Time：14:00-16:00, Friday, November 28 2025
Venue：E4-233
Speaker：Matteo Ruggiero, Université Paris Cité
Biography：Professor Matteo Ruggiero obtained his PhD degree from Scuola Normale Superiore di Pisa, Italy. Then he was a post-doc of the FMJH at the Ecole Polytechnique, France. Now he is a Maître de conférences at Université Paris Cité, France. Professor Ruggiero made fundamental contributions to Local and semi-local dynamics of analytic maps, complex geometry and compactification of orbit spaces, birational aspects of singularities of analytic spaces and maps, links with toric, tropical and non-archimedean geometry, and arithmetic dynamics aspects of polynomial endomorphisms.
Title：On the Dynamical Manin Mumford problem for polynomial endomorphisms of the plane
Abstract：The Dynamical Manin-Mumford problem is a dynamical question inspired by classical results from arithmetic geometry.In the setting of regular polynomial endomorphisms of C^2 of degree d>=2, it tasks to determine whether an algebraic curve containing infinitely many preperiodic points must be itself preperiodic.
In a work in collaboration with Romain Dujardin and Charles Favre, we prove this conclusion to hold, provided that: (★) the dynamics at infinity has no superattracting periodic points.
The proof is an interesting blend of techniques from arithmetic geometry and complex/non-archimedean dynamics.
Condition (★) is crucial for our approach: it ensures that we can work near the Julia set at infinity at some place, and that the set W where orbits converge at super-exponential speed d at a fixed point at infinity is a (invariant) curve.
If time allows, I will also present our recent results about the properties of W in the superattracting case.
