Mirzakhani’s Curve Counting and Geodesic Currents

Viveka Erlandsson is Lecturer of Mathematics at University of Bristol, UK. Her research interests include hyperbolic geometry, low-dimensional topology, and Teichmüller theory.Juan Souto is Directeur de Recherche at the CNRS, Université de Rennes 1, France. His research interests include hyperbolic geometry, low-dimensional topology, and mapping class groups.

“The book constitutes a great scientific homage to the work of M. Mirzakhani. Using a number of tools which they develop carefully, the authors give a new general proof of her curve counting theorem, valid for both simple and non-simple curves.” (José Javier Etayo, zbMATH 1514.32003, 2023)“The book is written with enthusiasm and delightful touches of informality. It includes beautiful illustrations by Hugo Parlier. The authors delight in Mirzakhani's ideas and the surprising applications, and they take pleasure in making the mathematics accessible. They keep the book self-contained … .” (Boris Hasselblatt, Mathematical Reviews, July, 2023)

• 1. Introduction.- 2. Read Me.- 3. Geodesic Currents.- 4. Train Tracks.- 5. Radallas.- 6. Subconvergence of Measures.- 7. Approximating the Thurston Measure.- 8. The Main Theorem.- 9. Counting Curves.- 10. Counting Square Tiled Surfaces.- 11. Statistics of Simple Curves.- 12. Smörgåsbord.- A. Radon Measures.- B. Computing Thurston Volumes.- References.- Index.