Roughly speaking, a vector bundle is a family of vector spaces that are 'attached' to a manifold in a smoothly varying way. C^∞ vector bundles arise naturally in modern physics, especially when the manifold M is space-time or some extension of it. In this talk, I will introduce the definitions of vector bundles and connections with a brief history of the development in physics. I will also define a finitely generated projective module over C* algebras and connections as a generalization of vector bundles.

If time allows, I will give a non-trivial example: the projective module over a non-commutative torus and Connes' generalization of Yang-Mills Theory.

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