Abstract

The notion of a ‘point’ in geometry seems fairly basic, but during the 20th century a generalised and abstracted notion provided much new insight and intuition. More recently, studying some of the mathematical ideas underlying theoretical physics has led mathematicians to consider so-called non-commutative geometry. ‘Spaces’ in this setting have algebras of functions on them which are non-commutative, but there is no actual classical geometric space underlying them! However, the generalised notion of a point does allow one to develop a certain amount of geometric intuition. In this talk, I shall explain the modern point of view on points in algebraic geometry, and explain in some examples how this allows one to study non-commutative algebras in a way that resembles what we might normally consider geometry.