Introduction to Symplectic Topology

Introduction

This is a semester-long Master course at the University of Bonn. This course aims at providing students with a sense of symplectic topology: how it comes from and gradually becomes an interesting and connects to various areas of mathematics. We will roughly follow the historical timeline, starting with the "main branch" of symplectic topology: from Arnold's classical mechanics perspective to Floer's revolutionary theory that is regarded as a common ground of symplectic topology. Then we will select and discuss some other selected branches of symplectic topology, particularly the study of flexibility, Viterbo's isomorphism and the resolution of nearby Lagrangian conjecture.

Prerequisites.

The students are required to have a basic understanding of differential geometry, particularly about smooth manifolds, differential forms, and Lie groups. Some knowledge of algebraic topology are also expected, including familiarity with homology and cohomology theories. Some basic knowledge of complex geometry will be helpful, but not required. Knowledges about partial differential equations, particularly on elliptic PDEs will also be very helpful.

Assessment.

The course will have no problem sets or assignments. The assessment will be based on an oral exam at the end of the course. Students registering for the oral exam should take one of the following two options:

1. General Exam. Students will be asked to solve some problems that are related to the course materials (listed below).
2. Specified Exam. Students will be asked to explain and understand one of the topics listed below as "problems". This usually involves paper reading but the exam will be mostly based on the topic and contents in the lecture related to the topic. Selecting this option, students are required to inform the examiner at least one week before the exam.

Schedule of the Course

Lecture will start 14 Apr., from 14 to 16 on Tues and Thurs. Below is a table of topics covered per lecture.

Notes are available here.

Oral Exam Problems

We will list model problems for the oral exam here, just for Students convenience. Solutions will not be shown on this website.

General Exam.

1. (a) Show that $G$ is a conserved quantity for $H$ if and only if $\{G, H\} = 0$.
   (b) Let $G$ be a conserved quantity for $H$, then any integral curve of the hamiltonian vector field $X_H$ lies on the level set $G^{-1}(c)$ for some real number $c$.
2. Show that $\mathbb{S}^6$ is not a symplectic manifold.
3. Run Moser's argument for any one of the neighbourhood theorems.
4. Show that an isotropic submanifold $N\subseteq M^{2n}$ has to have dimension at most $n$, a coisotropic submanifold has to have dimension at least $n$, and in particular, a Lagrangian submanifold $L\subseteq M$ has to have dimension $n$.
5. (a) Inside the cotangent bundle $T^{\ast}L$, find the equivalence condition for a graph $$\Gamma (\alpha) = \{(x, \alpha_x)\in T^{\ast}L | x\in L\}$$ of a differential $1$-form $\alpha\in C^{\infty} (L,T^{\ast}L)$ to be Lagrangian in $T^{\ast} L$.
   (b) Show that $\Gamma (\alpha)$, once being Lagrangian, is always Lagrangian isotopic to the zero section. Show that it's furthermore Hamiltonian isotopic to the zero section $L\hookrightarrow T^{\ast}L$ iff $\alpha$ is an exact $1$-form.
6. Show that there exists a volume-preserving embedding $D_r\hookrightarrow Z_R$ for any $r,R>0$, where $D_r$ is the open ball of radius $r$ in $\mathbb{R}^{2n}$ and $Z_R$ is the product of a disk of radius $R$ and an Euclidean space $\mathbb{R}^{2n-2}$.
7. Show that any complex structure $j$ on a Riemann surface $\Sigma$ is compatible with some symplecticc structure $\omega_{\Sigma}$ on $\Sigma$. In other words, $\Sigma$ is always Kähler.

Specified Exam.

1. Understand and explain the proof of the following theorem of Atiyah and Guillemin-Sternberg:
   Theorem. Let $(M, \omega)$ be a compact connected symplectic manifold with an effective hamiltonian $T$-action, then the image $\mu_T(M)$ of the moment map is a convex polytope in $\mathfrak{t}^*$, and the fibres of $\mu_T$ are connected.
   References: [MS17, Theorem 5.5.1] and [Aud91, Theorem 4.2.1].
2. A symplectic $4$-manifold is said to be minimal if it does not contain any symplectically embedded spheres with negative self-intersection number. Classify all compact minimal symplectic $4$-manifolds containing a symplectically embedded sphere with non-negative self-intersection number.
   References: [McD90].
3. Consider the symplectic manifold $(\mathbb{P}^1\times\mathbb{P}^1,a\omega_{FS}\oplus b\omega_{FS})$ where $a,b>0$ are not necessarily equal. Compute $H^{\ast}(\mathrm{Symp}_0 (\mathbb{P}^1\times\mathbb{P}^1,a\omega_{FS}\oplus b\omega_{FS});\mathbb{Z})$.
   Reference: [Abr98].
4. Compute the symplectic mapping class group of $T^{\ast}\mathbb{S}^2$.
   Reference: [Sei98].
5. As we have seen, Morse homology is built out from a sequence of spaces coming from trajectories between critical points of Morse functions. We should be able to get more information from the topology of these spaces than simply singular homology. In fact, there is a notion of flow category that can be built out from the data of Morse homology, and by taking "homology" of this flow category we can recover the Morse homology. Explain the construction of this flow category and how it recovers the Morse homology.
   Reference: [CJS95].
6. There is an interesting $A_{\infty}$-structure on the Morse complex $C_{\ast} (M,f;\Bbbk)$ for a given Morse function $f$, which requires a deeper look at the moduli spaces of trajectories. Explain this operadic structure and the construction of this $A_{\infty}$-structure on the Morse complex.
   Reference: [Fukaya97, Hic17].
7. Show that for semipositive symplectic manifolds there is a chain map from Floer chain complex of a given non-degenerate Hamiltonian to the Morse complex of the underlying manifold, with coefficients in the Novikov ring. This chain map is called the Piunikhin-Salamon-Schwarz (PSS) morphism. Show that this chain map is a quasi-isomorphism, and it's a quasi-isomorphism of rings if we equip the Morse complex with a certain product structure called quantum product, and equip the Floer chain complex with the pair-of-pants product.
   Reference: [PSS96].

References