What is…

(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_2

Symplectic Manifold

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}

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

What does this get you?

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)