What is... non-commutative geometry?

Philipp Schmitt will explain what non-commutative geometry is.

Abstract: The emergence of quantum mechanics, where the position and momentum operators do not commute, motivated the study of non-commutative operator algebras in the first half of the 20th century. About 50 years later, Alain Connes applied non-commutative methods successfully in other areas of Mathematics, for example to study quotient spaces of ill-behaved group actions or leaf spaces of foliations. More recent developments include the non-commutative standard model that predicted a Higgs mass in the correct order of magnitude.

In this talk, I would like to introduce C*-algebras and describe the starting point of non-commutative geometry: A contravariant equivalence between the categories of compact Hausdorff spaces and commutative unital C*-algebras. This equivalence allows to translate geometric questions about spaces into algebraic questions about their function algebras, and vice versa, and motivates the definition of compact non-commutative spaces as (the opposite of the category of) unital C*-algebras. Time permitting, I will outline some developments in non-commutative geometry, and how non-commutative geometry can help to solve problems in the commutative world.

"What is...?" is an accessible and non-technical seminar where speakers explain in one (short) lecture some object or theorem that they think is interesting. There will be drinks and snacks during the talk. See the seminar website for more information.