Alain Connes and NCG

One of the first things that I learned about Non-Commutative Geometry (NCG) was that Alain Connes is the founder of NCG. I  googled his name, and I found out that he has traveled to Iran. I download the PDF of the interview which had been done by Khosroshahi and Eaman Eftekhari at IPM. then by reading the interview, I found out that NCG has got roots in Operator Algebra (OA) and some other fields of mathematics, but mainly it is rooted in OA. I remember in some part of the interview Prof. Khosroshahi  asked Alain Connes: have you chosen a new name for an old branch of math? but everybody knows it is not true.

I was seraching some french university those has got research groups in NCG, and I faced with this amazing website: group de travail et recherche de géométrie non commutative (GDR 2947)

You can find so many usefull information about NCG and its conferences and also job positions by seeing this two websites:

  1. Alain Connes official website
  2. GDR NCG


Convex functions

Whenever we talk about convex functions, an inequality suddenly arise that presents an algebraic definition for a geometric content. I mean we have an image of convexity and from that imagination to this definition there is something to explain. The only thing I never have understood is that “what does it come from?” “Why we define a convex function as something that satisfies in such inequality?” The below image is just an example of a mathematical book (Rudin, complex and real analysis) but you can find it in every mathematical book, and in all cases a very same sentence is repeating even with no extra comma!? Can you tell me why we define a convex function as something that satisfies in that inequality?