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Topology

Topology is a major area of mathematics concerned with spatial properties that are preserved under continuous deformations of objects, for example deformations that involve stretching, but no tearing or gluing. Geometric topology is the science of mathematical shapes known as the manifolds. Examples of manifolds include the familiar Euclidean space that surrounds us and the four-dimensional space-time continuum from Einstein's theory of relativity. Topologists are interested in the qualitative properties of manifolds such as their degree of symmetry or the complexity of their positioning in the ambient space. These qualitative properties are not easy to measure, at least in quantifiable terms. Algebraic topology provides algebraic computational tools to study such properties. Quantum topology, on the other hand, is inspired by certain ideas from physics and uses representation theory of groups and algebras to propose a plethora of invariants that are useful to study knots and links, particularly in low dimensions.

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