“There are spaces that are more complicated (than ordinary geometry) because they are spaces where you not only look at the points of sets but you also look at the relations between the points. These new sets with relations can be described by Algebras, but these Algebras are non commutative.”
“After working on this for 10 years, I developed in full a new geometry called non-commutative geometry, in which one refines all the usual geometrical ideas and applies them to the new spaces. These spaces have amazing properties that generate their own time. Not only do they have their own time, but they have features which enable you to cool them down or warm them up. You can do thermodynamics with them. There is an entirely new part of geometry and algebra that is related to these new spaces, called non-commutative geometry, on which I have been working essentially all my life.”
– Extract: “Mathematicians – an outer view of the inner world”
The video series (Part 1 to 3) in French by Alain Connes.