Dirty hands seminar (Fall/Winter 2026)
The goal of this seminar is to become familiar with the essential techniques of closed pseudoholomorphic curves in closed symplectic 4-manifolds and to understand how these techniques are used to prove foundational theorems of symplectic topology due to Gromov and McDuff, from a somewhat more modern point of view.
In particular, we want to understand how holomorphic curves can be used to recognize rational and ruled symplectic 4-manifolds, the behaviour of exceptional spheres under deformation, and the symplectic analogue of Castelnuovo's criterion.
We will assume basic knowledge of symplectic geometry and review the basic theory of pseudoholomorphic curves, but keep most of the analysis in the background. Hopefully, the material covered will not only provide the necessary background for more advanced research topics in symplectic 4-manifolds but also provide a lot of “moral” motivation for many techniques in other areas of symplectic topology, such as Floer theory and SFT.
Our main reference will be Wendl's book on Holomorphic Curves in Low Dimensions and here is our preliminary schedule. For the original persepective please see the original paper The structure of rational and ruled symplectic 4-manifolds and its Erratum by D. McDuff. See also the survey article J-curves and the classification of rational and ruled symplectic 4-manifolds by F. Lalonde and D. McDuff.
If you wish to participate in the seminar or have trouble accessing the literature please don't hesitate to contact Vincent Humilière, Russell Avdek, Nikolas Adaloglou, Johannes Hauber or Luis Zegarra.
Schedule
Topic/Abstract: Basic facts and examples of symplectic 4-manifolds. Overview of the theorems that will be covered.
Literature/Background: Chapter 1 of Holomorphic Curves in Low Dimensions
Topic/Abstract: Basic notions of closed pseudo-holomorphic curves and getting acquainted with automatic transversality in dimension 4.
Topic/Abstract: Discussion of the complex blow-up and how to perform it in the symplectic category (mostly moral, no technical proofs). Introduction to basic notions of Lefschetz fibrations/pencils.
Links
Other seminars of interest in Paris include, amongst others, Symplectix, Séminaire de Topologie (IMJ-PRG) , and the Symplectic Zoominar (online).