Groups in Geometry and Topology

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AbstractAward: DMS 1604241, Principal Investigator: Michael KapovichGroups appear naturally as symmetries of geometric and physical objects, like wall-patterns, minerals, snowflakes and, ultimately, the entire universe. This project studies relation between algebraic properties of groups and geometry of spaces for which groups appear as symmetries. Another part of the project deals with geometry of mechanical devices such as bar-and-joint mechanisms and gear trains. The project will involve rigorous mathematical definitions of such devices and analysis of possible motion spaces (configuration spaces) of the devices.This project is a continuation of the principal investigator's research of previous years in the areas of geometry, topology, geometric group theory and theory of mechanical devices. The subjects of the planned research revolve around geometry of group actions on various spaces, geometry of buildings, interactions of algebraic geometry, hyperbolic geometry and geometric group theory. Another part of the project deals with geometry of mechanical devices such as gear trains. The goal here is to give mathematical definition of such devices, generalizing the definition of a mechanical linkage as a finite metric graph and proving a universality theorem for configuration spaces of these devices, namely that any algebraic partial differential relation can be realized as the configuration space of some device.