• In this naive example, squeezing $q_1$ comes at the cost of spreading out $p_1$.

• In our example $\mathcal{G} = 4$, assuming positions are drawn uniformly over the unit disk, and similarly for momenta.
• Remarkably, this theorem was not anticipated before the 1980's
• Gromov's proof introduced the method of pseudoholomorphic curves, which now form the basis of Gromov--Witten theory, Floer homology, quantum cohomology, Fukaya categories, embedded contact homology, symplectic field theory, ...
• Reminiscent of Heisenberg's uncertainy principle.

Hamiltonian dynamics

• Phase space $\mathbb{R}_{q,p}^{2n}$ with position coordinates $q_1,\dots,q_n$ and momentum coordinates $p_1,\dots,p_n$
• A Hamiltonian is a smooth function $H: \mathbb{R}^{2n} \rightarrow \mathbb{R}$ representing total energy
• symplectic gradient: $\nabla_\omega H := J \nabla H$, where $J = \begin{pmatrix} 0 & -\mathbb{1} \\ \mathbb{1} & 0 \end{pmatrix}$
• Hamiltonian flow $\fl_H: \mathbb{R}^{2n} \times \mathbb{R}_t \rightarrow \mathbb{R}^{2n}$ given by integrating $\nabla_\omega H$.
• For example, $H(q,p) = V(q) + K(p)$, with $K(p) = \tfrac{1}{2}p^2$ kinetic energy. The corresponding Hamiltonian flow is equivalent to Newton's second law $-\nabla V(q) = \ddot{q}$.

Symplectic geometry

• Hamiltonian flows preserve the standard symplectic two-form $\om = \sum_{i=1}^n dq_i \wedge dp_i$.
• A symplectomorphism of $\mathbb{R}^{2n}$ is a diffeomorphism $\Phi: \mathbb{R}^{2n} \rightarrow \mathbb{R}^{2n}$ which preserves $\om$.

Symplectic ellipsoid embeddings

• Given $\veca = (a_1,\dots,a_n) \in \mathbb{R}_{>0}^n$, put $$E(\veca) := \{(q,p) \in \mathbb{R}^{2n} \;|\; \sum_{i=1}^n\tfrac{\pi}{a_i}(q_i^2+p_i^2) \leq 1\}.$$
• Put $E(\veca) \hooksymp E(\vecb)$ if there exists a symplectomorphism $\Phi: \mathbb{R}^{2n} \rightarrow \mathbb{R}^{2n}$ satisfying $\Phi(E(\veca)) \subset E(\vecb)$.

First observations about ellipsoid embeddings

• By Liouville's theorem: $$E(\veca) \hooksymp E(\vecb) \Longrightarrow a_1\cdots a_n \leq b_1\cdots b_n.$$
• For $n=1$, we have $E(a_1) \hooksymp E(b_1)$ if and only if $a_1 \leq b_1$.
• Gromov's nonsqueezing theorem: $$E(\veca) \hooksymp E(\vecb) \Longrightarrow \min \{a_1,\dots,a_n\} \leq \min \{b_1,\dots,b_n\}.$$
• The case $n=2$ was completely worked out around 2010, or at least reduced to (nontrivial) combinatorics. Involves work by McDuff, Polterovich, Biran, Schlenk, Lalonde, Hutchings, Taubes, Hofer, and many others. Relies on special tools in dimension four, notably Seiberg--Witten theory.
• Ex: $E(1,5) \hooksymp E(2.51,2.51)$, but $E(1,5) \not\hooksymp E(2.49,2.49)$.
• The case $n \geq 3$ is largely open.

Stabilized ellipsoid embeddings

Question
When do we have $E(a_1,a_2,\infty,\dots,\infty) \hooksymp E(b_1,b_2,\infty,\dots,\infty)$?

Stabilized ellipsoid embeddings

Question
When do we have $E(a_1,a_2,\underbrace{\infty,\dots,\infty}_N) \hooksymp E(b_1,b_2,\underbrace{\infty,\dots,\infty}_N)$?
• Note that $E(a_1,a_2,\infty,\dots,\infty) = E(a_1,a_2) \times \mathbb{R}^{2N}$.
• In the special case $b_1 = b_2$, $E(b_1,b_2)$ is a round ball, and the answer is encoded by a single function \begin{align*} c_{B^4 \times \mathbb{R}^{2N}}(a) := \inf \{b\;|\; E(1,a) \times \mathbb{R}^{2N} \hooksymp E(b,b) \times \mathbb{R}^{2N}\}. \end{align*}

Theorem A

Theorem A

• Phase transition at $a = \tau^4 = \tfrac{7+3\sqrt{5}}{2}\approx 6.85$, from an “infinite staircase” to a rational function.
• Steps in first phase are described by ratios of Fibonacci numbers.
• Second phase is purely high dimensional.

Singular algebraic curves

• A polynomial $P(x,y) \in \C[x,y]$ gives an algebraic curve $V(P) := \{P(x,y) = 0\} \subset \C^2$.
• A homogenous polynomial $P(x,y,z) \in \C[x,y,z]$ gives an plane curve $V(P) := \{P(x,y,z) = 0\} \subset \CP^2$.
• A point $\pp = [x_0:y_0:z_0] \in V(P)$ is singular if $\bdy_x P(\pp) = \bdy_y P(\pp) = \bdy_z P(\pp) = 0$.
• For coprime integers $p > q \geq 2$, the $(p,q)$ cusp singularity $\mathcal{C}_{p,q}$ is modeled on $\{x^p = y^q\}$, and the ordinary double point $A_1$ is modeled on $\{x^2 = y^2\}$
• $V(P)$ is rational if it has genus zero, i.e. can be parametrized by $\CP^1$

Example of a singular plane curve

Example of a singular plane curve

Example of a singular plane curve

Classification problem for singular plane curves

• Adjunction formula: $\tfrac{1}{2}(d-1)(d-2) = \operatorname{gen}(C) + \sum\limits_{\pp \in C} \delta(\pp)$, where $\delta(\mathcal{C}_{p,q}) = \tfrac{1}{2}(p-1)(q-1)$ and $\delta(A_1) = 1$.

Sample classification result

Sample classification result

Sample classification result

Sample classification result (simplified version)

• Condition $p+q = 3d$ equivalent to curve being “rigid” after specifying the location and (jet) direction of the cusp.

Theorem B

• Phase transition at $p/q = \tau^4$.
• This solves the minimal degree problem for a $(p,q)$ cusp for “most” $(p,q)$ with $p+q \equiv 0 \;\text{mod}\; 3$.

From Theorem B to Theorem A

• We deduce Theorem A from Theorem B using techniques from symplectic field theory.
• Symplectic field theory is a rich framework for defining algebraic invariants of symplectic manifolds using moduli spaces of punctured pseudoholomorphic curves in symplectic manifolds.
• Important classes of symplectic manifolds include:
  • domains in $\mathbb{R}^{2n}$ (Hamiltonian dynamics)
  • smooth complex affine algebraic varieties (algebraic geometry)
  • cotangent bundles of smooth manifolds (smooth topology)

Proving Theorem B

• The proof of Theorem B relies on techniques from mirror symmetry for log Calabi--Yau surfaces.
• Using work of Gross--Pandharipande--Siebert, we show that the existence of the relevant curves is encoded by combinatorial objects called scattering diagrams
• Remarkably, these scattering diagrams can be analyzed using connections with quiver representation theory and cluster algebras.