Title: Differential calculus on unusual spaces

Abstract:

I will present two elementary structures that provide generalizations of differential calculus
beyond Cartesian spaces (R^n):
diffeology (a-la-Souriau), and differential structures (a-la-Sikorski).

A diffeology on a set declares which maps from Cartesian spaces to the set are smooth;
a differential structure declares which real-valued functions on the set are smooth.

In spite of their simplicity, these structures often capture surprisingly rich information.
They also lead to intriguing open questions. For example, for a linear flow of irrational slope on the 2-torus,
the topology of its quotient is trivial, but the diffeology of its quotient determines the slope
(up to an automorphism of the torus). On the other hand, we do not know if the differential structure
(or even the topology) of a trajectory of this flow also determines the slope.


Livestream/Recording Link: https://huji.cloud.panopto.eu/Panopto/Pages/Viewer.aspx?id=67849a77-6333-4519-8f64-b11d006d1204