### Clayton Shonkwiler PRO

Mathematician and artist

/ag23

this talk!

Aspects of Flag Manifolds with a View Towards Applications, July 11, 2023

National Science Foundation (DMS–2107700)

Simons Foundation (#709150 & #354225)

Florida State University

(The) Ohio State University

Johns Hopkins University

University of Georgia

Ochanomizu University

Kyoto University

Colorado State University

Université Paris-Saclay

Colorado State University

**Definition.**

If \(V\) is an \(n\)-dimensional vector space and \(1 \leq d_1 < \dots < d_k = n\) is an increasing list of integers, then \(F\ell(d_1, \dots , d_k)\) is the *flag manifold* of nested linear subspaces \(\mathcal{S}_1 \subset \dots \subset \mathcal{S}_k = V\) with \(\dim(\mathcal{S}_i) = d_i\).

\(F\ell(d,n) = \operatorname{Gr}(d,V)\) is usually called the *Grassmannian* or *Grassmann manifold* of

\(d\)-dimensional subspaces of \(V\).

When \(d=1\), this is the *projective space* \(\mathbb{P}(V)\) of lines in \(V\).

Sometimes \(F\ell(n_1, \dots , n_k)\) where \(\sum_{j=1}^i n_j = d_i\).

Flag manifolds are:

- Homogeneous Spaces (differential/Riemannian geometry)
- Projective Varieties (algebraic geometry)
- Coadjoint/Isotropy Orbits (Lie theory)

Applications of flag manifolds include:

Ye–Wong–Lim: flag manifolds are implicit in many tasks in numerical and statistical analysis, including mesh refinement, multiresolution analysis, and canonical correlation analysis.

\(A\) Hermitian, \(n \times n\).

A = U \Lambda U^\ast

with \(U\) unitary (i.e., \(U U^\ast = I_{n \times n}\)), and \(\Lambda = \text{diag}(\lambda_1, \dots , \lambda_n)\).

e.g., if \(\lambda_1 = \lambda_2 = 1\) and \(\lambda_3 = 0\), then \(\{u_1,u_2\}\) is an ONB for \(E_1\), \(\{u_3\}\) is an ONB for \(E_0\), but any other choice of ONBs would give a different valid \(U\).

What *is* unique is the flag \(E_1 \subset E_1 \oplus E_0\), which is a point in \(F\ell(2,3) = \operatorname{Gr}(2,\mathbb{C}^3)\).

**Proposition.**

For fixed \(\Lambda\), \(\{U \Lambda U^\ast : U \in U(n)\}\) is a flag manifold with signature determined by the multiplicities of the \(\lambda_i\).

Not unique!

A polymer in solution takes on an ensemble of random shapes, with topology as the unique conserved quantity.

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

– Alexander Grosberg

Protonated P2VP

Roiter/Minko

Clarkson University

Plasmid DNA

Alonso–Sarduy, Dietler Lab

EPF Lausanne

Knotted DNA

Wassermann et al.

*Science* 229, 171–174

*Random polygon* \(\Leftrightarrow\) *Point in some (nice!) configuration space*

**Framework** [with Deguchi and Cantarella]

By associating points in \(\operatorname{Gr}(2,\mathbb{C}^N)\) with piecewise-linear loops in \(\mathbb{R}^3\), get an \(E(3)\)-invariant representation. This was the first exactly-solvable model for loop random walks in \(\mathbb{R}^3\) with fixed scale.

Freely-jointed chain

\(\Longleftrightarrow\) point in \(\prod_{i=1}^N S^2 = \prod_{i=1}^N \mathbb{P}(\mathbb{C}^2)\)

**Theorem** [Kapovich–Millson, Hausmann–Knutson, with Cantarella]

Freely-jointed ring \(\Longleftrightarrow\) point in \(\prod_{i=1}^N \mathbb{P}(\mathbb{C}^2) /\!/SU(2) \simeq \operatorname{Gr}(2,\mathbb{C}^N)/\!/S(U(1)^N)\)

**Algorithm** [with Cantarella, Duplantier, and Uehara]

Sampling random freely-jointed rings in expected \(\Theta(N^{5/2})\) time.

\(\mathbb{K} = \mathbb{R}\), \(\mathbb{C}\), or \(\mathbb{H}\). A (finite) *frame* in \(\mathbb{K}^d\) is a spanning set \(f_1, \dots, f_N \in (\mathbb{K}^d)^N\).

\(F = [f_1 \, f_2 \, \ldots \, f_N ] \in \mathbb{K}^{d \times N}\)

Data representation: \(\mathbb{K}^d \ni v \mapsto F^\ast v \in \mathbb{K}^N\) (the *analysis operator*).

If \(\|f_i\|^2 = 1\) for all \(i\), the \(P_i := f_if_i^*\) are orthogonal projectors.

Unit-norm frames \(\Longleftrightarrow\) (certain) points in \(\prod_{i=1}^N \mathbb{P}(\mathbb{K}^d)\)

A frame is *Parseval* if \(v = \sum_i \langle f_i, v \rangle f_i = FF^\ast v\) for all \(v \in \mathbb{K}^d\).

i.e., \(FF^\ast = I_{d \times d}\), or equivalently \(F^\ast F\) is a rank-\(d\) orthogonal projector.

Parseval frames \(\Longleftrightarrow\) points in \(\operatorname{Gr}(d,\mathbb{K}^N)\).

In general, specifying the \(\|f_i\|^2\) determines a product of scaled projective spaces.

In general, specifying the spectrum of \(FF^\ast\) (or \(F^\ast F\)) determines a flag manifold.

**Generalized Frame Homotopy Theorem** [with Needham]

The space of frames in \(\mathbb{C}^d\) (or \(\mathbb{H}^d\)) with *any* specified \(\|f_i\|^2\) and *any* specified spectrum of \(FF^\ast\) is always path-connected.

**Theorem** [with Needham]

In the above spaces, either there are no frames with full spark (i.e., full Kruskal rank), or the full spark frames have full measure.

**Theorem** [with Mixon, Needham, and Villar]

In the space of unit-norm frames, every full spark frame flows by gradient descent to a unit-norm\(^\ast\) Parseval frame. (And similarly for fusion frames.)

\(^\ast\) For technical reasons, actually equal-norm, not unit-norm.

\(\psi(t)\)

\(\psi_{n,k}(t) := 2^{n/2}\psi(2^n t - k)\)

D_2(10) = \frac{1}{\sqrt{2}}\begin{bmatrix}
1 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 & 1 \\
0 & 0 & 0 & 0 & 1 \\
\end{bmatrix}

\(P_i(2n) = D_i(2n) D_i^T(2n)\)

\(2m \times 2n\) matrix \(A \mapsto P_i(2m) A P_j(2n)\)

\(\updownarrow\)

\(\mathcal{S} \in \operatorname{Gr}(n,\mathbb{R}^{2n})\)

\(D_2\)

\(D_4\)

Random

\(D_2\)

\(D_{24}\)

**Theorem** [with Collery and Peterson]

The Daubechies wavelets all live in a totally geodesic Schubert cycle in the Grassmannian.

Probability theory of random polygons from the quaternionic viewpoint

Jason Cantarella, Tetsuo Deguchi, and Clayton Shonkwiler

*Communications on Pure and Applied Mathematics* **67** (2014), no. 10, 1658–1699

Toric symplectic geometry and full spark frames

Tom Needham and Clayton Shonkwiler

*Applied and Computational Harmonic Analysis* **61** (2022), 254–287

By Clayton Shonkwiler

- 401