LTCC LECTURES ON NONCOMMUTATIVE DIFFERENTIAL GEOMETRY SHAHN MAJID Abstract. Noncommutative geometry is the idea that when geometry is done in terms of coordinate algebras, one does not really need the algebra to be commutative. We provide an introduction to the relevant mathematics from a constructive ‘differential algebra’ point of view that works over general fields and includes the noncommutative geometry of quantum groups as well as of finite groups. We also mention applications to models of quantum spacetime. Contents 1. Converting geometry to algebra 1 2. Quantum groups 5 3. Noncommutative differential forms 9 3.1. Differentials on algebras 9 3.2. Differentials on quantum groups 12 4. Noncommutative vector bundles 15 4.1. Projective modules and K-theory 15 4.2. K-theory and cyclic cohomology 18 5. Noncommutative Riemannian geometry 22 5.1. Quantum principal bundles and framing 22 5.2. Bimodule connections 26 Exercises 27 Further Reading 28 28 1. Converting geometry to algebra Noncommutative geometry of any flavour entails replacing a space and geomet- ric structures on it by an algebra with structures on that, inspired by a precise dictionary such as the one shown in the table below. The dictionary is a crutch which we eventually have to discard as we extend the structures on the algebra so as to make sense even when our algebra is noncommutative. The result is a more general conception of geometry that can even be useful when our algebra is in fact commutative. For example, noncommutative differential structures on a finite set correspond to directed graphs with the given set as vertices. Differentials here do not necessarily commute with functions even though the latter commute amongst Based on lectures at the London Taught Course Centre in 2011, revised January 2016. This is a preprint, now published in LTCC Lecture Notes Series, Vol 6 “Analysis and Mathematical Physics” eds. S. Bulllet, T. Fearn and F. Smith. World Scientific, 2017. pp 139–176. 1