Instructor: Ipsita Datta
Email: ipsita DOT datta AT math DOT ethz DOT ch
References:
An Introduction to Contact Topology - H. Geiges
Legendrian contact homology in $\mathbb{R}^3$ - J. Etnyre, L. Ng
Legendrian and Transversal Knots - J. Etnyre
| Week 1 |
| 1. Motivations:
Filling by holomorphic discs and its applications - Eliashberg Witten’s conjecture and Property P - Kronheimer-Mrowka Eliashberg's Proof of Cerf's Theorem - Geiges-Zehmisch Tight and Contact structures - Etnyre A Legendrian surgery presentation of contact 3–manifolds - Ding-Geiges 2. Basics of Contact structures: Definitions (contact structures, contactomorphisms), Examples, Orientability, Darboux's theorem 3. Legendrians: Definitions, Equivalence, 1-jet of a function is Legendrian. |
| Week 2 |
| Front projection, Reidemeister moves in front diagram, Lagrangian projection, Transverse knots |
| Week 3 |
| Thurston-Bennequin number, Rotation number |
| Week 4 |
| Definition of Chekanov-Eliashberg DGA (Algebra, Grading, Differential), Showed that the differential is a finite sum |
| Week 5 |
| Discussion on: the differential has degree 1, the differential squares to 0; Example computation of right handed trefoil |
| Week 6 |
| Invariance of stable tame isotopy class of the dga under Lagendrian isotopy and change of base point, Definition of Legendrian Contact Homology (LCH), Discussion of strengths and limitations of the LCH,
Definition of dga in the symplectization (Definition of symplectization, almost complex structure, grading, moduli spaces of holomorphic discs asymptotic to Reeb chords) |
| Week 6 |
| Definition of dga in the symplectization (differential)
Overview of other definitions of combinatorial LCH in other manifolds Augmentations (definition, implications of existence) Definition of linearized (Legendrian) contact homology |
PhD students may edit/create one of the following Wikipedia pages for credit:
1. Legendrian Knots
2. Transverse Knots
3. Chekanov-Eliashberg dga
4. Relative Contact Homology
5. Augmentations
6. Thurston-Bennequin number, Rotation number, Mountain Peaks
7. Rulings