## Some Applications of Symplectic Geometry

Clayton Shonkwiler

shonkwiler.org

/usna21

this talk!

### Take-Home Message

Symplectic geometry can sound scary, but it turns out to be useful in some surprising contexts.

### Symplecti-wha?

—Dusa McDuff & Dietmar Salamon

Introduction to Symplectic Topology

—Ana Cannas da Silva

Lectures on Symplectic Geometry

(\textcolor{12a4b6}{M},\textcolor{d9782d}{\omega})

A nice space, where multivariable calculus makes sense. Think spheres, tori, projective spaces, …

A rule for assigning a number to pairs of tangent vectors which transforms like (signed) area of the parallelogram they span.

\omega(v_1,v_2) = 1
\omega(v_1,v_2) = -1
v_1
v_1
v_2
v_2

### Symplectic Manifold

Must be closed (think: divergence-free) and non-degenerate (doesn’t annihilate any vector).

### Standard Example and Some Non-Examples

Example: $$(\mathbb{R}^2,dx \wedge dy) = (\mathbb{C},\frac{i}{2}dz \wedge d\bar{z})$$

dx \wedge dy \left( \textcolor{12a4b6}{a \frac{\partial}{\partial x} + b \frac{\partial}{\partial y}}, \textcolor{d9782d}{c \frac{\partial }{\partial x} + d \frac{\partial}{\partial y}} \right) = ad - bc
(a,b) = a \vec{e}_1 + b \vec{e}_2 = a \frac{\partial}{\partial x} + b \frac{\partial}{\partial y}
(c,d) = c \vec{e}_1 + d \vec{e}_2 = c \frac{\partial}{\partial x} + d \frac{\partial}{\partial y}

Non-Example: $$(\mathbb{R}^3, dx \wedge dy)$$

For any vector $$v$$, $$dx \wedge dy\left(\frac{\partial}{\partial z}, v\right) = 0$$, so $$dx \wedge dy$$ is degenerate.

Non-Example: $$(\mathbb{R}^4, dx_1 \wedge dy_1 + y_1 dx_2 \wedge dy_2)$$

Because of the coefficient $$y_1$$, this is not closed (“not divergence-free”).

### Examples

$$(S^2,d\theta\wedge dz)$$

$$(\mathbb{R}^2,dx \wedge dy) = (\mathbb{C},\frac{i}{2}dz \wedge d\bar{z})$$

$$(S^2,\omega)$$, where $$\omega_p(u,v) = (u \times v) \cdot p$$

$$(\mathbb{R}^2,\omega)$$ where $$\omega(u,v) = \langle i u, v \rangle$$

$$(\mathbb{C}^n, \frac{i}{2} \sum dz_k \wedge d\overline{z}_k)$$

$$(\mathbb{C}^{m \times n}, \omega)$$ with $$\omega(X_1,X_2) = -\operatorname{Im} \operatorname{trace}(X_1^* X_2)$$.

$$(T^* \mathbb{R}^n,\sum dq_i \wedge dp_i)$$

phase space

position

momentum

### Volume

$$\omega^{\wedge n} = \omega \wedge \dots \wedge \omega$$ is a volume form on $$M$$, and induces a measure

m(U) := \int_U \omega^{\wedge n}

called Liouville measure on $$M$$.

In particular, if $$M$$ is compact, this can be normalized to give a probability measure.

Theorem [Archimedes]

$$\mathrm{d}\theta \wedge \mathrm{d}z$$ is the standard area form on the unit sphere $$S^2$$.

\int_{S^2} \mathrm{d}\theta \wedge \mathrm{d}z = \int_{-1}^1 \int_0^{2\pi} \mathrm{d}\theta\, \mathrm{d}z = 4\pi

### A (Very) Classical Example

KoenB [   ] from Wikimedia Commons

### Hamiltonian Mechanics

“Evolution of a physical system follows the symplectic gradient”

If $$H: M \to \mathbb{R}$$ is smooth, then there exists a unique vector field $$X_H$$ so that $${dH = \iota_{X_H}\omega}$$, i.e.,

dH(\cdot) = \omega(X_H, \cdot)

($$X_H$$ is called the Hamiltonian vector field for $$H$$, or sometimes the symplectic gradient of $$H$$)

Example. $$H: (S^2, d\theta\wedge dz) \to \mathbb{R}$$ given by $$H(\theta,z) = z$$.

$$dH = dz = \iota_{\frac{\partial}{\partial \theta}}(d\theta\wedge dz)$$, so $$X_H = \frac{\partial}{\partial \theta}$$.

$$H$$ is constant on orbits of $$X_H$$:

$$\mathcal{L}_{X_H}(H) = dH(X_H)=\omega(X_H,X_H) = 0$$

### Some Mechanics

Suppose we have a particle of mass $$m$$ with position $$\vec{q} = (q_1,q_2,q_3)$$.

Suppose the particle is subject to a potential $$V(\vec{q})$$.

Newton’s Second Law: $$-\nabla V(\vec{q}) = m \frac{d^2 \vec{q}}{dt^2}$$.

Momenta $$p_i := m \frac{dq_i}{dt}$$.

$$(\vec{q},\vec{p})\in T^*\mathbb{R}^3 \cong \mathbb{R}^6$$; canonical symplectic form $$\omega = dq_1 \wedge dp_1 + dq_2 \wedge dp_2 + dq_3 \wedge dp_3$$.

### The Hamiltonian Formulation

Energy function (or Hamiltonian)

$$H(\vec{q},\vec{p}) := \frac{1}{2m}|\vec{p}|^2 + V(\vec{q})$$.

$$dH = \sum\left(\frac{\partial H}{\partial p_i} dp_i + \frac{\partial H}{\partial q_i}dq_i\right)$$,

$$(\vec{q}(t),\vec{p}(t))$$ is an integral curve for $$X_H$$ if and only if

$$\frac{dq_i}{dt} = \frac{\partial H}{\partial p_i} = \frac{1}{m} p_i$$

$$\frac{dp_i}{dt} = -\frac{\partial H}{\partial q_i} = -\frac{\partial V}{\partial q_i}.$$

This is Newton’s Second Law written as a first-order system!

$$X_H = \sum\left(\frac{\partial H}{\partial p_i} \frac{\partial}{\partial q_i} - \frac{\partial H}{\partial q_i} \frac{\partial}{\partial p_i}\right)$$.

### Noether’s Theorem

“Every continuous symmetry has a corresponding conserved quantity”

### Circle Actions

A circle action on $$(M,\omega)$$ determines a vector field $$X$$ by

X(p) = \left.\frac{d}{dt}\right|_{t=0}e^{i t} \cdot p

$$S^1=U(1)$$ acts on $$(S^2,d\theta \wedge dz)$$ by

e^{it} \cdot(\theta, z) = (\theta + t, z).

So $$X = \frac{\partial}{\partial \theta}$$.

### Symmetries and Conserved Quantities

Definition. A circle action on $$(M,\omega)$$ is Hamiltonian if there exists a map

\mu: M \to \mathbb{R}

so that $$d\mu = \iota_{X}\omega = \omega(X,\cdot)$$, where $$X$$ is the vector field generated by the circle action. In other words, $$X = X_\mu$$.

$$X = \frac{\partial}{\partial \theta}$$

$$\mu(\theta,z) = z$$

$$\iota_X\omega = \iota_{\frac{\partial}{\partial \theta}} d\theta \wedge dz = (d\theta \wedge dz)\left(\frac{\partial}{\partial \theta},\cdot \right) = dz$$

### Generalizing Archimedes

Theorem [Atiyah and Guillemin–Sternberg]

If $$(M,\omega)$$ is symplectic and $$\mathbb{T}=(S^1)^k$$ acts in a Hamiltonian fashion, the conserved quantities are recorded by $$\mu:M \to \mathbb{R}^k$$ with:

1. The level sets of $$\mu$$ are connected.
2. $$\mu(M)$$ is the convex hull of the images of the fixed points of the $$\mathbb{T}$$ action.

Theorem [Duistermaat–Heckman]

If $$\dim(M)=2n$$ and $$\dim(\mathbb{T})=k$$, then the pushforward measure on the moment polytope $$\mu(M)$$ is absolutely continuous w.r.t. Lebesgue measure, with continuous, piecewise-polynomial density of degree $$\leq n-k$$.

In particular, if $$n=k$$, then the pushforward measure is a constant multiple of Lebesgue measure.

### Example

Theorem (Archimedes)

Let $$\mu: S^2 \to \mathbb{R}$$ be given by $$\mu(x,y,z) = z$$. Pushing forward the uniform measure on $$S^2$$ to the image $$[-1,1]$$ gives Lebesgue measure.

### Example 2

Proposition. If we rotate both factors of $$S^2 \times S^2$$ simultaneously around the $$z$$-axis, the moment map is $$((x_1,y_1,z_1),(x_2,y_2,z_2))\mapsto z_1 + z_2$$, and the pushforward measure on $$[-2,2]$$ has density given by the triangle function.

### Complex Example

The $$\mathbb{T}^2 = U(1) \times U(1)$$ action on $$\mathbb{CP}^2$$ given by

(e^{i\theta_1},e^{i\theta_2})\cdot [z_0:z_1:z_2] = \left[z_0:e^{i\theta_1}z_1:e^{i\theta_2}z_2\right]

has moment map

\mu([z_0:z_1:z_2]) = -\frac{1}{2} \left(\frac{|z_1|^2}{|z_0|^2+|z_1|^2+|z_2|^2},\frac{|z_2|^2}{|z_0|^2+|z_1|^2+|z_2|^2}\right).

### Application to Polymers

In solution, linear polymers become crumpled:

Protonated P2VP

Roiter–Minko, J. Am. Chem. Soc. 127 (2005), 15688-15689

[CC BY-SA 3.0], from Wikimedia Commons

Modern polymer physics is based on the analogy between a polymer chain and a random walk.

### Lots of Symmetries!

Continuous symmetry $$\Rightarrow$$ conserved quantity

The space of equilateral $$n$$-gons has lots of symmetries...

### A Generalized Archimedes Theorem

Theorem [with Cantarella]

The standard volume form on polygon space is

$$\mathrm{d}\theta_1 \wedge \dots \wedge \mathrm{d}\theta_{n-3} \wedge \mathrm{d} r_1 \wedge \dots \wedge \mathrm{d}r_{n-3}$$

Corollary [with Cantarella, Duplantier, Uehara]

The first efficient, provably correct algorithm for sampling equilateral random polygons in $$\mathbb{R}^3$$.

### Application to Frame Theory

Signal: $$v \in \mathbb{C}^d$$

Design: $$f_1, \dots , f_n \in \mathbb{C}^d$$

Measurements: $$\langle f_1, v \rangle, \dots , \langle f_n, v \rangle$$.

Parseval’s Theorem

If $$f_1, \dots , f_d \in \mathbb{C}^d$$ is an orthonormal basis,

$$\|v\|^2 = \sum |\langle f_i, v \rangle |^2 = \|F^* v\|^2$$

for any $$v \in \mathbb{C}^d$$.

Even better,

$$v = \sum \langle f_i, v \rangle f_i = FF^* v$$.

If $$F = [f_1 \, f_2 \dots f_n]$$, the measurement vector is $$F^*v$$.

$$\Leftrightarrow FF^* = \mathrm{Id}_{d \times d}$$

Parseval frame

Fact: For any $$n \geq d$$, there exist Parseval frames $$f_1, \dots , f_n \in \mathbb{C}^d$$ so that $$\|f_i\|=\|f_j\|$$ for all $$i,j$$; these are the equal-norm Parseval frames (ENPs).

This is fragile! What if a measurement gets lost?

### Group Actions on Frames

Some nice group actions on $$\mathbb{C}^{d \times n}$$:

1. $$U(d)$$ acts on the left
2. $$U(n)$$ acts on the right
3. $$U(1)^d$$ acts on the left
4. $$U(1)^n$$ acts on the right
\mu_{U(d)}(F) = FF^*
\mu_{U(n)}(F) = -F^*F
\mu_{U(1)^d}\left(\begin{bmatrix} \rule[.8mm]{4mm}{.5px}\, f^1 \rule[.8mm]{4mm}{.5px}\\ \vdots \\ \rule[.8mm]{4mm}{.5px}\, f^d \rule[.8mm]{4mm}{.5px} \end{bmatrix}\right) = \left(\frac{1}{2}\|f^1\|^2 , \dots , \frac{1}{2}\|f^d\|^2\right)
\mu_{U(1)^n} \left(\begin{bmatrix} f_1 | \cdots | f_n \end{bmatrix}\right) = \left(-\frac{1}{2}\|f_1\|^2 , \dots , -\frac{1}{2}\|f_n\|^2\right)

Parseval frames

$$\mu_{U(d)}^{-1}(I_{d\times d})$$

equal-norm frames

$$\mu_{U(1)^N}^{-1}\left(-\frac{1}{2}r,\dots , -\frac{1}{2}r\right)$$

ENPs: $$\mu_{U(d)}^{-1}(I_{d \times d}) \cap \mu_{U(1)^n}^{-1}\left(-\frac{d}{2n}, \dots , -\frac{d}{2n}\right)$$

### A Generalization of the Frame Homotopy Conjecture

Theorem [with Needham]

There is a continuous interpolation between any two frames $$f_1, \dots , f_n$$ and $$g_1, \dots , g_n$$ with $$FF^* = GG^*$$ and $$\|f_i\| = \|g_i\|$$ preserving these conditions.

### Full Spark Frames

It is often desirable to require every minor of the $$d \times N$$ matrix $$F$$ to be invertible. Such an $$F$$ is said to have full spark.

Theorem [with Needham]

Fix the spectrum of $$FF^\ast$$ and fix $$\|f_1\|,\dots,\|f_N\|$$. There are three possibilities:

• It is impossible for a frame to have these data
• It is impossible for a frame with these data to have full spark
• A random frame with these data will have full spark with probability 1.

# Thank you!

Funding: National Science Foundation (DMS–2107700) and Simons Foundation

### References

Symplectic geometry and connectivity of spaces of frames

Tom Needham and Clayton Shonkwiler

Advances in Computational Mathematics 47 (2021), no. 1, 5

The symplectic geometry of closed equilateral random walks in 3-space

Jason Cantarella and Clayton Shonkwiler

Annals of Applied Probability 26 (2016), no. 1, 549–596

A fast direct sampling algorithm for equilateral closed polygons

Jason Cantarella, Bernard Duplantier, Clayton Shonkwiler, and Erica Uehara

Journal of Physics A: Mathematical and Theoretical 49 (2016), no. 27, 275202

#### Some Applications of Symplectic Geometry

By Clayton Shonkwiler

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