## Olivetti Club

The correspondence between geometric spaces and commutative algebras is a familiar and basic idea of algebraic geometry, which is motivated by the solutions of polynomial functions. In a similar manner, we see that the introduction of matrices is motivated by the solution of systems of linear equations, where we see non-commutative properties for the first time. In fact, it is quantum physics and matrix non-commutative mathematics that inspires A. Connes' introduction of non-commutative geometry in the 1980s. In this talk, I'll give a short review of matrix mechanics and how it gives rise to groupoids and non-commutative algebras, thus non-commutative spaces. In particular, I'll discuss the example of the non-commutative torus. In the end, I'll try to give a brief introduction to its application in K-theory and if time permits, its relation to other subjects.