Topological persistence in geometry and analysis

Part II: Persistence in symplectic geometry

Reading group - Mathematical Institute - Hillary term 2023/2024
Organised with Luis Scoccola

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Abstract: A central tool of symplectic geometry is Hamiltonian Floer homology, introduced by A. Floer to prove Arnold's conjecture on the minimal number of fixed points of Hamiltonian diffeomorphisms on symplectic manifolds. This homology theory can be regarded as an infinite-dimensional analogue of Morse homology. In particular, Floer homology is naturally filtered, and thus yields a persistence module similar to the classical sublevel-set persistent homology of a Morse function.

In this semester's reading group, we will study the applications of persistence theory to symplectic geometry and Hamiltonian dynamics. Depending on the aspirations of the audience, our journey may include: a proof of the celebrated Gromov's non-squeezing theorem using barcodes, a contribution of Polterovitch and Shelukhin to Palis's say "vector fields generate few diffeomorphisms", or an introduction to C^0 symplectic topology.

Potential references

[1] Topological persistence in geometry and analysis - Polterovitch, Rosen, Samvelyan, Zhang

[2] Persistent modules and Hamiltonian diffeomorphisms - Lecture by Polterovich (video)

[3] Autonomous Hamiltonian flows, Hofer's geometry and persistence modules - Polterovich, Shelukhin

[4] Barcodes and area-preserving homeomorphisms - Le Roux, Seyfaddini, Viterbo

[5] Morse Theory and Floer Homology - Audin, Damian

[6] Lectures on Symplectic Geometry - Cannas da Silva

[7] Floer-type bipersistence modules and rectangle barcodes - Koeda, Orita, Yashiro

Calendar

19/01/2024 - Week 1 - Crash course on differential and symplectic geometry [1, Chap. 7]
Room: L1   Speaker: Luis Scoccola

26/01/2024 - Week 2 - Introduction to Floer homology
Room: L1   Speaker: Dominic Joyce

02/02/2024 - Week 3 - Hamiltonian persistence modules [1, Sec. 8.2] [notes]
Room: L6   Speaker: V.L.

09/02/2024 - Week 4 - Hamiltonian persistence modules [1, Sec. 8.2] [notes]
Room: L1   Speaker: V.L.

16/02/2024 - Week 5 - Sources of Symplectic Geometry: From Dynamics to Linear Algebra
Room: L6   Speaker: Justin Curry
Abstract: In this talk I will briefly review Lagrangian and Hamiltonian mechanics as a way of motivating symplectic geometry. Because most non-linear dynamics must be analyzed locally, Hamiltonian mechanics leads naturally to symplectic linear algebra. After reviewing basic properties of matrix exponentials, I will turn to Floquet theory and the study of time-dependent linear ODEs. The talk will conclude with a stability analysis of the (forced) inverted pendulum due to Mark Levi, which examines how changing the forcing frequency necessitates intersection with a stable semi-analytic subset of the symplectic group of matrices.

23/02/2024 - Week 6 - A proof of Gromov's non-squeezing theorem using barcodes [1, Chap. 9]
Room: L1   Speaker: Uzu Lim

01/03/2024 - Week 7 - Floer-type bipersistence modules and rectangle barcodes [7]
Room: L6   Speaker: V.L.

08/03/2024 - Week 8 - Autonomous Hamiltonian flows, Hofer's geometry and persistence modules [3]
Room: L1   Speaker: David Beers