Grades.
The final grade will be based on weekly homework and on participation in class. Homework problems
should be solved independently and the usual code of conduct applies.
Core topics.
The course will develop a variety of topics, such as:
1. Symplectic linear algebra
2. Symplectic manifolds and cotangent bundles
3. Darboux's theorem
4. Contact manifolds
5. Sympletic capacities
6. Weinstein's conjecture
7. Viterbo's theorem
8. Gromov-Eliashberg rigidity
9. Gromov's Symplectic Width
10. J-Holomorphic curves
Calendar.
12 Jan (Tue): First Class
8,10 March (Tue, Th): Spring Break - no classes
28 Apr (Th): Last Class
Homework
Homework 1 (due January 28)
Consider the ODE: $x'=b(x)$, where $b:\mathbb{R}^k\to\mathbb{R}^k$
is a sufficiently smooth function. Denote the flow of this ODE by $\phi_t(x)$.
1. Prove that $\big(\nabla\phi_t(x)\big)b(x) = b(\phi_t(x))$.
2. Prove that $\partial_t \big(\det\nabla\phi_t(x)\big) = \big((\mbox{div}
b) (\phi_t(x))\big) \big(\det\nabla\phi_t(x)\big)$
3. Let $f:\mathbb{R}^k\to\mathbb{R}$ be a (smooth) scalar function on the phase
space $\mathbb{R}^k$ and define: $v(x,t) = f(\phi_t(x))$. Prove that
$v_t = \langle \nabla v, b\rangle$. This is called the Liouville theorem.
Homework 2 (due February 4)
1. Prove the dual formulation of the Courant-Hilbert formula, which
reads as follows. Denote
the eigenvalues of a symmetric matrix $B\in\mathcal{S}(k)$ by:
$\mu_1\leq\mu_2\ldots\leq\mu_k$. Then: $\mu_i = \sup_{\mbox{dim } V
= j-1} \inf_{x\in V^\perp\setminus\{0\}}\frac{\langle Bx,
x\rangle}{|x|^2}$.
2. Prove that eigenvalues of a Hamiltonian matrix $C\in Ham (n)$ where
$C=JB$ and $B>0$ is a symmetric matrix, are
purely imaginary and that they come in couples, namely:
$\mbox{Spec}(C) = \{\pm i\lambda_j\}_{j=1}^n$ for some
$\lambda_1\leq\lambda_2\ldots\leq\lambda_n$.
3. Let $T\in Sp(n)$ and write its unique polar decomposition: $T=PQ$ where
$P\in S(2n)$, $P>0$ and $Q\in \mathcal{O}(2n)$. Prove that both $P$ and
$Q$ are symplectic: $P, Q\in Sp(n)$.
4. Let $Q\in Sp(n)\cap\mathcal{O}(2n)$ and write $Q$ in the block
form: $Q=\left[\begin{array}{cc} A & B \\ C & D\end{array}
\right]$. Prove that $A=D$ and $B=-C$. Prove that $A-iB\in\mathbb{C}^{n\times n}$ is unitary.
5. Prove that $Q$ as in problem 4 above, is similar to
$\mbox{diag}\big((A+iB), (A-iB)\big)$. Deduce that $\det
Q>0$. Then, using problem 3 above deduce that for any $T\in Sp(n)$ there
must be: $\det T = 1$. Note that this is an alternative proof of
this statement, which does not use differential forms.
Homework 3 (due February 16)
1. Show that the characteristic
polynomial $P$ of a symplectic matrix is reflexive: $P(\lambda) = \lambda^{2n}P(\lambda^{-1})$.
Deduce that the eigenvalues of a symplectic matrix occur in
quadruples, namely: ($\lambda, \lambda^{-1}, \bar\lambda,
\bar\lambda^{-1}$).
2. Let $H$ be a positive definite quadratic function on
$\mathbb{R}^{2n}$. Show that there exists an antisymplectic linear map
$T$ such that for every $x= (q_1\ldots q_n, p_1\ldots p_n)$ there
holds: $H(Tx) = \sum_{j=1}^n \frac{q_j^2 + p_j^2}{r_j^2}$. Here,
$0 < r_1\leq r_2\ldots \leq r_n$ are the symplectic radii of $H$.
3. Let $E_1$ and $E_2$ be two ellipsoids in $\mathbb{R}^{2n}$. Show that they have the same
(vectors of) symplectic radii if and only if there is an
antisymplectic linear map $T$ such that $T(E_1) = E_2$.
4. Compute the determinant of an antysymplectic linear transformation
in $\mathbb{R}^{2n}$. What are the antisymplectic matrices in
$\mathbb{R}^{2\times 2}$?
5. Let $T$ be an invertible linear map on $\mathbb{R}^{2n}$ for
$n>1$. Assume that for every ellipsoid $E$, the second symplectic
radii of $E$ and $TE$ are the same. Show that $T$ must be symplectic
or antisymplectic.
Homework 4 (due March 1)
1. Let $T$ be an invertibe $n\times n$ matrix. Show that the following
conditions are equivalent:
(i) $T\in \mathcal{O}(n)$.
(ii) For every ellipsoid $E$ we have: $\mathbf{R}(E) =
\mathbf{R}(TE)$, where $\mathbf{R}$ stands for the vector of the
elliptic radii of a given ellipsoid.
(iii) Fix $i:1\ldots n$. For every ellipsoid $E$ we have: $\mathbf{R}_i(E) =
\mathbf{R}_i(TE)$. Recall that $\mathbf{R}_i$ stands for the $i$-th
elliptic radius (in order of magnitude) of a given ellipsoid.
2. Let $\varphi: \mathbb{R}^{2n}\to\mathbb{R}^{2n}$ be a symplectic
diffeomorphism: $\phi\in Sp(\mathbb{R}^{2n})$. Let
$H:\mathbb{R}^{2n}\to\mathbb{R}$ be a Hamiltonian. Prove that:
$\phi_t^{(H\circ\varphi)} = (\varphi^{-1})\circ \phi_t^H\circ
\varphi$. (Do not use the Cartan's formula!)
3. Prove that the Lie derivative of differential forms, defined as:
$\mathcal{L}_X = d\circ i_X + i_X\circ d$ satisfies the following
fundamental properties:
(i) $\mathcal{L}_X (\omega\wedge \eta) = \mathcal{L}_X (\omega)\wedge
\eta + \omega\wedge \mathcal{L}_X (\eta)$,
(ii) $\mathcal{L}_X (d\omega) = d (\mathcal{L}_X \omega)$.
4. Let $(M_1, \omega^1)$ and $(M_2, \omega^2)$ be two symplectic
manifolds. Prove that $(M_1\times M_2, \omega^1\times\omega^2)$ is a
symplectic manifold, where we define:
$(\omega^1 \times \omega^2)_{(x_1, x_2)} ((a_1, a_2), (b_1, b_2)) =
\omega^1_{x_1}(a_1, b_1) + \omega^2_{x_2}(a_2, b_2)$ for all $(x_1,
x_2)\in M_1\times M_2$ and all $a_1, b_i \in T_{x_1}M_1$, $a_2, b_2\in
T_{x_2}M_2$.
5. Let $N$ be a $n$-dimensional smooth manifold. Define $\lambda\in
\Lambda^1(T^*N)$ by: $\lambda_{(q,p)}(v,a) = p(v)$ for all
$(q,p)\in T^*N$ and $(v,a)\in T_{(q,p)}(T^*N)$. Prove that
$\omega=d\lambda$ is nondegenerate. This is done by defining
appropriate (natural) charts $\bar h: T^*U\to\mathbb{R}^{2n}$ on each
open cover set $U$ of $N$ such that: $(\bar h)^*\bar\lambda =
\lambda$, where $\bar\lambda\in \Lambda^1(\mathbb{R}^{2n})$ is the
usual: $\lambda = \langle p, dq\rangle$.
Homework 5 (due March 15)
1. Let $(M, \omega)$ be a $2n$ dimensional, closed (i.e. compact, without
boundary), symplectic manifold. Prove that for every $1\leq j \leq
n$ there exists a closed $2j$ form which is not exact.
2. Let $(M, \omega)$ be a symplectic manifold of dimension higher than
$2$. Let $\varphi:M\to M$ be a diffeomorphism and let
$f:M\to\mathbb{R}$ be a Hamiltonian. Prove that if $\varphi^*\omega =
f\omega$, then $f$ is constant.
3. Consider $M=\mathbb{R}^{2n-1}$ with $\alpha=\langle u, dx\rangle\in
\Lambda^1(M)$ given by some smooth vector field $u$ on $M$.
(i) When is $(M,\alpha)$ contact?
(ii) Assume that $(M,\alpha)$ contact and let
$H:\mathbb{R}^{2n}\to\mathbb{R}$ be a Hamiltonian function. Prove that
the $\alpha$-contact field associated with $H$, i.e.: $Z=\pi_{T_xM}
X_{\hat M}^{d\hat{\alpha}}$ is given by the formula:
$$Z= -\tilde C \nabla H + HR^\alpha,$$
where $R^\alpha$ is the associated Reeb vector
field and $\tilde C\in\mathbb{R}^{(2n-1)\times (2n-1)}$ is uniquely specified
by the two requirements: $(\tilde C)_{\mid u^\perp}$ is the inverse
of the linear map $(\nabla u) - (\nabla u)^T: u^\perp \to
(R^\alpha)^\perp$; and $\tilde C u = 0$.
(iii) Assume that $n=2$. Find what is $\tilde C$ in this case and
prove that there must be:
$Z = {\langle u, \mbox{curl} u\rangle}^{-1}\big(H\mbox{curl} u -
u\times \nabla H\big).$
4. Let $M$ be an odd-dimensional manifold and let $\alpha\in
\Lambda^1(M)$. Prove that $\alpha$ is contact if and only if
$\alpha\wedge (d\alpha)^{n-1}$ is a volume form on $M$.
5. Let $(M,\omega)$ be a symplectic manifold of dimension larger than
$2$. Let $\phi:M\to M$ be a diffeomorphism such that $\phi^*\omega =
f\omega$ for some scalar function $f:M\to\mathbb{R}$. Prove that $f$
must be constant.
Homework 6 (due March 31)
1. Let $M=\mathbb{T}^3$ be the $3$-dimensional torus and define $\hat M =
M\times (-\delta, \delta)$ for some $\delta>0$. For a vector $(\xi_1,
\xi_2, \xi_3)\in\mathbb{R}^3$ such that $\xi_3\neq 0$, consider the skew-symmetric
matrix:
$$ \bar C = \left[\begin{array}{cccc} 0 & 1 & 0 & \xi_1\\ -1 & 0 & 0 &
\xi_2 \\ 0 & 0 & 0 & \xi_3 \\ -\xi_1 & -\xi_2 & -\xi_3 &
0\end{array}\right].$$
Let $C= - \bar{C}^{-1}$ and define: $\omega = \sum_{i < j} c_{ij}
dx_i\wedge dx_j\in\Lambda^2(\hat M)$.
Finally, define the Hamilonian on $\hat M$ by: ${H}(x_1, x_2, x_3, x_4)=x_4$.
(i) Prove that $(\hat M, \omega)$ is a symplectic manifold.
(ii) Compute the Hamiltonian vector field $X_H^\omega$.
(iii) Assume that $\xi$ is irrational, that is for all $a\in
\mathbb{Z}^3\setminus \{0\}$ we have: $\langle \xi, a\rangle \neq
0$. Prove that $H$ has then no periodic orbit. Deduce that the submanifold $M$ has no
contact structure in $(\hat M, \omega)$.
2. Let $M=\partial D$ for an open, bounded domain $D\subset\mathbb{R}^{2n}$
that is star-shaped with respect to a ball $B_\epsilon(0)$. Prove that $M$ is a
contact manifold. [Hint: Check that $X(x) = \frac{1}{2}x$ is $\omega$-Liouville.]
3. Consider the contact manifold $(\mathbb{R}^{2n-1}, \bar\alpha)$,
where we write $x=(q,p,z)$ and $\bar\alpha = \langle p, dq\rangle +
dz$. Let $H=H(q,p,z)$ be a smooth Hamiltonian on
$\mathbb{R}^{n-1}$. Show that the related contact vector field $Z_H$
is given by: $Z_H = \big (\partial_pH , -\partial_qH + \partial_zH p , H -
\langle p, \partial_pH\rangle\big)$.
4. Consider the manifold
$M=\mathbb{S}^{2n-1}\subset\mathbb{R}^{2n}$. Let $\bar\mu\in
\Lambda^1(M)$ be the restriction of
$\bar\lambda=\langle p, dq\rangle\in \Lambda^1(\mathbb{R}^{2n})$,
where we wrire $x=(q,p)\in\mathbb{R}^{2n}$.
(i) Show that $(M, \bar\mu)$ is not a contact manifold.
(ii) Let $\mu = \frac{1}{2} \big(\langle p, dq\rangle - \langle q,
dp\rangle\big)\in \Lambda^1(M)$. Show that $(M, \mu)$ is a contact
manifold. Find its Reeb vector field.
5. Let $K=\bar{U}$ for an open, convex set $U\subset\mathbb{R}^{2n}$
containing $0$. Recall that the gauge function of $K$ is defined as: $g_K(x)
= \inf\{r>0; ~ \frac{x}{r}\in K\}$.
(i) Prove that $g_K$ is a Banach functional.
(ii) Compute the Legendre transform of $g_K$.
(iii) Prove that $K=\{y; ~ \langle x, y\rangle\leq\sup_{z\in K}
\langle x, z\rangle \quad \forall x\in \mathbb{R}^{2n}\}$.
Homework 7 (due April 12)
1. Consider the Hamiltonian $H:\mathbb{R}^2\to\mathbb{R}$ given by:
$H(x) = \pi |x|^2$ and the periodic solutions of $\dot
x=J\nabla H(x)$ on the level set $\{H=\frac{1}{2}\}$.
(i) Show that $H=H_K$ for some set $K$. What is $K$?
(ii) Show that $\inf_{\Lambda} \mathcal{C} = 0$.
(iii) Show that $\sup_{\Lambda} \mathcal{C} = \infty$.
2. Let $K\subset \mathbb{R}^n$ be the closure of an open set $U$, that
is convex and symmetric with respect to $0$. Show that
the "Minkowski Billiard" Hamiltonian flow starting from $(0, \nabla
g_K(q))\in K\times K^o$ for $q\in\partial K$, induces a periodic orbit whose action equals
$4$. Deduce that the symplectic width of the set $K\times
K^o\subset\mathbb{R}^{2n}$ satisfies: $\mathbb{c}(K\times K^o) \leq 4$.
3. Prove that for every symplectic manifold $(M, \omega)$ there holds:
$\underline{\mathbb{c}}(M, \omega) \leq \mathbb{c}(M,
\omega)\leq \overline{\mathbb{c}}(M, \omega)$.
4. Show that $\mathbb{c}_{HZ}(B_{2n}(0,1), \bar\omega) \geq \pi$.
5. Show that $\mathbb{c}_{HZ}(K, \bar\omega) \geq \mathbb{c}_0(K)$ for
every bounded convex set $K\subset\mathbb{R}^{2n}$ that is a closure
of an open set.
Homework 8 (due April 26)
1. Verify the symplectomorphism monotonicity and the scaling
properties in the definition of symplectic capacity for
$\mathbb{c}_{HZ}$.
2. Assume that $F\in\mathcal{C}^1(\mathbb{R}^n,\mathbb{R})$ satisfies
$\lim_{|x|\to\infty}F(x) = +\infty$.
(i) Show that $F$ automatically satisfies the Palais-Smale condition.
(ii) Assume that the flow of $x'=-\nabla F(x)$ is well defined for all
$t\geq 0$. Let $x_1, x_2$ be two distinct relative minima of
$F$. Prove that there exists yet another critical point
$x_3\not\in\{x_1, x_2\}$.
3. Consider the functon $F(x,y) = e^{-x} - y^2$.
(i) Show that it does not satisfy the Palais-Smale condition.
(ii) Denote $E^\pm = \{(x,y); ~ F(x,y)\leq 0, ~ \pm y \geq 0\}$, and
let $\mathcal{F}$ be a family of subsets of $\mathbb{R}^2$ given by:
$\mathcal{F}= \{\gamma[0,1]; ~ \gamma:[0,1]\to\mathbb{R}^2, ~
\gamma(0)\in E^-, ~\gamma(1)\in E^+, ~ \gamma \mbox{ continuous}\}.$
Show that $\mathcal{F}$ is positively invariant with respect to the
flow of $(x,y)' = -\nabla F(x,y)$ and that $\alpha(F, \mathcal{F}) = 0$, but $F$ has no critical point.