Quantised Calculus in One Dimension

Abstract: 

In his 1994 book Noncommutative Geometry, Alain Connes introduced the quantised calculus. This is a means of doing calculus which makes sense in a high degree of generality, and has many properties analogous to the classical notion of "infinitesimal calculus". This thesis is an in-depth study of quantised calculus in two cases: on the real line and on the circle.