Symplectic Geometry and Frame Theory

Clayton Shonkwiler

Colorado State University

http://shonkwiler.org

/ag19

This talk!

Take Home Message

Symplectic geometry is a powerful set of tools which is useful for the study of frames

Frames

A frame in \(\mathbb{C}^d\) is an ordered spanning set \(\{\phi_1,\dots ,\phi_N\}\), which we often represent as a short, fat matrix

\Phi = \begin{bmatrix} \phi_1 | \phi_2 | \dots | \phi_N \end{bmatrix} \in \mathbb{C}^{d \times N}

Symplectic Geometry

Definition. A symplectic manifold is a smooth manifold \(M\) together with a closed, non-degenerate 2-form \({\omega \in \Omega^2(M)}\).

Example: \((S^2,d\theta\wedge dz)\)

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

Any smooth (affine or projective) complex variety is a symplectic manifold.

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

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

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

dx \wedge dy \left( a \frac{\partial}{\partial x} + b \frac{\partial}{\partial y}, c \frac{\partial }{\partial x} + d \frac{\partial}{\partial y} \right) = ad - bc

Maps

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 \(X_H = \frac{\partial}{\partial \theta}\).

Lie Group Actions

Let \(G\) be a Lie group, and let \(\mathfrak{g}\) be its Lie algebra. If \(G\) acts on \((M,\omega)\), then each \(V \in \mathfrak{g}\) determines a vector field \(X_V\) on \(M\) by

X_V(p) = \left.\frac{d}{dt}\right|_{t=0}\exp(t V) \cdot p

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

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

For \(r \in \mathbb{R} \simeq \mathfrak{u}(1)\), \(X_r = r \frac{\partial}{\partial \theta}\).

Momentum Maps

Definition. An action of \(U(1)\) on \((M,\omega)\) is Hamiltonian if there exists a map

\mu: M \to \mathfrak{u}(1)^* \simeq \mathbb{R}

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

\(X = \frac{\partial}{\partial \theta}\)

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

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

Momentum Maps

Definition. An action of \(G\) on \((M,\omega)\) is Hamiltonian if each one-parameter subgroup action is Hamiltonian. Equivalently, there exists a map

\mu: M \to \mathfrak{g}^*

so that \(\omega_p(X_V, X) = D_p \mu(X)(V)\) for each \(p \in M\), \(X \in T_pM\), and \(V \in \mathfrak{g}\).

\(X_{V_{(a,b,c)}}(x,y,z) = (a,b,c) \times (x,y,z)\)

\((\iota_{X_{V_{(a,b,c)}}}\omega)_{(x,y,z)} = a dx + b dy + c dz \)

\(\mu(x,y,z)(V_{(a,b,c)}) = (x,y,z)\cdot(a,b,c)\)

Group Actions on Frames

What compact Lie groups act nicely 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)}(\Phi) = \Phi \Phi^*
\mu_{U(N)}(\Phi) = -\Phi^*\Phi
\mu_{U(1)^d}\left(\begin{bmatrix} \rule[.8mm]{4mm}{.5px}\, \phi^1 \rule[.8mm]{4mm}{.5px}\\ \vdots \\ \rule[.8mm]{4mm}{.5px}\, \phi^d \rule[.8mm]{4mm}{.5px} \end{bmatrix}\right) = \left(\frac{1}{2}\|\phi^1\|^2 , \dots , \frac{1}{2}\|\phi^d\|^2\right)
\mu_{U(1)^N} \left(\begin{bmatrix} \phi_1 | \cdots | \phi_N \end{bmatrix}\right) = \left(-\frac{1}{2}\|\phi_1\|^2 , \dots , -\frac{1}{2}\|\phi_N\|^2\right)

Parseval frames

\(\mu_{U(d)}^{-1}(I_d)\)

unit-norm frames

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

FUNTFs

The (finite) unit-norm tight frames (FUNTFs) are the level set

Let \(\mu\) be the momentum map of the product group

U(d) \times U(1)^N
\mu^{-1}\left(\frac{N}{d} I_d, \left(-\frac{1}{2},\dots , -\frac{1}{2}\right)\right)= \mu^{-1}\left(\frac{N}{d}I_d, -\vec{\frac{1}{2}}\right)

The Frame Homotopy Conjecture

Theorem [Cahill–Mixon–Strawn ’17]

The space of length-\(N\) FUNTFs in \(\mathbb{C}^d\) is path-connected for all \(N \geq d\geq 1\).

A New Proof

FUNTF space is connected if and only if

\mu^{-1}\!\left(\frac{N}{d}I_d, -\vec{\frac{1}{2}}\right)\!/(U(d)\times U(1)^N) =: \mathbb{C}^{d \times N} /\!/\!_{\left(\frac{N}{d}I_d, -\vec{\frac{1}{2}}\right)}(U(d)\times U(1)^N)
\simeq \left(\mathbb{C}^{d \times N}/\!/\!_{\frac{N}{d}I_d}U(d)\right)/\!/\!_{-\vec{\frac{1}{2}}}U(1)^N
\simeq \operatorname{Gr}_d(\mathbb{C}^N)/\!/\!_{-\vec{\frac{1}{2}}} U(1)^N

Theorem [Atiyah] 

Torus reductions of compact, connected symplectic manifolds are connected.

is.

A Double Generalization

Theorem [with Needham]

For any invertible, Hermitian matrix \(S\) and any \(r_1, \dots , r_N \geq 0\), the space of frames \(\Phi = [\phi_1 | \cdots | \phi_N]\) with

\Phi \Phi^* = S
\|\phi_i\|^2 = r_i

and

is either empty or path-connected.

Symplectic Geometry and Frame Theory

This seems to be a pretty general viewpoint, with a number of potential applications...

See, for example, Tom Needham’s talk.

Thank you!

Funding: Simons Foundation

References

The geometry of constrained random walks and an application to frame theory

Clayton Shonkwiler

2018 IEEE Statistical Signal Processing Workshop (SSP), 343–347

Symplectic Geometry and Frame Theory

By Clayton Shonkwiler

Symplectic Geometry and Frame Theory

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