# Ignasi Mundet i Riera

Ignasi Mundet i Riera (University of Barcelona) visited Laboratory of Algebraic Geometry in

Abstract: Let X be a compact symplectic manifold and let Symp(X) (resp. Ham(X)) denote the group of symplectomorphisms (resp. hamiltonian diffeomorphisms) of X. I will talk about the following results: Theorem 1: Ham (X) is Jordan. More precisely, there exists a constant C (depending only on the topology of X) such that any finite subgroup G of Ham(X) has an abelian subgroup whose index in G is at most C. Theorem 2: if b_1(X)=0 then Theorem 1 holds true replacing Ham(X) by Symp(X). Ham(X) by Symp(X) and "abelian" by "abelian or 2-step nilpotent". In the first part of the seminar I will explain the context of these results. In particular, I will talk on the Jordan property for diffeomorphism groups, putting emphasis on situations where these theorems imply that the finite transformation groups in the symplectic category are much more restricted than in the smooth category. In the second part of the seminar I will explain the main ideas in the proofs of the theorems.

**April 2018**at the invitation of Research Fellow Konstantin Shramov.**On April 13th 2018**Ignasi Mundet i Riera gave a talk "Finite subgroups of Ham and Symp" at the seminar of Laboratory.Abstract: Let X be a compact symplectic manifold and let Symp(X) (resp. Ham(X)) denote the group of symplectomorphisms (resp. hamiltonian diffeomorphisms) of X. I will talk about the following results: Theorem 1: Ham (X) is Jordan. More precisely, there exists a constant C (depending only on the topology of X) such that any finite subgroup G of Ham(X) has an abelian subgroup whose index in G is at most C. Theorem 2: if b_1(X)=0 then Theorem 1 holds true replacing Ham(X) by Symp(X). Ham(X) by Symp(X) and "abelian" by "abelian or 2-step nilpotent". In the first part of the seminar I will explain the context of these results. In particular, I will talk on the Jordan property for diffeomorphism groups, putting emphasis on situations where these theorems imply that the finite transformation groups in the symplectic category are much more restricted than in the smooth category. In the second part of the seminar I will explain the main ideas in the proofs of the theorems.

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