Mathematical Application Research Team Meeting #13

The Mathematical Application Research Team is pleased to welcome Chiyaha Jibiki, who will join the team as an SPDR in April. He will give a talk at this meeting prior to his official start date. Everyone is welcome to join the meeting.

Title: Left-Orders and Dynamics: Applications to Hyperbolic Geometry and Low-Dimensional Topology

Abstract: A left-order on a group is a total order that is invariant under left-multiplication. The concept dates back to the early 20th century, when Dedekind and Hölder characterized the natural order on the real line purely in terms of group actions.
In the 21st century, this theory has evolved significantly through connections with discrete dynamical systems. Currently, there is active research linking geometric structures to left-orders on geometrically significant groups, such as fundamental groups and mapping class groups.
In this talk, I will introduce the framework of left-order theory, starting from the simple perspective of binary operations on infinite sets. I will then provide an overview of the field by presenting, as much as possible, the diverse applications arising from its high versatility. In particular, I will discuss the speaker's recent results concerning the structure of the space of left-orders (comparisons between left-orders) and methods for constructing specific left-orders.
This talk includes joint work with Shuhei Maruyama (Kanazawa University).