# Joint IAS/Princeton/Montreal/Paris/Tel-Aviv Symplectic Geometry Zoominar

## Three 20-min Research Talks

**Julien Dardennes:**The Coarse Distance from Dynamically Convex to Convex:Chaidez and Edtmair have recently found the first examples of dynamically convex domains in $R^4$ that are not symplectomorphic to convex domains, answering a long-standing open question.

In this talk, we present new examples of such domains without referring to Chaidez-Edtmair’s criterion. We also show that these domains are arbitrarily far from the set of symplectically convex domains in $R^4$ with respect to the coarse symplectic Banach-Mazur distance by using an explicit numerical criterion for symplectic non-convexity (joint work with J. Gutt, V. Ramos and J. Zhang).**Arnaud Maret:**Complex Projective Spaces Via Surface Groups Representations:My plan is to explain how complex projective spaces can be identified with components of totally elliptic representations of the fundamental group of a punctured sphere into $PLS(2,R)$. I will explain how this identification realizes the pure mapping class group of the punctured sphere as a subgroup of the group of Hamiltonian diffeomorphisms of the complex projective space.

**Luya Wang:**Deformation Inequivalent Symplectic Structures and Donaldson's Four-six QuestionStudying symplectic structures up to deformation equivalences is a fundamental question in symplectic geometry. Donaldson asked: given two homeomorphic closed symplectic four-manifolds, are they diffeomorphic if and only if their stabilized symplectic six-manifolds, obtained by taking products with $CP^1$ with the standard symplectic form, are deformation equivalent? I will discuss joint work with Amanda Hirschi on showing how deformation inequivalent symplectic forms remain deformation inequivalent when stabilized, under certain algebraic conditions. This gives the first counterexamples to one direction of Donaldson’s “four-six” question and the related Stabilizing Conjecture by Ruan.