## (Applied) Symplectic Geometry

Clay Shonkwiler

shonkwiler.org

/sg

this talk!

### 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 acts 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

### 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 $$\omega(u,v) = \langle i u, v \rangle$$

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

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

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

(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}

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

### 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]

$$d\theta \wedge dz$$ is the standard area form on the unit sphere $$S^2$$.

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

### A (Very) Classical Example

KoenB [   ] from Wikimedia Commons

### Noether’s Theorem

Every 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.

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

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

$$\iota_X\omega = \iota_{\frac{\partial}{\partial \theta}} d\theta \wedge dz = dz$$

### Extremely Deep Fact

Archimedes’ Theorem can be suitably generalized to any symplectic manifold with a Hamiltonian circle action (or, better yet, many commuting Hamiltonian circle actions).

See: Atiyah, Guillemin–Sternberg, Duistermaat–Heckman, and toric symplectic manifolds (and toric varieties)

### “​(Applied)”…?

I am interested in mathematical models of topological polymers, especially ring-shaped biopolymers and synthetic materials…

…and in frame theory and statistical signal processing.

### Examples of Results

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

### Examples of Results

Improved stick number bounds on ≈38% of the knots in the Rolfsen table.

### Examples of Results

A broad generalization of the Frame Homotopy Conjecture, which says the space of unit-norm tight frames is path-connected.