and unifies various themes that run throughout mathematical experience. Physicists may be most familiar with groups, but group theory pervades many parts of mathematics and arose to solve a primarily mathematical problem. Category theory is a relatively new birth that arose from the study of cohomology in topology and quickly broke free of its shackles to that area and became a powerful tool that currently challenges set theory as a foundation of mathematics, although category theory requires more mathematical experience to appreciate and cannot in its current state be reasonably used to introduce mathematics. It has also found many applications in the physical sciences.
- Wikipedia: Group theory
- Wikipedia: Ring theory
- Wikipedia: Field theory
- Wikipedia: Category theory
- Wikipedia: Representation theory
- Wikipedia: Non-associative algebra
- Wikipedia: Multilinear algebra