Date/Time : 20 May 2026 14:00
Venue : S17 #04-04 (Seminar Room 3)
Speaker : Doan Nhat Minh, Vietnam Academy of Science and Technology

Simple closed geodesics on a once-punctured torus are naturally organized by primitive homology classes, and their lengths can be studied through the McShane-Rivin norm ball. In this talk, I will explain how estimates on the normal turn of the boundary of this norm ball give a new polylogarithmic bound for length multiplicities: On every complete finite-area hyperbolic once-punctured torus X, the number of simple closed geodesics of any fixed length L is at most C_X(log L)^2. For the modular torus, this translates into a bound of order (log(log(3m)))^2 for Markoff fibers, improving previously known logarithmic estimates. If time permits, I will also discuss how the same normal-turn estimates give some new quantitative information about the local geometry of McShane-Rivin norm balls. This is joint work with Xiaobin Li and Van Nguyen.

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