A pseudoholomorphic curve can be defined as a map from a Riemann surface to a symplectic manifold that satisfies a nonlinear PDE generalizing the standard Cauchy-Riemann equations of complex analysis. In a seminal paper [3] in 1985, Gromov showed that the topology of the space of holomorphic curves in a symplectic manifold can be used to prove many deep results about the manifold's symplectic structure, such as his famous "non-squeezing" theorem. In the early 90's, the work of Floer, Eliashberg [2] and Hofer [4] extended the applications of holomorphic curves into the odd-dimensional cousin of symplectic geometry, known as contact geometry. Contact manifolds arise naturally as regular level surfaces of Hamiltonians in symplectic manifolds, or as boundary components of symplectic cobordisms: the latter furnish a natural setting in which to study punctured holomorphic curves with cylindrical ends, producing a more TQFT-style picture that yields invariants of contact structures.

The goal of this lecture series will be to give an overview of the basic theory and a few of its classic applications in symplectic and contact topology. I will spend the first three lectures surveying the necessary technical apparatus, which involves a mixture of global analysis with methods from the theory of elliptic PDEs. The fourth lecture will then explain a beautiful application due to McDuff [5], in which the presence of a single symplectically embedded sphere in a closed symplectic 4-manifold allows one to deduce what that manifold is up to symplectomorphism. In the final lecture, I will sketch the extension of these methods to study contact manifolds, including applications to the Weinstein conjecture and symplectic fillability.

Most (but not all) of the material we'll discuss is covered in the lecture notes [7], and some further references are listed below. In particular, the book [6] is an invaluable resource for learning the subject.

- [1] F. Bourgeois, Y. Eliashberg, H. Hofer, K. Wysocki and E. Zehnder, "Compactness results in symplectic field theory", Geom. Topol. 7 (2003), 799--888
- [2] Y. Eliashberg, "Filling by holomorphic discs and its applications", Geometry of low-dimensional manifolds, 2 (Durham, 1989), London Math. Soc. Lecture Note Ser., vol. 151, Cambridge Univ. Press, Cambridge, 1990, pp. 45--67.
- [3] M. Gromov, "Pseudoholomorphic curves in symplectic manifolds", Invent. Math. 82 (1985), no. 2, 307--347.
- [4] H. Hofer, "Pseudoholomorphic curves in symplectizations with applications to the Weinstein conjecture in dimension three", Invent. Math. 114 (1993), no. 3, 515--563.
- [5] D. McDuff, "The structure of rational and ruled symplectic 4-manifolds", J. Amer. Math. Soc. 3 (1990), no. 3, 679--712
- [6] D. McDuff and D. Salamon, "J-holomorphic curves and symplectic topology", American Mathematical Society, 2004
- [7] C. Wendl, "Lectures on Holomorphic Curves in Symplectic and Contact Geometry", Preprint http://arxiv.org/abs/1011.1690

A symplectic form on an even-dimensional manifold is a non-degenerate closed 2-form.
These structures naturally appear in cotangent spaces (the phase space of classical mechanics),
or in algebraic geometry, in the framework of projective, or more generally Kahler, manifolds.
Although locally, symplectic structures are very flexible, without invariant, the poor
compatibility between closedness and non-degeneracy tend to rigidify the global structure.
This competition *local flexibility* versus * global rigidity* leaves the structure
halfway between a purely topological object (like a volume form) or a much more rigid one (like a Riemanian structure). The pseudo-holomorphic curves, introduced by Gromov in 1985, are a central tool to understand this competition.

This lecture is an introduction to symplectic topology, that is global problems apearing in symplectic geometry. I will illustrate the above-mentioned competition and the use of pseudo-holomorphic curves on various kind of problems about the topology of symplectic manifolds.