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Seminars

Number Theory Seminar

Monodromy in Arithmetic Dynamics

2026/09/18 (Fri.) 10:30~11:45
  • Date : 2026/09/18 (Fri.) 10:30~11:45
  • Location : Seminar Room 617, Institute of Mathematics (NTU Campus)
  • Speaker : Ophelia Adams (Academia Sinica)
  • Organizer : Julie Tzu-Yueh Wang (AS)
  • Abstract :
    In this talk, I'll discuss how monodromy (i.e. Galois theory) manifests in arithmetic dynamics. Given a (separable) rational map over a field K, one can construct a "profinite iterated monodromy group". This group has a self-similar, or fractal, structure which reflects the dynamics, and is further equipped with an outer action of the absolute Galois group of K. It is a natural analogue of the Tate module of an abelian variety, but unlike Tate modules, we do not yet have a full understanding of their structure, nor of the associated Galois action. Nevertheless, a great deal of progress has been made recently. I'll introduce self-similar groups and some of their main properties, then discuss my work, joint with Trevor Hyde, on unicritical polynomials, where the groups and the Galois action have been completely determined, and then some of my more recent results constraining the group-theoretic and fractal structure of the groups and associated galois actions without necessarily determining the profinite iterated monodromy group. At the end, I'll comment on some ongoing work, and share a few questions and conjectures.
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