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I tried to understand Connes' approach several times but eventually I got stuck all the time. Does anybody know an introduction / review paper which explains the basic ideas, results and open issues?
Non-commutative geometry is a branch of mathematics that studies non-commutative algebraic structures, which are structures where the order in which operations are performed affects the outcome. It is a generalization of traditional geometry where the commutative property holds, meaning that the order of operations does not affect the outcome.
Non-commutative geometry has applications in various fields such as physics, where it is used to study quantum mechanics and general relativity. It also has applications in number theory, topology, and algebraic geometry.
Non-commutative geometry is a generalization of traditional geometry, meaning that it includes traditional geometry as a special case. In traditional geometry, the commutative property holds, while in non-commutative geometry, this property is relaxed, allowing for more general structures to be studied.
Some key concepts in non-commutative geometry include non-commutative algebras, non-commutative spaces, and non-commutative invariants. Non-commutative algebras are algebraic structures where the order of operations matters. Non-commutative spaces are spaces that are described by non-commutative algebras. Non-commutative invariants are mathematical objects that are used to classify non-commutative spaces.
Non-commutative geometry has been used in theoretical physics to study quantum mechanics and general relativity, providing a framework for understanding the quantum nature of space-time. It also has applications in string theory, which is a proposed theory of everything that aims to unify all fundamental forces and particles in the universe. Additionally, non-commutative geometry has also been used in mathematical models of the universe, providing insights into the structure and dynamics of the universe.