Seminars on Geometric Structures and Integrable Systems: Integral Geometry in the Three-Dimensional Heisenberg Group: Crofton, Kinematic, and Buffon-Type Formulas for Heisenberg Group（Yen-Chang Huang）

Time： 14:45–16:15 

Place：Faculty of Science Building  #4, Room 501

Speaker：Yen-Chang Huang (National Yang Ming Chiao Tung University)

Title：Integral Geometry in the Three-Dimensional Heisenberg Group: Crofton, Kinematic, and Buffon-Type Formulas for Heisenberg Group

Abstract ：In this talk, I will survey recent joint work with Hung-Lin Chiu and Sin-Hua Lai on classical problems in integral geometry within the three-dimensional Heisenberg group. The first part concerns the geometry of curves, including curvature properties and a Fundamental Theorem of Curves. As an application, we obtain a Crofton-type formula relating surface area to an integral over suitable families of intersecting lines. The second part focuses on the kinematic density of rigid motions in the Heisenberg group. Using this density, we derive a formula for computing the volume of convex domains through integrals involving chord lengths along certain lines. We further apply this framework to a Heisenberg-group analogue of Buffon’s needle problem, determining the probability that a fixed-length needle randomly placed in a convex domain either remains inside the domain or crosses its boundary.

Organizer：Jun-ichi Inoguchi and Shimpei Kobayashi