Preface

Summary

My first encounter with differential spaces was in the mid 1980s. At a conference in Toruń, I presented the notion of algebraic reduction of symmetries of a Hamiltonian system. After the lecture, Constantin Piron asked me if my reduced spaces were the differential spaces of Sikorski. I had to admit that I did not know what Sikorski's differential spaces were. To this Piron replied something like “You should be ashamed of yourself! You are a Pole and you do not know what are differential spaces of Sikorski!” During the lunch break I went to the library to consult Sikorski's work. In the afternoon session, I told Piron that the spaces we were dealing with were not the differential spaces of Sikorski. At that time I did not realize that they were differential schemes.

Around the same time, Richard Cushman was working out his examples of singular reduction. I was fascinated by his pictures of reduced spaces with singularities. However, I had not the faintest idea what he was really doing. Since Richard was spending a lot of time in Calgary working on his book with Larry Bates, I had a chance to ask him to explain singular reduction to me. It took me a long time to realize that he was talking the language of differential spaces without being aware of it. From conversations with Richard, it became clear that differential spaces provided a convenient language for the description of the reduction of symmetries for proper actions of symmetry groups.