Geometry and Dynamics of Domains in the Cotangent Bundles of Tori

In Euclidean spaces, star-shaped domains (also known as Liouville domains) are fundamental objects in modern symplectic geometry. Several important subclasses have been introduced and studied, including dynamically convex domains, geometrically convex domains, and toric domains. These classes are not only interesting in their own right but also exhibit deep interrelations. In this talk, we extend this discussion to (fiberwise) star-shaped domains in the cotangent bundles of tori. In particular, we introduce analogous subclasses in this new setting and explore classical topics such as large-scale geometric properties and embedding problems. This talk is based on joint work with Antong Zhu.