Equivalently,

\(0 = AA^\ast - A^\ast A = [A,A^\ast]\).

Define the non-normal energy \(\operatorname{E}:\mathbb{C}^{d \times d} \to \mathbb{R}\) by

\(\operatorname{E}(A) := \|[A,A^\ast]\|^2.\)

Obvious Fact.

The normal matrices are the global minima of \(\operatorname{E}\).

Theorem [with Needham]

The only critical points of \(\operatorname{E}\) are the global minima; i.e., the normal matrices.

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

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

This action is Hamiltonian with associated momentum map \(\mu: \mathbb{C}^{d \times d} \to \mathscr{H}_0(d)\) given by

\(\mu(A) := [A,A^\ast]\).

So \(\operatorname{E}(A) = \|\mu(A)\|^2\).

Definition.

\(\{f_1,\dots, f_n\}\subset \mathbb{C}^d\) is a Parseval frame if \(\operatorname{Id}_{d\times d}=FF^*=f_1f_1^*+\dots+f_nf_n^*\).

An equal-norm Parseval frame (ENP frame) is a Parseval frame \(f_1,\dots , f_n\) with \(\|f_i\|^2=\|f_j\|^2\) for all \(i\) and \(j\).

Proposition [cf. Welch]

The equal-norm Parseval frames are exactly the global minima of \(\operatorname{FP}|_{\text{equal norm}}\).

Theorem [Benedetto–Fickus]

As a function on equal-norm frames with fixed \(d\) and \(n\), \(\operatorname{FP}\) has no spurious local minima.

Theorem [with Needham; CodEx 2022 video]

Same for fusion frames.

Proposition [Bodmann–Haas]

The equal-norm Parseval frames are exactly the global minima of \(\operatorname{FE}|_{\text{Parseval}}\).

Theorem [Caine]

On the space of Parseval frames, consider the initial value problem

\(\widetilde{\Gamma}(F_0,0) = F_0 \qquad \frac{d}{dt}\widetilde{\Gamma}(F_0,t) = -\operatorname{grad} \operatorname{FE}(\widetilde{\Gamma}(F_0,t))\).

If \(F_0\) is full spark, then \(\lim_{t \to \infty} \widetilde{\Gamma}(F_0,t)\) is an ENP frame.

Frame energy: \(\operatorname{FE}(F) = \sum_j \|f_j\|^4\)

Definition.

For \(\vec{r} \in \mathbb{R}_+^n\), the total frame energy of a frame is

\(E_{\vec{r}}(F) :=\|FF^* - \mathbb{I}_d \|_{\text{Fr}}^2 + \frac{1}{4}\sum_j \left(\frac{\|f_j\|^2}{r_j}-1 \right)^2\)

Proposition.

The Parseval frames with \(\|f_j\|^2 = r_j\) for \(j=1,\dots , n\) are exactly the global minima of \(E_{\vec{r}}\).

Theorem [with Caine and Needham]

Let \(\vec{r} \in \mathbb{Q}_+^n\) such that \(\operatorname{PF}_d^{\mathbb{K}}(\vec{r}) \neq \emptyset\). Consider

\(\Gamma_{\vec{r}}(F_0,0) = F_0,\quad \frac{d}{dt} \Gamma_{\vec{r}}(F_0,t) = -\nabla E_{\vec{r}}(\Gamma_{\vec{r}}(F_0,t))\).

If \(F_0 \in \mathbb{K}^{d \times n}\) is full spark, then \(\lim_{t \to \infty} \Gamma_{\vec{r}}(F_0,t)\) is in \(\operatorname{PF}_d^{\mathbb{K}}(\vec{r})\).

Theorem [Cahill, Mixon, Strawn 2017]

The Frame Homotopy Conjecture is true.

Theorem [with Needham 2022; CodEx 2021 video]

\(\operatorname{PF}_d^{\mathbb{H}}(\vec{r})\) is path-connected for all admissible \(\vec{r} \in \mathbb{R}_+^n\).

Theorem [with Needham 2021; CodEx 2021 video]

\(\operatorname{PF}_d^{\mathbb{C}}(\vec{r})\) is path-connected for all admissible \(\vec{r}\in \mathbb{R}_+^n\).

Theorem [Mare 2024]

\(\operatorname{PF}_d^{\mathbb{R}}(\vec{r})\) is path-connected for certain \(\vec{r}\) with entries repeating in special patterns.

Theorem [Mare 2025]

\(\operatorname{PF}_d^{\mathbb{C}}(\vec{r})\) is simply-connected for admissible \(\vec{r} \in \mathbb{R}_+^n\) so that all \(r_{i_1} + \dots + r_{i_{n-k}} \geq 1\).

Corollary [with Caine and Needham]

Let \(q \in \mathbb{Z}_{\geq 0}\) and 

\(d \geq \begin{cases} q+2 & \text{if } \mathbb{K}= \mathbb{R}\\ \frac{q+2}{2} & \text{if } \mathbb{K} = \mathbb{C}.\end{cases}\)

Then the space of ENP frames is \(q\)-connected for all \(n \geq \frac{d}{d-1}(d+q+1)\).

Corollary [with Caine and Needham]

Let \(d \geq 2\). Then there exists \(\epsilon > 0\) so that, for any admissible \(\vec{r} \in \mathbb{Q}_+^n\) with \(\left|r_i - \frac{d}{n}\right| < \epsilon\) for all \(i\), the space \(\operatorname{PF}_d^{\mathbb{R}}(\vec{r})\) is path-connected.