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The Lumenary · Research pathways

Symplectic geometry and topology

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Date: ______________________

You will learn to

Define symplectic manifolds, Hamiltonian vector fields and symplectomorphisms, and derive Hamilton's equations from $\iota_{X_H}\omega = dH$
Prove that Hamiltonian flows preserve $H$ and $\omega$ , hence volume (Liouville), and that closed symplectic manifolds have non-zero even cohomology
Prove Darboux's theorem using Moser's trick, and explain why symplectic manifolds have no local invariants
Prove linear non-squeezing, state Gromov's non-squeezing theorem and the Arnold conjecture, and explain the idea of Floer homology, separating proved cases from open ones

1.
On $\R^2$ with $\omega = dq \wedge dp$ and the convention $\iota_{X_H}\omega = dH$ , what is the Hamiltonian vector field of $H = \frac{p^2}{2m} + V(q)$ ?
- A. $X_H = \frac{p}{m}\,\partial_q - V'(q)\,\partial_p$
- B. $X_H = -\frac{p}{m}\,\partial_q + V'(q)\,\partial_p$
- C. $X_H = V'(q)\,\partial_q + \frac{p}{m}\,\partial_p$
- D. $X_H = \frac{p}{m}\,\partial_q + V'(q)\,\partial_p$
2.
Which of these closed manifolds admit a symplectic form? (Choose all that apply.)
- A. $S^2$
- B. $T^4$
- C. $S^4$
- D. $S^2 \times S^2$
- E. $\mathbb{CP}^2$
- F. $S^1 \times S^3$
- G. $S^3$
3.
What is the dimension of the Lie group $\operatorname{Sp}(4, \R)$ ? (Hint: its Lie algebra is $\{X : X^{\mathsf T} J + J X = 0\}$ , i.e. $X = JS$ with $S$ symmetric.)
4.
In Moser's proof of Darboux's theorem we define a time-dependent vector field $X_t$ by $\iota_{X_t}\omega_t = -\sigma$ . Which property of $\omega_t$ guarantees that this equation has a (unique) solution $X_t$ ?
- A. $\omega_t$ is non-degenerate
- B. $\omega_t$ is closed
- C. $\omega_1 - \omega_0$ is exact
- D.The ball is compact
5.
In coordinates $(x_1, y_1, x_2, y_2)$ on $\R^4$ with $\omega_0 = dx_1\wedge dy_1 + dx_2 \wedge dy_2$ , which diagonal linear map is volume-preserving and maps the unit ball into the cylinder $\{x_1^2 + y_1^2 \le \frac14\}$ — showing that Gromov's theorem fails for volume-preserving maps?
- A. $\operatorname{diag}(\frac12, \frac12, 2, 2)$
- B. $\operatorname{diag}(\frac12, 2, \frac12, 2)$
- C. $\operatorname{diag}(\frac12, \frac12, \frac12, \frac12)$
- D. $\operatorname{diag}(2, \frac12, 2, \frac12)$
6.
Let $\varphi$ be a Hamiltonian diffeomorphism of the torus $T^2$ with standard area form, all of whose fixed points are non-degenerate. What lower bound for $\#\operatorname{Fix}(\varphi)$ does the (proved) Arnold conjecture give?
7.
Which of the following are established theorems (not conjectures)? (Choose all that apply.)
- A.Gromov: $B^{2n}(r)$ embeds symplectically into $B^2(R) \times \R^{2n-2}$ only if $r \le R$ .
- B.For every closed symplectic manifold, a Hamiltonian diffeomorphism with non-degenerate fixed points has at least $\sum_k \dim H_k(M; \Q)$ fixed points.
- C.For every closed symplectic manifold, a Hamiltonian diffeomorphism has at least as many fixed points as the minimal number of critical points of a smooth function on $M$ .
- D.Homological mirror symmetry holds for every Calabi–Yau manifold.
- E.Every Reeb vector field on a closed contact $3$ -manifold has a periodic orbit.
- F.Among convex domains of given volume in $\R^{2n}$ , the ball maximises every normalised capacity (Viterbo).
8.
The Lean theorem hamiltonian_energy_conserved assumes hω : ∀ v, ω v v = 0 and hX : ∀ z w, ω (X z) w = dH z w, and concludes H (γ s) = H (γ t) for a flow line γ of X. Which modification would make the statement false?
- A.Replace hω by ∀ v w, ω v w = ω w v (ω symmetric).
- B.Add a hypothesis that ω is non-degenerate.
- C.Specialise V from a general normed space to ℝ⁴.
- D.Conclude instead that H ∘ γ has derivative zero at every time.
9.
A piece of the round sphere is never isometric to a piece of the flat plane, but any two area forms on surfaces are locally equivalent. Which theorem gives the second statement, in all dimensions?
- A.Darboux's theorem
- B.Gauss's Theorema Egregium
- C.Gromov's non-squeezing theorem
- D.Liouville's theorem
10.
For the harmonic oscillator $H = \frac12(q^2 + p^2)$ , one step of symplectic Euler is $p' = p - hq$ , then $q' = q + hp'$ ; one step of explicit Euler is $q' = q + hp$ , $p' = p - hq$ . Write symplectic_euler_matrix(h) and explicit_euler_matrix(h) returning the $2 \times 2$ matrices (lists of rows, acting on the column $(q, p)$ ), and det2(m). The tests use exact fractions.
Computational — Practice

Answer key — Symplectic geometry and topology

1.
Write $X_H = a\,\partial_q + b\,\partial_p$ . Then $\iota_{X_H}(dq \wedge dp) = a\,dp - b\,dq$ , while $dH = V'(q)\,dq + \frac{p}{m}\,dp$ . So $a = \frac{p}{m}$ and $b = -V'(q)$ , i.e. $\dot q = p/m$ , $\dot p = -V'(q)$ — Hamilton's equations, and Newton's law $m \ddot q = -V'(q)$ .
2.
$S^2$ , $T^4$ , $S^2 \times S^2$ and $\mathbb{CP}^2$ are symplectic (area forms, the standard form on the torus, a product form, and the Fubini–Study form). $S^3$ is odd-dimensional. $S^4$ and $S^1 \times S^3$ have $H^2 = 0$ , but if $\omega$ were symplectic on a closed $4$ -manifold and $\omega = d\beta$ , then $\omega^2 = d(\beta \wedge \omega)$ and $\int \omega^2 = 0$ , contradicting that $\omega^2$ is a volume form.
3.
Differentiating $A^{\mathsf T} J A = J$ at $A = I$ gives $X^{\mathsf T} J + J X = 0$ , i.e. $JX$ is symmetric. So the Lie algebra is isomorphic to the symmetric $4 \times 4$ matrices, of dimension $\frac{4 \cdot 5}{2} = 10$ . In general $\dim \operatorname{Sp}(2n, \R) = n(2n+1)$ .
4.
At each point the map $X \mapsto \iota_X \omega_t = \omega_t(X, \cdot)$ is linear from $T_xM$ to $T_x^*M$ , spaces of the same dimension; it is injective, hence bijective, exactly when $\omega_t$ is non-degenerate. That is why the proof first shrinks the ball so that all $\omega_t$ are non-degenerate. Closedness enters in Cartan's formula, exactness of $\omega_1 - \omega_0$ produces $\sigma$ , and compactness lets the flow run to time $1$ .
5.
$\operatorname{diag}(\frac12, \frac12, 2, 2)$ has determinant $1$ and sends $(x_1, y_1, x_2, y_2)$ with $|x|^2 \le 1$ to a point with $x_1'^2 + y_1'^2 = \frac14(x_1^2 + y_1^2) \le \frac14$ . It is not symplectic: $\omega_0(Ae_{x_1}, Ae_{y_1}) = \frac14$ . The maps in (b) and (d) are symplectic and so, by (linear) non-squeezing, cannot fit the unit ball into a cylinder of radius $\frac12$ ; (c) shrinks volume. This is the Lean declaration squeeze.
6.
The Arnold bound is $\sum_k b_k(T^2) = 1 + 2 + 1 = 4$ , proved for tori by Conley and Zehnder (1983). The Lefschetz number of a map isotopic to the identity is $\chi(T^2) = 0$ , which gives no information. The Hamiltonian hypothesis matters: a translation of $T^2$ is symplectic and has no fixed points.
7.
Theorems: Gromov's non-squeezing theorem (1985), the rational Arnold conjecture for non-degenerate Hamiltonian diffeomorphisms (Fukaya–Ono, Liu–Tian), and the three-dimensional Weinstein conjecture (Taubes, 2007). Open: Arnold's strong form with $\operatorname{crit}(M)$ , and homological mirror symmetry in general. Disproved: Viterbo's volume–capacity conjecture (Haim-Kislev–Ostrover, 2024).
8.
The proof computes $(H\circ\gamma)' = dH_{\gamma}(X(\gamma)) = \omega(X(\gamma), X(\gamma))$ and uses hω to make this zero. If $\omega$ is symmetric — say an inner product — then hX makes $X$ the gradient of $H$ and $(H \circ \gamma)' = |\nabla H|^2 \ge 0$ , with strict inequality away from critical points, so energy is not conserved. Adding hypotheses or specialising cannot falsify a theorem, and (d) is a step of the proof.
9.
Darboux's theorem: near every point there are coordinates with $\omega = \sum dx_i \wedge dy_i$ . Proved with Moser's trick in the lesson. In Riemannian geometry, by contrast, curvature is a local invariant (Theorema Egregium).
10.
```
def symplectic_euler_matrix(h):
    return [[1 - h * h, h], [-h, 1]]


def explicit_euler_matrix(h):
    return [[1, h], [-h, 1]]


def det2(m):
    return m[0][0] * m[1][1] - m[0][1] * m[1][0]
```
Symplectic Euler gives $q' = (1-h^2)q + hp$ , $p' = -hq + p$ , with determinant $(1 - h^2) + h^2 = 1$ : area is preserved exactly, for every step size. Explicit Euler has determinant $1 + h^2$ , so after $N$ steps areas grow by $(1+h^2)^N$ and the numerical orbit spirals outwards.