- New York University
The first part of this extended endeavor developed a new approach to the foundations of geometry, and particularly of topology, called the “Theory of Linear Structures.” The main aim is the analysis of the structure of physical space-time. The purely mathematical part of this project has been completed, and was published in 2014. The second part is a companion volume, applying the new mathematical language to physics. One objective is to express some familiar space-time structures (classical and relativistic) in this new form. Another is to investigate some novel proposals for space-time structure that can be articulated in this mathematical language, namely discrete relativistic space-times. A common mathematical language in which both continuous and discrete geometries can be described allows a close comparison of these different geometries. This language also sheds light on the central role of time and the direction of time in space-time geometry.