MAT 645, Spring 2014

MAT 645: Symplectic Topology

Stony Brook Spring 2014

Announcements

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).

Course Instructor

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

Tentative Schedule

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	Spring Break; SCGP Workshop
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	[MS04]: Sections 3.0,3.1 [Z1]: Sections 1-2.3,3
04/01, Tu	Universal moduli space	[MS04]: Section 3.2
04/03, Th	Universal moduli space
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-invariants SCGP 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	Quantum cohomology	[MS]: Sections 11.0,11.1

References

[ISh] S. Ivashkovich and V. Shevchishin, Pseudo-holomorphic curves and envelopes of meromorphy of two-spheres in CP², math.CV/9804014
- 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.

Last modified: May 6, 2014.