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Trace polynomials for closed curves on the enlarged modular torus: Monotonicity, positivity, and log-concavity

  • Speaker:Ying Zhang (Soochow University)
  • TIME:January 13, 2023 16:00-17:00 Beijing time (11:00-12:00 Moscow time)
  • LOCATION:online

Recording: https://disk.pku.edu.cn:443/link/A07BDF7E34AC9C1BD87EB37BF6B9E789
Valid Until: 2027-02-28 23:59


Abstract: The modular torus is a once-cusped hyperbolic torus with the maximal order of symmetry. The traces of the simple closed geodesics on the modular torus give the geometric version of the classical Markoff numbers. We study trace polynomials for the closed curves on the enlarged modular torus, with the single variable the enlargement factor. We obtain partial ordering of the polynomials for certain simple closed curves. We obtain positivity of the polynomials for all closed curves. We have observed log-concavity for these polynomials and confirmed log-concavity of the polynomials for the simple closed curves. We also mention the problem to enlarge an arbitrary complete one-holed hyperbolic torus. This is joint work with Xiangfei Li.


Bio: Ying Zhang obtained his Ph.D. from the National University of Singapore. He is currently a professor of mathematics at Soochow University. His research interests include hyperbolic geometry and low-dimensional topology. He has contributed to the geometry of hyperbolic cone-surfaces, generalizations of McShane identity, dynamics and topology of the character varieties of surfaces, trigonometry of 4-dimensional hyperbolic space etc.