On the semantics of non-commutative geometry and exotic summation formulas.

Abtract. The well-known duality of classical algebraic geometry between affine varieties  and their co-ordinate rings has a perfect analogue in the theory of
commutative C^*-algebras, which can be seen by the Gel'fand-Naimark theorem as the algebras of continuous complex-valued functions on a compact Hausdorff space. 
We interpret this as the Syntax-Semantics duality. 
In modern geometry and physics one deals with much more advanced generalisations of co-ordinate algebras, such as schemes, stacks and non-commutative 
C^*-algebras, where a geometric counterpart is no longer readily available and in many cases is believed impossible.
I will discuss some results of a model-theoretic project which challenges this point of view. This will be illustrated by an application calculating classically non-convergent infinite sum.