Algebra

Elementary Algebra covers the fundamentals of algebra, focusing on operations with variables, simplifying expressions, and solving basic equations and inequalities.

Abstract Algebra involves studying algebraic structures such as groups, rings, and fields, which are key conceptual tools in understanding mathematical phenomena.

Linear Algebra deals with vector spaces and linear mappings between spaces, including matrices, determinants, eigenvalues, and systems of linear equations.

Commutative Algebra focuses on commutative rings, their ideals, and modules over them, serving as a foundation for algebraic geometry and number theory.

Algebraic Geometry is the study of geometric properties of solutions to polynomial equations, combining abstract algebra, especially commutative algebra, with geometry.

Homological Algebra explores the methods of algebraic topology with chain complexes, homology, and cohomology theories.

Universal Algebra studies algebraic structures themselves, not examples of algebraic structures or properties of particular algebraic structures.

Algebraic Number Theory examines the algebraic structures related to algebraic integers, giving insight into number theory through the lens of abstract algebra.

Algebraic Combinatorics investigates abstract algebraic structures with combinatorial methods, often utilizing group theory and graph theory.

Computational Algebra, or Computer Algebra, deals with algorithms and software for manipulating algebraic expressions and solving algebraic equations.

Algebraic Coding Theory applies algebraic techniques to coding theory, designing algorithms to improve the reliability of data transmission.

Algebraic Topology applies concepts from abstract algebra to study topological spaces, investigating invariant properties under continuous deformations.

Noncommutative Algebra looks at algebraic structures wherein the commutative property does not hold, such as noncommutative rings.

Homotopy Theory is an aspect of algebraic topology that studies spaces and the algebraic properties that are invariant under homotopies.

Representation Theory examines abstract algebraic structures by representing their elements as linear transformations of vector spaces.

Algebraic K-Theory is involved in the study of modules over a ring, which links to other fields such as number theory and topology.