Practice: Symplectic geometry and topology

Symplectic geometry and topology

Warm-upHuman

On $\R^2$ with $\omega = dq \wedge dp$ and the convention $\iota_{X_H}\omega = dH$ , what is the Hamiltonian vector field of $H = \frac{p^2}{2m} + V(q)$ ?

Choose one

X_H = \frac{p}{m}\,\partial_q - V'(q)\,\partial_p

X_H = -\frac{p}{m}\,\partial_q + V'(q)\,\partial_p

X_H = V'(q)\,\partial_q + \frac{p}{m}\,\partial_p

X_H = \frac{p}{m}\,\partial_q + V'(q)\,\partial_p

Symplectic geometry and topology

CoreHuman

Which of these closed manifolds admit a symplectic form? (Choose all that apply.)

Choose all that apply

Symplectic geometry and topology

Warm-upHuman

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.)

Symplectic geometry and topology

CoreHuman

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 $X_t$ ?

Choose one

\omega_t

is non-degenerate

\omega_t

is closed

\omega_1 - \omega_0

is exactThe ball is compact

Symplectic geometry and topology

CoreHuman

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?

Choose one

\operatorname{diag}(\frac12, \frac12, 2, 2)

\operatorname{diag}(\frac12, 2, \frac12, 2)

\operatorname{diag}(\frac12, \frac12, \frac12, \frac12)

\operatorname{diag}(2, \frac12, 2, \frac12)

Symplectic geometry and topology

CoreHuman

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?

Symplectic geometry and topology

StretchHuman

Which of the following are established theorems (not conjectures)? (Choose all that apply.)

Choose all that apply

Gromov:

B^{2n}(r)

embeds symplectically into

B^2(R) \times \R^{2n-2}

only if

r \le R

.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.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

.Homological mirror symmetry holds for every Calabi–Yau manifold.Every Reeb vector field on a closed contact

3

-manifold has a periodic orbit.Among convex domains of given volume in

\R^{2n}

, the ball maximises every normalised capacity (Viterbo).

Symplectic geometry and topology

CoreFormal

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?

Choose one

Replace hω by ∀ v w, ω v w = ω w v (ω symmetric).Add a hypothesis that ω is non-degenerate.Specialise V from a general normed space to ℝ⁴.Conclude instead that H ∘ γ has derivative zero at every time.

Symplectic geometry and topology

Warm-upHuman

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?

Choose one

Darboux's theoremGauss's Theorema EgregiumGromov's non-squeezing theoremLiouville's theorem

Symplectic geometry and topology

StretchComputational

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.

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