Research Projects - William Chuang

Research Projects

Simple Geodesics on Hyperbolic Surfaces: Theory and Applications

My recent note on simple geodesics explores various techniques for understanding geodesics on hyperbolic surfaces. For further details, see the full document Simple Geodesics on Hyperbolic Surfaces: Theory and Applications.

In this survey, we explore the fascinating interplay between number theory, geometry, and dynamical systems. To set the stage, we begin by recalling the classical Prime Number Theorem which describes the asymptotic distribution of prime numbers. This fundamental result motivates analogous asymptotic counting problems in geometry, such as the enumeration of closed geodesics on hyperbolic surfaces.

Several key works form the backbone of our approach. Mirzakhani's groundbreaking study established deep connections between the asymptotic growth of simple closed geodesics on hyperbolic surfaces and the geometry of moduli spaces, while Arana-Herrera provides a modern ergodic-theoretic perspective on counting problems ranging from primitive integer points to simple closed curves. Foundational background on surface topology and mapping class groups is supplied by Farb and Margalit’s A Primer on Mapping Class Groups as well as Martelli’s An Introduction to Geometric Topology. Comprehensive treatments of hyperbolic geometry and its spectral theory are available in Ratcliffe’s Foundations of Hyperbolic Manifolds, Borthwick’s Spectral Theory of Infinite-Area Hyperbolic Surfaces, and Dal’Bo’s work on geodesic and horocyclic trajectories. For additional background in measure theory and the geometry of numbers, see Cassels and Einsiedler--Ward.

References

Dal'Bo, F. (2011). Geodesic and horocyclic trajectories. Springer-Verlag London, Ltd. DOI: 10.1007/978-0-85729-073-1.
Ratcliffe, John. G. (2019). Foundations of Hyperbolic Manifolds. Springer. DOI: 10.1007/978-3-030-31597-9.
Borthwick, D. (2016). Spectral Theory of Infinite-Area Hyperbolic Surfaces. Birkhäuser/Springer. DOI: 10.1007/978-3-319-33877-4.
Martelli, B. (2016). An Introduction to Geometric Topology. arXiv:1610.02592.
Farb, B., & Margalit, D. (2012). A Primer on Mapping Class Groups. Princeton University Press.
Mirzakhani, M. (2004). Simple geodesics on hyperbolic surfaces and the volume of the moduli space of curves. Harvard University.
Arana-Herrera, F. (2022). Counting problems from the viewpoint of ergodic theory: from primitive integer points to simple closed curves. arXiv:2202.04156.