Symplectic Topology Student Seminar (Fall 2024)

Topic: Symplectic topology of cotangent bundles

Organizers: Shuhao Li and Frank Zheng

Description: Given a closed smooth manifold \(Q\), does the symplectic topology on \(T^*Q\) determine the smooth topology of \(Q\)? The attempt to answer this question leads to the study of exact embedded Lagrangians in \(T^*Q\) (“nearby Lagrangians”), culminating in the striking result by Abouzaid-Kragh that nearby Lagrangians share the same simple homotopy type as the base manifold. These results followed from (and lead to) developments of categorical symplectic geometry, which also has applications in various other problems.

Schedule

Date Speaker Topic References
Sep. 6 Shuhao Cotangent bundles of exotic spheres, I [Abo12a], notes
Sep. 13 Shuhao Cotangent bundles of exotic spheres, II [Abo12a], notes
Sep. 20 Frank Liouville domains and symplectic cohomology [Gan13] §3, [Rit13], notes
Sep. 27 Ceyhun Lagrangian Floer homology and Fukaya categories [Aur13], [Gan13] §3 and §5, [Rit13], notes
Oct. 4 Yichen \(A^\infty\) categories, I [Gan13] §2.1–2.8, [Kel01]
Oct. 11 Yichen \(A^\infty\) categories, II [Gan13] §2.9–2.13
Oct. 18 Open-closed maps [Gan13] §4–5
Oct. 25 Ceyhun Based loops [Abo12b]
Nov. 1 Generation criterion, I [Abo10]
Nov. 8 Generation criterion, II [Abo11]
Nov. 15 Homotopy equivalence of nearby Lagrangians [Abo12c]

References

[Abo10]
M. Abouzaid. A geometric criterion for generating the Fukaya category. Publ. IHES (2010). arXiv:1001.4593.
[Abo11]
M. Abouzaid. A cotangent fibre generates the Fukaya category. Adv. Math. (2011). arXiv:1003.4449.
[Abo12a]
M. Abouzaid. Framed bordism and Lagrangian embeddings of exotic spheres. Ann. Math. (2012). arXiv:0812.4781.
[Abo12b]
M. Abouzaid. On the wrapped Fukaya category and based loops. J. Symplectic Geom. (2012). arXiv:0907.5606.
[Abo12c]
M. Abouzaid. Nearby Lagrangians with vanishing Maslov class are homotopy equivalent. Invent. Math. (2012). arXiv:1005.0358.
[Aur13]
D. Auroux. A beginner's introduction to Fukaya categories. Preprint. (2013). arXiv:1301.7056.
[Gan13]
S. Ganatra. Symplectic cohomology and duality for the wrapped Fukaya category. Preprint. (2013). arXiv:1304.7312.
[Kel01]
B. Keller. Introduction to A-infinity algebras and modules. Homology Homotopy Appl. (2001). arXiv:math/9910179.
[Rit13]
A. Ritter. Topological quantum field theory structure on symplectic cohomology. J. of Topol. (2013). arXiv:1003.1781.