At the end of a conference given by Alain Connes in 2000 ([here](http://www.canal-u.tv/video/universite_de_tous_les_savoirs/mathematiques_du_monde_quantique.1033) is a video in French), a member of the audience asked a question. I transcribed and translated it for you below: 


**Audience:** You showed that noncommutative geometry greatly simplifies the physics in a very elegant way, by actually losing a property that seems a bit simple. Similary, if we work with algebras, complex numbers, quaternions and octonions..., are there people investigating some nonassociative geometries?
**Alain Connes:** Ok, I can explain you about this right now...


The video cuts off at that time, even before Alain finishes his explanation, so I post this question here : 



> 
> What is about nonassociative geometry
>  ?
> 
> 
> 


**Sir Michael Atiyah** ([here](https://youtu.be/zCCxOE44M_M) at 58'): Connes' theory is very beautiful, but it only deals with associative algebras, so it can't deals with the octonions, so that's why I think this theory is not quite finished.



What is about nonassociative geometry?
There is progress in this direction by mathematicians here: [Jordan operator algebra](http://en.wikipedia.org/wiki/Jordan_operator_algebra) 


(see also this *Physics* post : [Non-associative operators in Physics](https://physics.stackexchange.com/questions/67266/non-associative-operators-in-physics)) 


**Warning**: the Jordan operator algebras *exchange* the associativity by the commutativity, in fact their product $\circ$, given by $a\_1 \circ a\_2 = \frac{1}{2}(a\_1\*a\_2+a\_2\*a\_1)$, is commutative nonassociative, whereas $\*$ is noncommutative associative. So it's an advanced but it's not really satisfying.  

A satisfying advanced would be with nonassociative **and** noncommutative algebras.


