Math 697: Symplectic and Combinatorial Methods in Low-Dimensional Topology (Winter 2021)

Math 697: Symplectic and Combinatorial Methods in Low-Dimensional Topology

Winter 2021

Time and location: Mondays, Wednesdays and Fridays, at 2:00-2:50pm, on Zoom

Instructor: Linh Truong (tlinh@umich.edu)

Office hours: Fridays after class, and by appointment, on Zoom

Course Information: Syllabus

Schedule

Date Topics References
1/20 Overview and goals
1/22 Morse theory [Mil] p. 1-39
1/25 Morse homology [McD] §1.1-1.2
[Hut] §2.1-2.4
1/27 Morse homology: examples and applications [AD] §3, §4
1/29 Symplectic Manifolds, almost complex structures [AD] §5
[Aur] §1
2/3 Lagrangian Floer homology [Aur] §1
2/5 Heegaard diagrams [GS] p. 112-115
[OS-5] §2
2/8 Heegaard moves [OS-5] §2
2/10 Symmetric product of surfaces [OS-1] §2.1-2.4
2/12 Disks in symmetric product, totally real tori [OS-5] §4, §5
2/15 Heegaard Floer homology: definition [OS-5] §7
2/19 Turaev reformulation of spin^c structures [OS-5] §6
2/22 Heegaard Floer homology: Invariance I [OS-1]
2/24 university well-being break (no class)
2/26 Invariance II: Holomorphic triangles and rectangles [OS-1]
3/1 Heegaard Floer variants: minus, plus, infinity
Surgery Exact Triangle: statement and applications
[OS-5] §8
[OS-6] §1
3/3 Surgery Exact Triangle: Proof I [OS-6] §2
3/5 Surgery Exact Triangle: Proof II [OS-6] §2
3/8 Applications: L-spaces and Branched Double Covers [OS-6] §1
3/10 Cobordism maps [OS-6] §3
3/12 Absolute gradings, d-invariants, and computations [OS-8]
3/15 Examples, nice diagrams and combinatorial HF [OS-7] §2.2
[SW]
3/17 Khovanov homology [Kho]
3/19 Lee homology [Lee]
3/22 no class
3/24 no class
3/26 student presentation (Rasmussen s-invariant) [R]
3/29 Proof of Rasmussen's slice genus bound, part I [R]
3/31 Proof of Rasmussen's slice genus bound, part II [R]
4/2 HF of branched double covers [OS-covers]
4/5 Closed 4-manifold invariant [OS-4]
4/7 student presentation: L-spaces and taut foliations [OS-genus]
4/9 Knot Floer homology: Definition [Hom]
4/12 Knot Floer homology: Examples
Definition of the tau invariant
[Hom]
4/14 Heegaard Floer homology of knot surgery and computations [Hom]
4/16 student presentation: A Khovanov Homotopy Type [LS]
4/19 Grid homology [Grid Homology]
[example]
4/21 Knot concordance group homomorphisms [DHST]