Скачать или смотреть Quiver Geometry with Tali Beynon

Continuous geometry serves as a substrate for a deeper understanding of many topics in science and engineering. Understanding how families of objects form an abstract space can allow us to exploit familiar intuitions as well as powerful mathematical tools. An incomplete list of such abstract spaces includes the phase spaces of physical systems, the parameter spaces of learned models, the state spaces of systems under active control, spaces of continuous distributions in information geometry, and configuration spaces of mechanical robots.

Continuous geometry is also the oldest playground of rational thought and argument, dating back to Euclid. Over the last two hundred years, sophisticated mathematical frameworks have been developed to organize in a unified way the geometrical phenomena that occur in multiple different contexts. However, the same maturity of effort does not exist for discrete spaces, for which the underlying mathematical descriptions, the graph and the hypergraph, are relatively recent inventions. This missing body of theory hobbles efforts to develop a deeper understanding of these discrete spaces where they occur, and apply standardized tools to understand them.

I'll try to present some of my efforts in this direction, using the very flexible formalism of rewriting systems to show how causality and geometry relate, and how seemingly continuous notions like curvature can have a completely discrete origin and interpretation. I'll also outline where I think machine learning, program synthesis, and other disciplines might enjoy benefits from these kinds of tools as they mature.