Exploring Geometry Through Geodesic Currents: Congratulations to Jiajun Shi

Congratulations to Jiajun Shi on the successful completion of his dissertation!

In his PhD thesis “Geodesic Currents, Rigidity, and Finsler Metrics on Flat Surfaces”, he explores how different geometries on the same surface can be compared using geodesic currents — mathematical objects that capture the length information of closed curves and open new ways to study geometric structures.

Jiajun investigates length functions and geodesic currents for singular flat and compatible Finsler geometries on closed surfaces. In the first part, he studies flat geometries arising from holomorphic quadratic differentials. By comparing the lengths of simple closed curves, he introduces an asymmetric length-ratio function and prove a volume-rigidity theorem: if the length function of one unit-area flat structure dominates that of another for every simple closed curve, then the two structures must, in fact, determine the same marked flat metric.

The second part focuses on compatible Finsler geometries on translation surfaces. Here, Jiajun shows that certain straight geodesics minimize Finsler length within their relative homotopy classes. A central result is the construction of geodesic currents associated with measured multi-foliations. These currents provide the building blocks for Liouville currents of compatible Finsler metrics. In particular, for every such metric, Jiajun constructs a current whose intersection with any closed curve recovers the curve's Finsler length.

Together, these results demonstrate how geodesic currents can provide a powerful framework for encoding and comparing geometric information on surfaces.

Jiajun Shi completed his Bachelor's degree at the University of the Chinese Academy of Sciences and his Master's degree at the University of Bonn. He initially began his PhD at the University of Heidelberg before moving to the Max Planck Institute for Mathematics in the Sciences and Leipzig University, where he completed his dissertation.

Jiajun will soon begin a postdoctoral position at Seoul National University in South Korea.

We warmly congratulate him on this achievement and wish him all the best for the next chapter of his research and for his future in Seoul!