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Microseminar 2:

Pseudo-Holomorphic Curves

Introduction

Let $(X,\omega)$ be a symplectic manifold. A linear-algebraic computation shows that we could always find a compatible almost complex structure $J$ so that $(X,\omega ,J)$ gives a Riemannian metric $g(v,w)=\omega (v,Jw)$. A $J$-holomorphic curve is simply a map $\phi\colon (\Sigma ,j)\to (X,\omega ,J)$, where $(\Sigma ,j)$ is a Riemann surface with a fixed complex structure, whose differential commutes with these chosen almost complex structures, i.e. $\mathrm{d}\phi\circ j =J\circ\mathrm{d}\phi$. The moduli space of such pseudo-holomorphic curves is a symplectic invariant and was applied to the symplectic embedding problem by Gromov [Gro85] in 1985, who showed that the symplectic ball of radius $1$ cannot be symplectically embedded into a symplectic solid cylinder of radius smaller than $1$. This is the so-called "Gromov's Non-squeezing Theorem" and has attracted many attention by symplectic topologists. This technique has been deeply studied since then and now it is the fundation of modern symplectic topology. This microseminar aims to go over the analysis of the moduli space of $J$-holomorphic curves, following Mcduff-Salamon's book [McD12], especially the first five to seven chapters and the appendix. At the end of the seminar, we'll visit some classical papers on symplectic geometry using pseudo-holomorphic curves. The list of papers is shown here.

Schedule

Time Contents Speakers References
May 24thpseudo-holomorphic curvesBoxi Hao[McD12], chapter 2
May 26thpseudo-holomorphic curves, continuedBoxi Hao[McD12], chapter 2
May 31stRegularity, ⅠSiyang Liu[McD12], chapter 3
June 2ndRegularity, ⅡSiyang Liu[McD12], chapter 3
June 7thGeneralized Riemann-Roch Theorem, ⅠDavid O'Connor[McD12], appendix C
June 9thGeneralized Riemann-Roch Theorem, Ⅱ and Mobiüs transformDavid O'Connor[McD12], appendix C and D
June 14thGromov Compactness, ⅠJonathan Michala[McD12], chapter 4
June 16thGromov Compactness, ⅡJonathan Michala[McD12], chapter 4
June 19th - June 23rdKylerrec 2022
June 28thGenus 0 Stable CurvesBoxi Hao[McD12], appendix D
June 30thStable MapsBoxi Hao[McD12], chapter 5
July 8thModuli Space of Stable Maps, ⅠShuhao Li[McD12], chapter 6
July 11th - July 22ndMSRI Floer Homotopy Theory
July 26thModuli Space of Stable Maps, Ⅱ Shuhao Li[McD12], chapter 6
July 28thGromov-Witten Invariants, ⅠBoxi Hao[McD12], chapter 7
Aug. 2ndGromov-Witten Invariants, ⅡBoxi Hao[McD12], chapter 7
Aug. 4thThe Structure of Rational and Ruled Symplectic 4-ManifoldsSiyang Liu[McD90]
Aug. 8th - Aug. 12thSYNC Early Career Workshop

References

  1. [Gro85] Gromov, M. (1985). Pseudo holomorphic curves in symplectic manifolds. Invent. Math., 82(2), 307–347. doi: 10.1007/BF01388806.
  2. [McD90] McDuff, D. (1990). The Structure of Rational and Ruled Symplectic 4-Manifolds on JSTOR. J. Amer. Math. Soc., 3(3), 679–712.
  3. [McD94] McDuff, D., & Polterovich, L. (1994). Symplectic packings and algebraic geometry. Invent. Math., 115(1), 405–429. doi: 10.1007/BF01231766
  4. [Abr00] Abreu, M., & Mcduff, D. (2000). Topology of Symplectomorphism Groups of Rational Ruled Surfaces. J. Amer. Math. Soc., 13(4), 971–1009.
  5. [Hof03] Hofer, H., Wysocki, K., & Zehnder, E. (2003). Finite Energy Foliations of Tight Three-Spheres and Hamiltonian Dynamics. Ann. Of Math., 157(1), 125–255.
  6. [McD12] McDuff, D., & Salamon, D. (2012). J-holomorphic Curves and Symplectic Topology. American Mathematical Society.
  7. [Abb14] Abbas, C. (2014). An Introduction to Compactness Results in Symplectic Field Theory. Springer. doi: 10.1007/978-3-642-31543-5