Recommended Papers

Here're some recommended papers for the microseminar on pseudo-holomorphic curves.

1. Pseudo holomorphic curves in symplectic manifolds by M. Gromov.
   • Abstract: This paper set up the basic theory of pseudo-holomorphic curves and gave several results on the topology of symplectic manifolds using pseudo-holomorphic curves. In particular, he proved the so-called "Gromov compactness theorem", which described the moduli space of pseudo-holomorphic curves, and one of the famous corollary in his paper is the "non-squeezing theorem", which showed that the symplectic ball of radius $\lambda$ cannot embed into a symplectic cylinder of the form $B(0,r)\times\mathbb{R}^{2n-2}$ with $r>\lambda$.
2. Symplectic manifolds with contact-type boundaries by D. Mcduff.
   • Abstract: This paper constructed an example of a symplectic four-manifold with disconnected contact-type boundary and, using pseudo-holomorphic curves, gave some conditions on whether the contact-type boundary is connected. As a Corollary, she showed that a symplectically aspherical symplectic manifold with boundary $\mathbb{S}^{2n-1}$ equipped with the standard contact structure is symplectomorphic to $\mathbb{B}^{2n}$ with the standard symplectic structure.
3. Symplectic packings and algebraic geometry by D. Mcduff and L. Polterovich
   • Abstract: In this paper they studied symplectic ball-packing problems. They proved that under various volume conditions there always exists a ball-packing with a given number of balls of prescribed volumes.
4. The structure of rational and ruled symplectic 4-manifolds by D. Mcduff
   • Abstract: In this paper she gave a complete classification of rational and ruled symplectic 4-manifolds. More precisely, she showed that all minimal symplectic 4-manifolds with a rational curve of non-negative self-intersection are symplectomorphic either to the complex projective space $\mathbb{C}P^2$ or a $\mathbb{S}^2$-bundle over a compact surface $M$. As a corollary, she showed that any lens space $L_p$ with $p\geq 1$ admits minimal symplectic fillings, and if $p\neq 4$, the minimal fillings are diffeomorphic, and when $p=4$, $L_4$ has exactly two non-diffeomorphic minimal symplectic fillings.
5. Topology of symplectomorphism groups of rational ruled surfaces by M. Abreu and D. Mcduff
   • Abstract: In this paper, they determined the cohomology ring of the symplectomorphism group of all symplectic rational ruled 4-manifolds, using the classification of Mcduff. The proof is based on a detailed study of the moduli space of pseudo-holomorphic curves on symplectic rational ruled surfaces.
6. Lefschetz pencils and the canonical class for symplectic four-manifolds by S. Donaldson and I. Smith
   • Abstract: In this paper they reproved a theorem of Taubes on the existence of symplectic hypersurfaces of a given symplectic four-manifold Poincaré dual to the canonical class $K_X$. The proof is based on a count of pseudo-holomorphic sections of a bundle with fiber symmetric product. This lead to the Donaldson-Smith invariant and is equivalent to Taubes' Gromov invariant for symplectic 4-manifolds.
7. Symplectic hypersurfaces and transversality in Gromov-Witten theory by Cieliebak and Mohnke
   • Abstract: In this paper they gave an alternative approach to the Gromov-Witten invariant using Donaldson's construction of symplectic hypersurfaces.