## MAT 645: Pseudoholomorphic Curves
## Stony Brook Spring 2022 |
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**[Z4]**, 05/30/22: Extensive updates described in Section 1.1

**[Z1]**, 04/24/22: Section 6.2 added (bubbling on thin necks)

03/26/22: Proposition 6.2 added (bubbling with marked points colliding)

03/15/21: Section 6.1 added

03/14/21: Section 2.2 and Examples 2.7,2.8 added

03/10/21: statement and proof of Lemma 5.4 sharpened;
Section 5.3 redone, justifying the termination in finitely many steps carefully.

Date |
Topic | Read |

01/24, M | Introduction lecture notes |
[MS04]: Preface, Chapter 1 [MS98]: Introduction, Chapter 1 [Z1]: Section 1 |

01/26, W | Energy of J-holomorphic curves | [Z1]: Sections 2,4.1 [MS04]: Sections 2.0-2.2,4.0,4.1 |

01/31, M | Mean Value Inequality | [Z1]: Sections 4.1 [MS04]: Section 4.3 |

02/02, W | Removal of Singularities | [Z1]: Section 5.1 [MS04]: Sections 4.1,4.5 |

02/07, M | Carleman Similarity Principle | [Z1]: Sections 3.1,3.2 [MS04]: Section 2.3 |

02/09, W | The Monotonicity Lemma | [Z1]: Section 3.3 |

02/14, M | Bubbling | [Z1]: Section 5.2 [MS04]: Sections 4.2,4.6 |

02/16, W | Isoperimetric Inequality | [Z1]: Section 4.2 [MS04]: Section 4.4 |

02/21, M | Energy distribution | [Z1]: Section 4.4 [MS04]: Section 4.7 |

02/23, W | Gromov's Convergence | [Z1]: Section 5.3 [MS04]: Section 4.7 |

02/28, M | Stable curves | [Z1]: Section 1.1 [Z6]: Section 4.1 [MS04]: Sections 5.1-5.4 |

03/02, W | Sequences and topologies | [Z1]: Section 5.5 [MS04]: Section 5.6 |

03/07, M | Moduli spaces of stable curves | [MS04]: Appendix D |

03/09, W | Convergence of stable maps | [Z1]: Section 1.2 [MS04]: Sections 5.0-5.3 |

03/21, M | Topology of moduli space of stable maps | |

03/23, W | Implicit Function Theorem for Banach manifolds | [MS04]: Appendix A.3 |

03/28, M | Linearization of dbar-operator and examples | [Z1]: Sections C.3-4 [MS04]: Section 3.1 |

03/30, W | Expansion of dbar-operator | |

04/04, M | Analytic properties of linearization of dbar-operator | |

04/06, W | Serre Duality | [ISh]: Section 2 |

04/11, M | Transversality at reduced maps | [MS04]: Section 3.2 [Z2]: Section 3 |

04/13, W | Universal moduli space | |

04/18, M | The determinant line bundle | [Z4] |

04/20, W | Pseudocycles and integral homology | [Z3] |

04/25, M | GW-invariants for semipositive manifolds | [MS04]: Sections 6.4,7.0,7.1 |

04/27, W | Gromov's nonsqueezing theorem | [MS04]: Section 9.3 |

05/02, M | Quantum cohomology
and enumerative geometry by Shuhao and Spencer |
[MS]: Sections 11.0,11.1; [Z6] |

05/04, W |

- [ISh] S. Ivashkovich and V. Shevchishin,
*Pseudo-holomorphic curves and envelopes of meromorphy of two-spheres in CP*, math.CV/9804014^{2}- Lemma 2.3.2 is Serre duality for real Cauchy-Riemann operators.

- [LT] J.~Li and G.~Tian,
*Virtual moduli cycles and Gromov-Witten invariants of general symplectic manifolds*, math.AG/9608032- Section 3 contains gluing for pseudoholomorphic maps without exponentially weighted norms on long cylinders.

- [MS98] D. McDuff and D. Salamon,
*Introduction to Symplectic Topology*, Oxford Mathematical Monographs, 1998/2017- Introduction and Chapter 1: an overiew of symplectic topology and its connections with classical mechanics; just read through this without worrying about the details.

- [MS04] D. McDuff and D. Salamon,
*J-Holomorphic Curves and Symplectic Topology*, AMS Colloquium Publications 52, 2004/2012/2017- Preface and Chapter 1: a thorough overview of the book, giving a flavor of J-holomorphic curves techniques; just read through this without worrying about the details.
- Chapter 2 establishes basic properties of J-holomorphic maps, some of which depend on the compatibility with a symplectic form and some do not.
- Chapter 4 concerns the
**key**rigidity and compactness type properties of J-holomorphic maps that depend heavily on the compatibility with a symplectic form (though some preliminary statements do not depend on the compactability). In a sense, this chapter establishes the foundations of the theory of pseudo-holomorphic maps; the remainder of the book is about re-packaging them for specific applications.

- [Sh] V. Shevchishin,
*Pseudoholomorphic curves and the symplectic isotopy problem*, math/0010262- Lemma 2.4.1 extends the usual twisting up/down construction of complex geometry to real Cauchy-Riemann operators.

- [TZ] M. Tehrani and A. Zinger,
*Absolute vs. relative invariants*- This compares the two invariants when one might hope for them to be equal.

- [Z1] A. Zinger,
Notes on J-holomorphic maps
- More systematic version of Chapters 2 and 4 of [MS04].

- [Z2] A. Zinger,
Transversality for J-holomorphic maps:
a complex-geometric perspective, in progress
- More systematic analogue of Chapters 3 and 6 of [MS04].

- [Z3] A. Zinger,
*Basic Riemannian geometry and Sobolev estimates used in symplectic topology*- This contains analytic estimates people in symplectic topology regularly use, but do not like bothering with their details too much.

- [Z4] A. Zinger,
*The determinant line bundle for Fredholm operators: construction, properties, and classification*- This fully constructs a compatible system of determinant line bundles and describes the choices that specify such a system.

- [Z5] A. Zinger,
*Pseudocycles and integral homology*- This shows that pseudocycles and integral homology classes are essentially the same thing.

- [Z6] A. Zinger,
*Counting plane rational curves: old and new approaches*- Section 4 contains a proof of Kontsevich's recursion for plane rational curves, excluding the gluing part.

This page is maintained by Aleksey Zinger.