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What is the relationship between topology, functional analysis, and group theory? All three seem to overlap, and I can't quite see how to distinguish them / what they're each for.
Topology is a branch of mathematics that studies the properties of geometric objects that are preserved under continuous deformations, such as stretching or bending. It is concerned with the study of abstract spaces and their properties, rather than specific geometric shapes.
Functional analysis is a branch of mathematics that is concerned with the study of vector spaces and operators acting on them. It uses tools and concepts from calculus, linear algebra, and topology to study the properties of these spaces and operators.
Group theory is a branch of mathematics that studies the algebraic structures known as groups. A group is a set of elements that can be combined together in a specific way, and group theory studies the properties and structures that arise from these combinations.
Topology, functional analysis, and group theory are all branches of mathematics that are concerned with the study of abstract structures and their properties. They are closely related and often used together to study and solve problems in mathematics and other fields.
Topology, functional analysis, and group theory have a wide range of applications in various fields, including physics, engineering, and computer science. Some specific applications include the study of dynamical systems, data analysis, and cryptography.