Reveal.js Slideshow

Now suppose that after releasing the beaker we try to localize the gas using a force field
We will assume that the force is conservative: $F = - \nabla V (q_{1}, q_{2}),$ where $V (q_{1}, q_{2})$ is a potential function and $(q_{1}, q_{2})$ are the position coordinates.
For example, $V (q_{1}, q_{2}) = C (q_{1}^{2} + q_{2}^{2})$ is the “harmonic oscillator” potential.

V (q_{1}, q_{2}) = 3 (q_{1}^{2} + q_{2}^{2})

Observe that as the position coordinates

(q_{1}, q_{2})

localize, the momentum coordinates

(p_{1}, p_{2})

spread out.

Theorem (Liouville, cerca 1838)

The volume in phase space

R_{q_{1}, q_{2}, p_{1}, p_{2}}^{4}

is preserved by any Hamiltonian transformation.

In particular, we cannot simultaneously squeeze all four coordinates $q_{1}, q_{2}, p_{1}, p_{2}$ .

Question

Could we squeeze our gas in the

q_{1}, p_{1}

coordinates, at the cost of spreading out

q_{2}, p_{2}

V (q_{1}, q_{2}) = 4 q_{1}^{2}

In this naive example, squeezing $q_{1}$ comes at the cost of spreading out $p_{1}$ .

Theorem (Gromov 1985)

The area of the projection to

R_{q_{1}, p_{1}}^{2}

cannot be squeezed below a constant

G

In our example $G = 4$ , 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 $R_{q, p}^{2 n}$ with position coordinates $q_{1}, \dots, q_{n}$ and momentum coordinates $p_{1}, \dots, p_{n}$
A Hamiltonian is a smooth function $H : R^{2 n} \to R$ representing total energy
symplectic gradient: $\nabla_{ω} H := J \nabla H$ , where $J = (\begin{matrix} 0 & - 1 \\ 1 & 0 \end{matrix})$
Hamiltonian flow ${Fl}_{H} : R^{2 n} \times R_{t} \to R^{2 n}$ given by integrating $\nabla_{ω} H$ .
For example, $H (q, p) = V (q) + K (p)$ , with $K (p) = \frac{1}{2} p^{2}$ 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 $ω = \sum_{i = 1}^{n} d q_{i} \land d p_{i}$ .
A symplectomorphism of $R^{2 n}$ is a diffeomorphism $Φ : R^{2 n} \to R^{2 n}$ which preserves $ω$ .

Question

What is the border between flexibility and rigidity for symplectomorphisms of

R^{2 n}

Symplectic ellipsoid embeddings

Given $\vec{a} = (a_{1}, \dots, a_{n}) \in R_{> 0}^{n}$ , put $E (\vec{a}) := {(q, p) \in R^{2 n} | \sum_{i = 1}^{n} \frac{π}{a_{i}} (q_{i}^{2} + p_{i}^{2}) \leq 1} .$
Put $E (\vec{a}) \overset{s}{↪} E (\vec{b})$ if there exists a symplectomorphism $Φ : R^{2 n} \to R^{2 n}$ satisfying $Φ (E (\vec{a})) \subset E (\vec{b})$ .

Question

When do we have

E (\vec{a}) \overset{s}{↪} E (\vec{b})

First observations about ellipsoid embeddings

By Liouville's theorem: $E (\vec{a}) \overset{s}{↪} E (\vec{b}) ⟹ a_{1} \dots a_{n} \leq b_{1} \dots b_{n} .$
For $n = 1$ , we have $E (a_{1}) \overset{s}{↪} E (b_{1})$ if and only if $a_{1} \leq b_{1}$ .
Gromov's nonsqueezing theorem: $E (\vec{a}) \overset{s}{↪} E (\vec{b}) ⟹ min {a_{1}, \dots, a_{n}} \leq min {b_{1}, \dots, b_{n}} .$
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: $E (1, 5) \overset{s}{↪} E (2.51, 2.51)$ , but $E (1, 5) ⧸ \overset{s}{↪} E (2.49, 2.49)$ .
The case $n \geq 3$ is largely open.

Stabilized ellipsoid embeddings

Question

When do we have

E (a_{1}, a_{2}, \infty, \dots, \infty) \overset{s}{↪} E (b_{1}, b_{2}, \infty, \dots, \infty)

Stabilized ellipsoid embeddings

Question

When do we have

E (a_{1}, a_{2}, \underset{N}{\underset{⏟}{\infty, \dots, \infty}}) \overset{s}{↪} E (b_{1}, b_{2}, \underset{N}{\underset{⏟}{\infty, \dots, \infty}})

Note that $E (a_{1}, a_{2}, \infty, \dots, \infty) = E (a_{1}, a_{2}) \times R^{2 N}$ .
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 $\begin{array}{r} c_{B^{4} \times R^{2 N}} (a) := inf {b | E (1, a) \times R^{2 N} \overset{s}{↪} E (b, b) \times R^{2 N}} . \end{array}$

Theorem A

Theorem (joint with D. McDuff)

The function

c_{B^{4} \times R^{2 N}} (a)

is given as follows:

Theorem A

Theorem (joint with D. McDuff, 2024)

The function

c_{B^{4} \times R^{2 N}} (a)

is given for any

N \in Z_{\geq 1}

as follows:

Phase transition at $a = τ^{4} = \frac{7 + 3 \sqrt{5}}{2} \approx 6.85$ , 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 $p = [x_{0} : y_{0} : z_{0}] \in V (P)$ is singular if $\partial_{x} P (p) = \partial_{y} P (p) = \partial_{z} P (p) = 0$ .
For coprime integers $p > q \geq 2$ , the $(p, q)$ cusp singularity $C_{p, q}$ 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

Classification problem for singular plane curves

Questions

What are the possible combinations of singularities for a plane curve of degree $d$ and genus $g$ ?
For given coprime $p, q$ , what is the minimal degree of a plane curve with a $(p, q)$ cusp singularity?

Adjunction formula: $\frac{1}{2} (d - 1) (d - 2) = gen (C) + \sum_{p \in C} δ (p)$ , where $δ (C_{p, q}) = \frac{1}{2} (p - 1) (q - 1)$ and $δ (A_{1}) = 1$ .

Sample classification result

Theorem (Bobadilla et al)

There exists a rational plane curve of degree

d

with a

(p, q)

cusp singularity and no other singularities if and only if

(d, p, q)

is one of the following:

(a) $(p, q) = (d, d - 1)$ for $d \geq 3$
(b) $(p, q) = (2 d - 1, d / 2)$ for $d \geq 4$ even
(c) $(p, q) = ({Fib}_{k + 2}^{2}, {Fib}_{k}^{2})$ and $d = {Fib}_{k + 2} {Fib}_{k}$ for $k \geq 3$ odd
(d) $(p, q) = ({Fib}_{k + 4}, {Fib}_{k})$ for $d = {Fib}_{k + 2}$ for $k \geq 3$ odd
(e) $(p, q) = (22, 3)$ and $d = 8$
(f) $(p, q) = (43, 6)$ for $d = 16$ .

Sample classification result

Theorem (Bobadilla et al)

There exists a rational plane curve of degree

d

with a

(p, q)

cusp singularity and no other singularities if and only if

(d, p, q)

is one of the following:

(a) $(p, q) = (d, d - 1)$ for $d \geq 3$
(b) $(p, q) = (2 d - 1, d / 2)$ for $d \geq 4$ even
(c) $(p, q) = ({Fib}_{k + 2}^{2}, {Fib}_{k}^{2})$ and $d = {Fib}_{k + 2} {Fib}_{k}$ for $k \geq 3$ odd
(d) $(p, q) = ({Fib}_{k + 4}, {Fib}_{k})$ for $d = {Fib}_{k + 2}$ for $k \geq 3$ odd
(e) $(p, q) = (22, 3)$ and $d = 8$
(f) $(p, q) = (43, 6)$ for $d = 16$ .

Sample classification result

Theorem (Bobadilla et al)

There exists a rational plane curve of degree

d

with a

(p, q)

cusp singularity and no other singularities if and only if

(d, p, q)

is one of the following:

(a) $(p, q) = (d, d - 1)$ for $d \geq 3$
(b) $(p, q) = (2 d - 1, d / 2)$ for $d \geq 4$ even
(c) $(p, q) = ({Fib}_{k + 2}^{2}, {Fib}_{k}^{2})$ and $d = {Fib}_{k + 2} {Fib}_{k}$ for $k \geq 3$ odd
(d) $(p, q) = ({Fib}_{k + 4}, {Fib}_{k})$ for $d = {Fib}_{k + 2}$ for $k \geq 3$ odd
(e) $(p, q) = (22, 3)$ and $d = 8$
(f) $(p, q) = (43, 6)$ for $d = 16$ .

Sample classification result (simplified version)

Theorem (Kashiwara, Orevkov, folklore?)

There exists a rational plane curve of degree

d

with a

(p, q)

cusp singularity and no other singularities, with

p + q = 3 d

, if and only if

(p, q) = ({Fib}_{k + 4}, {Fib}_{k})

and

d = {Fib}_{k + 2}

for some

k \geq 3

odd.

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

Theorem B

Theorem (joint with D. McDuff, 2024)

There exists a rational plane curve of degree

d

with a

(p, q)

cusp singularity and possibly other singularities, with

p + q = 3 d

, if and only if one of the following holds:

$(p, q) = ({Fib}_{k + 4}, {Fib}_{k})$ for some $k \in Z_{\geq 3}$ odd
$p / q > τ^{4}$ .

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

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 $R^{2 n}$ (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.

Symplectic Nonsqueezing and Singular Algebraic Curves

Kyler Siegel

Hamiltonian dynamics

Symplectic geometry

Symplectic ellipsoid embeddings

First observations about ellipsoid embeddings

Stabilized ellipsoid embeddings

Stabilized ellipsoid embeddings

Theorem A

Theorem A

Singular algebraic curves

Example of a singular plane curve

Example of a singular plane curve

Example of a singular plane curve

Classification problem for singular plane curves

Sample classification result

Sample classification result

Sample classification result

Sample classification result (simplified version)

Theorem B

From Theorem B to Theorem A

Proving Theorem B