The area of research of this project is the topology of closed symplectic manifolds of dimension 4. A symplectic 4-manifold is a smooth manifold endowed with a closed and non-degenerate (symplectic) 2-form. Symplectic manifolds arise naturally in various areas of mathematics, and their study has relevant consequences also in physics and engineering. Kaehler manifolds are natural examples of such manifolds, and until 30 years ago it was an open question whether other examples exist. Since then, wealth of other constructions have appeared, and important results (often arising from the interaction of physics and mathematics) have started to unveil the richness of this class of manifolds. This project investigates the properties of certain natural submanifolds of a symplectic $4$-manifold, namely lagrangian submanifolds (where the restriction of the symplectic form vanishes) and symplectic submanifolds (where the restriction of the symplectic form is again symplectic). The common theme underlying the various parts of the project is the use of a technique, developed by R.Fintushel and R.Stern, that allows one to construct symplectic $4$-manifolds and study their submanifolds. The first goal is to study symplectic representatives of the canonical homology class of a symplectic manifold; it is known, by a result of C.Taubes, that such representatives exist, but little is known about their topology. The author wants to show that there exist manifolds whose canonical class admit several representatives distinguished by the number of components. This would lead to a better understanding of how to define invariants of a symplectic manifold starting from the study of its canonical. The second goal is to analyze the existence of lagrangian submanifolds that are homologous but not isotopic, i.e. connected by a sequence of embeddings. Results of this kind have been obtained in the past for symplectic submanifolds, but little is known in the lagrangian case. The third goal is to study the relation between certain symplectic invariants (recently introduced by P.Seidel for a particular class of symplectic manifolds) and the Seiberg-Witten invariants. Other applications of these techniques to other problems of symplectic and smooth topology of 4-manifolds are expected.
Symplectic manifolds of dimension 4 lie at the intersection between two areas of research. Both the first, the topology of low dimensional manifolds, and the second, symplectic topology, are traditional areas of investigations that in the last 20 years have experienced a tremendous progress. This progress is due to new and exciting interaction with theoretical physics, that has served as a source of models and conjectures, and a renewed relation of topology with other areas of mathematics that has produced in both directions fruitful applications and consequences. Topology in dimension 4 attracts great interest from both mathematics and physics as it investigates the dimension of the space-time; symplectic topology involves the properties of certain manifolds that appear naturally in the description of problems arising from physics and engineering. A growing number of researchers, with original and stimulating approaches, is increasing in a spectacular way our comprehension on the subject, verifying or disproving old conjectures, and suggesting new relations with other areas of mathematics and physics. The investigator wants to continue his research in this field, using techniques of topology that originate from classical results in theory of knots, in conjunction with modern tools as the invariants of 4-manifolds introduced by the physicists Seiberg and Witten. In particular, the aim of the research is to better understand the submanifolds of a symplectic 4-manifold.
