## MAT 645: Symplectic Topology
## Stony Brook Spring 2014 |
---|

Because of the related activities at the Simons Center, the class will take place on Tuesdays, 9:55-11:05, and Thursdays, 9:55-11:25.

Here is general information about the course (updated 01/26).

*Name:*
Aleksey Zinger
*E-mail:* azinger@math
*Phone:* 632-8288

*Office:* Math Tower 3-111
*Office Hours:* W 9-10 in P-143, 10-12 on 3-111

Date |
Topic | Read |

01/28, Tu | Introduction | [MS04]: Preface, Chapter 1 [MS98]: Introduction, Chapter 1 |

01/30, Th | Energy of J-holomorphic curves | [MS04]: Sections 2.0-2.2,4.0,4.1 Notes |

02/04, Tu | Mean Value Inequality | [MS04]: Section 4.3; Notes |

02/06, Th | Removal of Singularities: Continuity | [MS04]: Sections 2.3,4.5; Notes |

02/11, Tu | Local Structure of J-Holomorphic Maps | [MS04]: Section 2.4; Notes |

02/13, Th | snow day | |

02/18, Tu | Moduli Spaces and Invariants | [MS04]: Sections 4.0-4.2 |

02/20, Th | The Monotonicity Lemma | Notes |

02/25, Tu | Moduli Spaces and Invariants | [MS04]: Sections 4.0-4.2 |

02/27, Th | The Monotonicity Lemma | [MS04]: Section 4.4,4.7; Notes |

03/04, Tu | Gromov's Convergence | [MS04]: Section 4.7 |

03/06, Th | Stable Maps | [MS04]: Sections 5.0-5.3 |

03/11, Tu | Gromov-Witten Invariants | [MS04]: Sections 6.6,7.0,7.1 |

03/13, Th | Relative Stable Maps | |

03/18, Tu |
Spring Break;
SCGP Workshop
| |

03/20, Th | ||

03/25, Tu | Linearizations of Bundle Sections | [MS04]: Sections 3.0,3.1 [Z1]: Sections 1-2.3,3 |

03/27, Th | Linearizations of dbar-operator | |

04/01, Tu | Universal moduli space | [MS04]: Section 3.2 |

04/03, Th | ||

04/08, Tu | Regularity of simple maps | |

04/10, Th | Simple vs. somewhere injective maps | [MS04]: Section 2.5 |

04/15, Tu | The determinant line bundle | [Z2] |

04/17, Th | Simple nodal maps | [MS04]: Sections 6.0-6.3 |

04/22, Tu | GW-invariants for semipositive manifolds | [MS04]: Sections 6.4,7.0,7.1 [Sh]: Lemma 2.4.1; [ISh]: Lemma 2.3.2 |

04/24, Th | Pseudocycles | [MS04]: Section 6.5; [Z3] |

04/29, Tu | Absolute vs. relative GW-invariantsSCGP 313 @ 11:15 and 2:30 | [TZ] |

05/01, Th | Enumeration of plane rational curves | [MS04]: Sections 7.4,7.5 [Z4]: Section 4 |

05/06, Tu | Quantum cohomology | [MS]: Sections 11.0,11.1 |

05/08, Th |

- [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.

- [MS98] D. McDuff and D. Salamon,
*Introduction to Symplectic Topology*, Oxford Mathematical Monographs, 1998- 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- 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,
*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.

- [Z2] 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.

- [Z3] A. Zinger,
*Pseudocycles and integral homology*- This shows that pseudocycles and integral homology classes are essentially the same thing.

- [Z4] 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.