Erwin reminded us how excited he was by the fact that string theory provides us with a quantum generalization of the rules of geometry. What does it mean and how does it work?

Well, all previous theories in physics have used the classical manifold geometry (whose definition will be sketched momentarily) as one of the basic prerequisites that the theories had to accept and elaborate upon. This made the classical manifold geometry and its calculations directly relevant for all these theories and the rules of the geometry were therefore rigid dogmas.

In other words, the theories followed the template:

Dear theory, listen, here you have a classical manifold with some shape.And the theories just couldn't do anything else. They were

What can you achieve with this pre-existing shape?

*dependent*on the geometry of a classical manifold. If there were no manifold, there was no physical theory. And if two manifolds were geometrically different, the physical theories on them had to be distinguishable, too.

Before the discovery of special relativity, physics was also dividing spacetime to the absolute time and the space that exists with it. That meant that the "spacetime" as we understood it today had to be basically factorized to \(\RR \times M^3\) where \(\RR\) was the real axis representing time and \(M^3\) was a purely spatial manifold (OK, some time-dependent fibration with a different \(M^3(t)\) at each moment time was sometimes allowed, too). At most, you could have picked time-dependent coordinates on that \(M^3\) in order to celebrate the Galilean relativity.